TL;DR
This paper establishes non-asymptotic excess risk bounds for statistical learning with nuisance parameters, leveraging Neyman orthogonality to achieve second-order impact of nuisance estimation errors, applicable to complex models.
Contribution
It introduces a general framework for excess risk guarantees in nuisance-robust learning, accommodating arbitrary algorithms and complex classes with weaker assumptions.
Findings
Second-order impact of nuisance estimation errors.
Applicable to complex nonparametric classes.
Achieves oracle rates under certain entropy conditions.
Abstract
We provide non-asymptotic excess risk guarantees for statistical learning in a setting where the population risk with respect to which we evaluate the target parameter depends on an unknown nuisance parameter that must be estimated from data. We analyze a two-stage sample splitting meta-algorithm that takes as input arbitrary estimation algorithms for the target parameter and nuisance parameter. We show that if the population risk satisfies a condition called Neyman orthogonality, the impact of the nuisance estimation error on the excess risk bound achieved by the meta-algorithm is of second order. Our theorem is agnostic to the particular algorithms used for the target and nuisance and only makes an assumption on their individual performance. This enables the use of a plethora of existing results from machine learning to give new guarantees for learning with a nuisance component.…
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Code & Models
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Orthogonal Statistical Learning
Abstract
We provide non-asymptotic excess risk guarantees for statistical learning in a setting where the population risk with respect to which we evaluate the target parameter depends on an unknown nuisance parameter that must be estimated from data. We analyze a two-stage sample splitting meta-algorithm that takes as input arbitrary estimation algorithms for the target parameter and nuisance parameter. We show that if the population risk satisfies a condition called Neyman orthogonality, the impact of the nuisance estimation error on the excess risk bound achieved by the meta-algorithm is of second order. Our theorem is agnostic to the particular algorithms used for the target and nuisance and only makes an assumption on their individual performance. This enables the use of a plethora of existing results from machine learning to give new guarantees for learning with a nuisance component. Moreover, by focusing on excess risk rather than parameter estimation, we can provide rates under weaker assumptions than in previous works and accommodate settings in which the target parameter belongs to a complex nonparametric class. We provide conditions on the metric entropy of the nuisance and target classes such that oracle rates of the same order as if we knew the nuisance parameter are achieved.
Contents
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4 Instantiating the Main Results: Plug-In Empirical Risk Minimization
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5 Instantiating the Main Results: Sufficient Conditions for Oracle Rates
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E Sufficient Conditions for Theorems and : Single Index Losses
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H.2 Plug-in Empirical Risk Minimization: Refined Guarantees for VC Classes
1 Introduction
Predictive models based on modern machine learning methods are becoming increasingly widespread in policy making, with applications in healthcare, education, law enforcement, and business decision making. Most problems that arise in policy making, such as attempting to predict counterfactual outcomes for different interventions or optimizing policies over such interventions, are not pure prediction problems, but rather are causal in nature. It is important to address the causal aspect of these problems and build models that have a causal interpretation.
A common paradigm in the search of causality is that to estimate a model with a causal interpretation from observational data—that is, data not collected via randomized trial or via a known treatment policy—one typically needs to estimate many other quantities that are not of primary interest, but that can be used to de-bias a purely predictive machine learning model by formulating an appropriate loss. One example of such a nuisance parameter is the propensity for taking an action under the current policy, which can be used to form unbiased estimates for the reward for new policies, but is typically unknown in datasets that do not come from controlled experiments.
To make matters more concrete, let us walk through an example for which certain variants have been well-studied in machine learning (Dudík et al., 2011; Swaminathan and Joachims, 2015a; Nie and Wager, 2021; Kallus and Zhou, 2018). Suppose a decision maker wants to estimate the causal effect of some treatment on an outcome as a function of a set of observable features ; the causal effect will be denoted as . Typically, the decision maker has access to data consisting of tuples , where is the observed feature for sample , is the treatment taken, and is the observed outcome. Due to the partially observed nature of the problem, one needs to create unbiased estimates of the unobserved outcome. A standard approach is to make an unconfoundedness assumption (Rosenbaum and Rubin, 1983) and use the so-called doubly-robust formula, which is a combination of direct regression and inverse propensity scoring. Let denote the potential outcome for treatment in sample , and let f_{0}(t,x_{i})\vcentcolon=\operatorname{\mathbb{E}}\big{[}Y_{i}(t)\mid x_{i}\big{]} and . If , then the following is an unbiased estimator for the \revoneeditconditional mean potential outcome (given covariates):
[TABLE]
Given such an estimator, we can estimate the treatment effect by running a regression between the unbiased estimates and the features, i.e. solve over a target parameter class . In the population limit, with infinite samples, this corresponds to finding a parameter that minimizes the population risk \operatorname{\mathbb{E}}\big{[}(\widehat{Y}_{i}(1)-\widehat{Y}_{i}(0)-\theta(X))^{2}\big{]}. Similarly, if the decision maker is interested in policy optimization rather than estimating treatment effects, they can use these unbiased estimates to solve over a policy space of functions mapping features to . However, when dealing with observational data, the functions and are not known, and must be estimated if we wish to evaluate the proxy labels . Since these functions are only used as a means to learn the target parameter , we may regard them as nuisance parameters. The goal of the learner is to estimate a target parameter that achieves low population risk when evaluated at the true nuisance parameters as opposed to the estimated nuisance parameters, since only then does the model have a causal interpretation.
This phenomenon is ubiquitous in causal inference and motivates us to formulate the abstract problem of statistical learning with a nuisance component: Given i.i.d. examples from a distribution , a learner is interested in finding a target parameter so as to minimize a population risk function . The population risk depends not just on the target parameter, but also on a nuisance parameter whose true value is unknown to the learner. The goal of the learner is to produce an estimate that has small excess risk evaluated at the unknown true nuisance parameter:
[TABLE]
Depending on the application, such an excess risk bound can take different interpretations. For many settings, such as treatment effect estimation, it is closely related to mean squared error, while in policy optimization it typically corresponds to regret. Following the tradition of statistical learning theory (Vapnik, 1995; Bousquet et al., 2004), we make excess risk the primary focus of our work, independent of the interpretation. We develop algorithms and analysis tools that generically address Eq. 2, then apply these tools to a number of applications of interest.
The problem of statistical learning with a nuisance component is strongly connected to the well-studied semiparametric inference problem (Levit, 1976; Ibragimov and Has’Minskii, 1981; Pfanzagl, 1982; Bickel, 1982; Klaassen, 1987; Robinson, 1988; Bickel et al., 1993; Newey, 1994; Robins and Rotnitzky, 1995; Ai and Chen, 2003; van der Laan and Dudoit, 2003; van der Laan and Robins, 2003; Ai and Chen, 2007; Tsiatis, 2007; Kosorok, 2008; van der Laan and Rose, 2011; Ai and Chen, 2012; Chernozhukov et al., 2022a; Belloni et al., 2017; Chernozhukov et al., 2018a), which focuses on providing so-called “-consistent and asymptotically normal” estimates for a low-dimensional target parameter (which may be expressed as a population risk minimizer or a solution to estimating equations) in the presence of a typically nonparametric nuisance parameter. Unlike the semiparametric inference problem, statistical learning with a nuisance component does not require a well-specified model, nor a unique minimizer of the population risk. Moreover, we do not ask for parameter recovery or asymptotic inference (e.g., asymptotically valid confidence intervals). Rather, we are content with an excess risk bound, regardless of whether there is an underlying true parameter to be identified. As a consequence, we provide guarantees even in the presence of misspecification, and when the target parameter belongs to a large, potentially nonparametric class. For example, one line of previous work gives semiparametric inference guarantees when the nuisance parameter is a neural network (Chen and White, 1999; Farrell et al., 2021); by focusing on excess risk we can give guarantees for the case where the target parameter is a neural network.
The case where the target parameter belongs to an arbitrary class has not been addressed at the level of generality we consider in the present work, but we mention some prior work that goes beyond the low-dimensional/parametric setup for special cases. Athey and Wager (2017) and Zhou et al. (2023) give guarantees based on metric entropy of the target class for the specific problem of treatment policy learning. For estimation of treatment effects, various nonparametric classes have been used for the target class on a case by case basis, including kernels (Nie and Wager, 2021), random forests (Athey et al., 2019; Oprescu et al., 2019; Friedberg et al., 2020), and high-dimensional linear models (Chernozhukov et al., 2017, 2018b). Other results allow for fairly general choices for the target parameter class in specific statistical models (Rubin and van der Laan, 2005, 2007; Díaz and van der Laan, 2013; van der Laan and Luedtke, 2014; Kennedy et al., 2017, 2019; Künzel et al., 2019). Our work unifies these directions into a single framework, and our general tools lead to improved or refined results when specialized to many of these individual settings.
Our approach is to reduce the problem of statistical learning with a nuisance component to the standard formulation of statistical learning. We build on a recent thread of research on semiparametric inference known as “double” or “debiased” machine learning (Chernozhukov et al., 2022a, 2017, 2018a, 2018c, 2018b), which leverages sample splitting to provide inference guarantees under weak assumptions on the estimator for the nuisance parameter. Rather than directly analyzing particular algorithms and models for the target parameter (e.g., regularized regression, gradient boosting, or neural network estimation), we assume a black-box guarantee for the excess risk in the case where a nuisance value is fixed. Our main theorem asks only for the existence of an algorithm that, for any given nuisance parameter and data set , achieves low excess risk with respect to the population risk , i.e. with probability at least ,
[TABLE]
Likewise, we assume the existence of a black-box algorithm to estimate the nuisance component from the data, with the required estimation guarantee varying from problem to problem.
Given access to the two black-box algorithms, we analyze a simple sample splitting meta-algorithm for statistical learning with a nuisance component, presented as Meta-Algorithm 1. We can now state the main question addressed in this paper: When is the excess risk achieved by sample splitting robust to nuisance component estimation error?
In more technical terms, we seek to understand when the two-stage sample splitting meta-algorithm achieves an excess risk bound with respect to , in spite of error in the estimator output by the first-stage algorithm. Robustness to nuisance estimation error allows the learner to use more complex models for nuisance estimation and—under certain conditions on the complexity of the target and nuisance parameter classes—to learn target parameters whose error is, up to lower order terms, as good as if the learner had known the true nuisance parameter in advance. Such a guarantee is referred to as achieving an oracle rate in semiparametric inference.
Overview of results
We use Neyman orthogonality (Neyman, 1959, 1979), a key tool in inference in semiparametric models (Newey, 1994; van der Vaart, 2000; Robins et al., 2008; Zheng and van der Laan, 2010; Belloni et al., 2017; Chernozhukov et al., 2018a), to provide oracle rates for statistical learning with a nuisance component. We show that if the population risk satisfies a functional analogue of Neyman orthogonality, the estimation error of has a second order impact on the overall excess risk (relative to ) achieved by . To gain some intuition, Neyman orthogonality is weaker condition than double-robustness, albeit similar in flavor, (see, e.g., Chernozhukov et al. (2022a)) and is satisfied by both the treatment effect loss and the policy learning loss described in the introduction. In more detail, our variant of the Neyman orthogonality condition asserts that a functional cross-derivative of the loss vanishes when evaluated at the optimal target and nuisance parameters. Prior work provides a number of means through which to construct Neyman orthogonal losses whenever certain moment conditions are satisfied by the data generating process (Chernozhukov et al., 2018a, 2022a, b). Indeed, orthogonal losses can be constructed in settings including treatment effect estimation, policy learning, missing and censored data problems, estimation of structural econometric models, and game-theoretic models.
We identify two regimes of excess risk behavior:
Fast rates. When the population risk is strongly convex with respect to the prediction of the target parameter (e.g., the treatment effect estimation loss), then typically so-called fast rates (e.g., rates of order of for parametric classes) are optimal if the true nuisance parameter is known. Letting denote the estimation error of the nuisance component, in this setting we show that orthogonality implies that the first stage error has an impact on the excess risk of the order of (in particular, -RMSE rates for the nuisance suffice when the target is parametric). 2. 2.
Slow rates. Absent any strong convexity of the population risk (e.g., for the treatment policy optimization loss), typically slow rates (e.g. rates of order for parametric classes) are optimal if the true nuisance parameter is known. For this setting, we show that the impact of nuisance estimation error is of the order so, once again, RMSE rates for the nuisance suffice when the target is parametric.
To make the conditions above concrete for arbitrary classes, we give conditions on the relative complexity of the target and nuisance classes—quantified via metric entropy—under which the sample splitting meta-algorithm achieves oracle rates, assuming the two black-box estimation algorithms are instantiated appropriately. This allows us to extend several prior works beyond the parametric regime to complex nonparametric target classes. Our technical results extends the works of Yang and Barron (1999); Rakhlin et al. (2017), which provide minimax optimal rates without nuisance components and utilize the technique of aggregation in designing optimal algorithms.
The flexibility of our approach allows us to instantiate our framework with any machine learning model and algorithm of interest for both nuisance and target parameter estimation, and to utilize the vast literature on generalization bounds in machine learning to establish refined (e.g., data-dependent or dimension-independent) rates for several classes of interests. For instance, our approach allows us to leverage recent work on size-independent generalization error of neural networks.
Moving beyond black-box results, we use our main theorems as a starting point to provide sharp analyses for certain general-purpose statistical learning algorithms for target estimation in the presence of nuisance parameters. First, we provide a new analysis for empirical risk minimization with plug-in estimation of nuisance parameters, wherein we extend the classical local Rademacher complexity analysis of empirical risk minimization (Koltchinskii and Panchenko, 2000; Bartlett et al., 2005) to account for the impact of the nuisance error (leveraging orthogonality). Second, in the slow rate regime we give a new analysis of variance-penalized empirical risk minimization with plug-in nuisance estimation, which allows us to recover and extend several prior results in the literature on policy learning. Our result improves upon the variance-penalized risk minimization approach of Maurer and Pontil (2009) by replacing the dependence on the metric entropy at a fixed approximation level with the critical radius, which is related to the entropy integral.
As a consequence of focusing on excess risk, we obtain oracle rates under weaker assumptions on the data generating process than in previous works. Notably, we obtain guarantees even when the target parameter is misspecified and the target parameters are not identifiable. For instance, for sparse high-dimensional linear classes, we obtain optimal prediction rates with no restricted eigenvalue assumptions. We highlight the applicability of our results to four settings of primary importance in the literature: 1) estimation of heterogeneous treatment effects from observational data, 2) offline policy optimization, 3) domain adaptation, 4) learning with missing data. For each of these applications, our general theorems allow for the use of arbitrary estimators for the nuisance and target parameter classes and provide robustness to the nuisance estimation error.
1.1 Related work
General frameworks for learning/inference with nuisance parameters
The work of van der Laan and Dudoit (2003) and subsequent refinements and extensions (van der Laan et al., 2006, 2007) develops cross-validation methodology for a similar risk minimization setting in which the target risk parameter depends on an unknown nuisance parameter. van der Laan and Dudoit (2003) analyze a cross-validation meta-algorithm in which the learner simultaneously forms a nuisance parameter estimator and a set of candidate target parameter estimators using a set of training samples, then selects a final estimate for the target parameter by minimizing an empirical loss over a validation set. The train and validation splits may be chosen in a general fashion that encompasses -fold and Monte Carlo validation. They provide finite-sample oracle rates for the excess risk in the case where the target parameter belongs to a finite class (in particular, rates of the type for a class of square losses and for general losses), and also extend these guarantees to linear combinations of basis functions via pointwise -nets (in our language, such classes are parametric). Overall, our approach offers several new benefits:
- •
By completely splitting nuisance estimation and target estimation into separate stages and taking advantage of orthogonality, we can provide meta-theorems on robustness that are invariant to the choice of learning algorithm both for the first and second stage, which obviates the need to assume the target class is finite or admits a linear representation (Section 3).
- •
When we do specialize to algorithms such as ERM and variants, we can provide finite-sample guarantees for rich classes of target parameters in terms of sharp learning-theoretic complexity measures such as local Rademacher complexity and empirical metric entropy (Section 4). In particular, we can provide conditions under which oracle rates are attained under very general complexity assumptions on the target and nuisance parameters (Section 5).
The methodology of van der Laan and Dudoit (2003) can be used to directly estimate a target parameter or to select the best of many candidate nuisance estimators in a data-driven fashion. van der Laan et al. (2007) refers to the use of this cross-validation methodology to perform data-adaptive estimation of nuisance parameters as the “super learner”, and subsequent work has advocated for its use for nuisance estimation within a framework for semiparametric inference known as targeted maximum likelihood estimation (TMLE). TMLE (Scharfstein et al., 1999; van der Laan and Rubin, 2006; Zheng and van der Laan, 2010; van der Laan and Rose, 2011) and its more general variant, targeted minimum loss-based estimation, are general frameworks for semiparametric inference which—like our framework—employ empirical risk minimization in the presence of nuisance parameters. TMLE estimates the target parameter by repeatedly minimizing an empirical risk (typically the negative log-likelihood) in order to refine an initial estimate. This approach easily incorporates constraints, and can be used in tandem with the super learning technique. The analysis leverages orthogonality, and is also agnostic to how the nuisance estimates are obtained. However, the main focus of this framework is on the classical semiparametric inference objective; minimizing a population risk is not the end goal as it is here.
Specific instances of risk minimization with nuisance parameters
A number of prior works employ empirical risk minimization with nuisance parameters for specific statistical models (Rubin and van der Laan, 2005, 2007; Díaz and van der Laan, 2013; van der Laan and Luedtke, 2014; Kennedy et al., 2017, 2019; Künzel et al., 2019). These results allow for general choices for the target class and nuisance class (typically subject to Donsker conditions, or with guarantees in the vein of van der Laan and Dudoit (2003)), and the main focus is semiparametric inference rather than excess risk guarantees.
Nonparametric target parameters
Outside of the risk minimization-based approaches above and the examples in the prequel (Athey et al., 2019; Nie and Wager, 2021; Athey and Wager, 2017; Zhou et al., 2023; Oprescu et al., 2019; Friedberg et al., 2020; Chernozhukov et al., 2017, 2018b), a number of other results also consider inference for nonparametric target parameters in the presence of nuisance parameters. In van der Vaart and van der Laan (2006), the target is a Lipschitz function over (the marginal survival function) and an estimation rate of is given. Wang et al. (2010) consider estimation of smooth nonparametric target parameters in the presence of missing outcomes, and give algorithms based on kernel smoothing. Robins and Rotnitzky (2001); Robins et al. (2008) consider settings where the target parameter is scalar, but the optimal rate is nonparametric due to the presence of complex nuisance parameters.
Sample splitting
While our use of sample splitting is directly inspired by recent use of the technique in double/debiased machine learning (Chernozhukov et al., 2022a, 2018a), the basic technique dates back to the early days of semiparametric inference and it has found use in many other works to remove Donsker conditions for estimation in the presence of nuisance parameters (Bickel, 1982; Klaassen, 1987; van der Vaart, 2000; Robins et al., 2008; Zheng and van der Laan, 2010).
Limitations
Our results are quite general, but there are some applications that go beyond the scope of our framework. For example, while we consider only plug-in estimation for the nuisance parameters, several works attain refined results by using specialized estimators van der Laan and Rubin (2006); Hirshberg and Wager (2021); Chernozhukov et al. (2018c); Ning et al. (2020). While our focus is on methods based on loss minimization, some problems such as nonparametric instrumental variables (Newey and Powell, 2003; Hall et al., 2005; Blundell et al., 2007; Chen and Pouzo, 2009, 2012, 2015; Chen and Christensen, 2018) are more naturally posed in terms of conditional moment restrictions.111In fact, nonparametric IV can be cast as a special case of the setup in Eq. 4, but we do not know of any estimators for this problem that satisfy the conditions required to apply our main theorems.
Another direction where our results leave room for future improvement concerns the reliance on Neyman orthogonality. While Neyman orthogonality is a fairly general condition which allows one to handle many nuisance parameters simultaneously, many problems admit additional structure which can lead to more refined guarantees. For example, in the context of treatment effect estimation, subsequent work of Kennedy (2020) uses the doubly robust structure of the problem to give guarantees that accommodate the case where different nuisance components (regression functions and propensity scores) are estimated at different rates.
1.2 Organization
The first part of this paper presents our main results. Section 2 contains technical preliminaries and definitions, and Section 3 presents our main theorems concerning the excess risk of Section 1. Section 3 also includes basic examples in which we apply these theorems to treatment effect estimation and policy learning.
Our main results are stated at a high level of generality, and consider generic estimation algorithms for the target and nuisance parameters. In the second part of the paper, we make matters more concrete and focus on specific algorithms. We leverage the main theorems to give explicit bounds based on the statistical capacity of the target and nuisance class. In particular:
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Section 4 (Plug-in Empirical Risk Minimization) provides explicit bounds for plug-in empirical risk minimization as the second stage of the meta-algorithm.
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Section 5 (Sufficient Conditions for Oracle Rates) considers aggregation based algorithms that go beyond empirical risk minimization, and gives sufficient conditions (as a function of the statistical capacity of the nuisance and target class) under which Section 1 can be configured such that oracle excess risk bounds are achieved.
We conclude with discussion in Section 6. Additional results are deferred to the appendix, which is split into three parts. Part I contains experiments, and Part II contains supplementary theoretical results, including sufficient conditions for Neyman orthogonality, applications of our main results to specific settings, and further guarantees for specific algorithms and function classes. Part III contains proofs for our main results.
2 Framework: Statistical Learning with a Nuisance Component
We work in a learning setting in which observations belong to an abstract set . We receive a sample set where each is drawn i.i.d. from an unknown distribution over . Define variable subsets ; the restriction is not strictly necessary but simplifies notation. We focus on learning parameters that come from a target parameter class and nuisance parameter class , where and are finite dimensional vector spaces of dimension and respectively, equipped with norms and . Note that since our results are fully non-asymptotic, the classes and may be taken to grow with .
Given an example , we write and to denote the subsets of that act as arguments to the nuisance and target parameters respectively. For example, we may write for or for . We assume that the function spaces and are equipped with \vseditpre-norms and respectively, which need to satisfy non-negativity and , but not necessarily the triangle inequality nor absolute homogeneity. In our applications, both pre-norms take the form for functions , where .
We measure performance of the target predictor through the real-valued population loss functional , which maps a target predictor and nuisance predictor to a loss. The subscript in denotes that the functional depends on the underlying distribution . For all of our applications, has the following structure, in line the classical statistical learning setting: First define a pointwise loss function , then define . Our general framework does not explicitly assume this structure, however.
Let be the unknown true value for the nuisance parameter. Given the samples , and without knowledge of , we aim to produce a target predictor that minimizes the excess risk evaluated at
[TABLE]
As discussed in the introduction, we will always produce such a predictor via the sample splitting meta-algorithm (Section 1), which makes uses of a nuisance predictor .
When the infimum in the excess risk is obtained, we use to denote the corresponding minimizer, in which case the excess risk can be written as
[TABLE]
We occasionally use the notation to refer to a particular target parameter with respect to which the second stage satisfies a first-order condition, e.g. . If and the population risk is convex, then we can take without loss of generality, but we do not assume this, and in general we do not assume existence of a such a parameter .
Notation
We let denote the standard inner product. will denote the norm over and will denote the spectral norm over .
Unless otherwise stated, the expectation , probability , and variance operators will be taken with respect to the underlying distribution . We define empirical analogues , , and with respect to a sample set , whose value will be clear from context. For a vector space with norm and function , we define for , with referring to the special case where . For a sample set , we define the empirical variant . When , we drop the first argument and write and . We extend these definitions to in the natural way.
For a subset of a vector space, will denote the convex hull. For an element , we define the star hull via
[TABLE]
and adopt the shorthand .
Given functions where is any set, we use non-asymptotic big- notation, writing if there exists a numerical constant such that for all and if there is a numerical constant such that . We write as shorthand for .
3 Orthogonal Statistical Learning
In this section we present our main results on orthogonal statistical learning, which state that under certain conditions on the loss function, the error due to estimation of the nuisance component has higher-order impact on the prediction error of the target component. The results in this section, which form the basis for all subsequent results, are algorithm-independent, and only involve assumptions on properties of the population risk . To emphasize the high level of generality, the results in this section invoke the learning algorithms in Section 1 only through “rate” functions and which respectively bound the estimation error of the first stage and the excess risk of the second stage.
Definition 1** (Algorithms and Rates).**
The first and second stage algorithms and corresponding rate functions are defined as follows:
- a)
Nuisance algorithm and rate.* The first stage learning algorithm , when given a sample set from distribution , outputs a predictor for which*
[TABLE]
with probability at least . 2. b)
Target algorithm and rate.* The second stage learning algorithm , when given sample set from distribution and any outputs a predictor for which*
[TABLE]
with probability at least .
\revoneedit
We let denote any function class (fixed a-priori) for which almost surely. We denote worst-case variants of the rates by and .
Observe that if one naively applies the algorithm for the target class using the nuisance predictor as a plug-in estimate for , the rate stated in Definition 1 will only yield a “pseudo”-excess risk bound of the form
[TABLE]
This clearly does not match the desired bound Eq. 4, which concerns the excess risk evaluated at rather than the plug-in estimate . The bulk of our work is to show that orthogonality can be used to correct this mismatch.
Definition 1 and subsequent results are stated in terms of a class containing , which in general may have . This extra level of generality serves two purposes. First, it allows for refined analysis in the case where , which is encountered when using algorithms based on regularization that do not impose hard constraints on, e.g., the norm of the estimator. Second, it permits the use of improper prediction, i.e. , which in some cases is required to obtain optimal rates for misspecified models (Audibert, 2008; Foster et al., 2018).
Recall that for a sample set , the empirical loss is defined via . Many classical results from statistical learning can be applied to the double machine learning setting by minimizing the empirical loss with plug-in estimates for , and we can simply cite these results to provide examples of for the target class . Note however that this structure is not assumed by Definition 1, and we indeed consider algorithms that do not have this form \revoneedit(cf. Section 5). \revoneeditLet us highlight that we allow the function to depend on both the target estimator and the nuisance estimator ; this extra level of generality is useful for deriving algorithm-specific guarantees (cf. Section 4).
Fast rates and slow rates
The rates presented in this section fall into two distinct categories, which we distinguish by referring to them as either fast rates or slow rates. The meaning of the word “fast” or “slow” here is two-fold: First, for fast rates, our assumptions on the loss imply that when the target class is not too large (e.g. a parametric or VC-subgraph class) prediction error rates of order are possible in the absence of nuisance parameters. For our slow rate results, the best prediction error rate that can be achieved is , even for small classes. This distinction is consistent with the usage of the term fast rate in statistical learning (Bousquet et al., 2004; Bartlett et al., 2005; Srebro et al., 2010), and we will see concrete examples of such rates for specific classes in later sections (Section 4, Section 5).
The second meaning of “fast” versus “slow” refers to the first stage: When estimation error for the nuisance is of order , the impact on the second stage in our fast rate results is of order , while for our slow rate results the impact is of order . The fast rate regime—particularly, the -type dependence on the nuisance error—will be the more familiar of the two for readers accustomed to semiparametric inference. While fast rates might at first seem to strictly improve over slow rates, these results require stronger assumptions on the loss. Our results in Section 5 show that which setting is more favorable will in general depend on the precise relationship between the complexity of the target parameter class and the nuisance parameter class.
3.1 Fast Rates Under Strong Convexity
We first present general conditions under which the sample splitting meta-algorithm obtains so-called fast rates for prediction. Our assumptions are stated in terms of directional derivatives with respect to the target and nuisance parameters.
Definition 2** (Directional Derivative).**
Let be a vector space of functions. For a functional , we define the derivative operator D_{f}F(f)[h]=\frac{d}{dt}F(f+th)\Big{|}_{t=0} for a pair of functions . Likewise, we define D_{f}^{k}F(f)[h_{1},\ldots,h_{k}]=\frac{\partial^{k}}{\partial{}t_{1}\ldots\partial{}t_{k}}F(f+t_{1}{}h_{1}+\ldots+t_{k}{}h_{k})\Big{|}_{t_{1}=\cdots=t_{k}=0}. When considering a functional in two arguments, e.g. , we write and to make the argument with respect to which the derivative is taken explicit.
\revoneedit
To present our results, we fix a representative . In general, the minimizer may not be unique—indeed, by focusing on excess risk, we can provide guarantees even when parameter recovery is impossible. Thus, we assume that a single fixed representative is used throughout all the assumptions stated in this subsection.
Our first assumption is the starting point for this work, and asserts that the population loss is orthogonal in the sense that the certain pathwise derivatives vanish.
Assumption 1** (Orthogonal Loss).**
The population risk is Neyman orthogonal:
[TABLE]
\revoneedit
Note that while Assumption 1 is stated in terms of the risk , it is typically satisfied by choosing a particular point-wise loss function whose expectation equals the risk; examples are given in the sequel. The construction of such a point-wise loss is typically achieved by adding a de-biasing correction term to some “initial” loss, whose minimizer is the target quantity (see Appendix D for details on automated orthogonal loss construction). The de-biasing correction reduces the impact of errors in the nuisance function estimates on the gradient of the loss, and is related to the notion of an efficient influence function in semi-parametric inference (however, our estimand is not necessarily pathwise differentiable, and hence violates the basic premise of most semi-parametric inference theory).
Beyond orthogonality, our main theorem requires three additional assumptions, all of which are fairly standard in the context of fast rates for statistical learning. We require a first-order optimality condition for the target class, and require that the population risk is both smooth and strongly convex with respect to the target parameter.
Assumption 2** (First Order Optimality).**
The minimizer for the population risk satisfies the first-order optimality condition:
[TABLE]
Remark 1**.**
The first-order condition is typically satisfied for models that are well-specified, meaning that there is some variable in that identifies the target parameter . More generally, it suffices to “almost” satisfy the first-order condition, i.e. to replace Eq. 8 by the condition
[TABLE]
The first-order condition is also satisfied whenever is star-shaped around , i.e. .
Assumption 3** (Higher-Order Smoothness).**
There exist constants and such that the following derivative bounds hold:
- a)
Second-order smoothness with respect to target.* For all and all :*
[TABLE] 2. b)
Higher-order smoothness.* \revoneeditThere exists such that for all , and :*
[TABLE]
Assumption 4** (Strong Convexity).**
The population risk is strongly convex with respect to the target parameter: \revoneeditThere exist constants such that for all and ,
[TABLE]
where is as in Assumption 3.
Assumption 3 and Assumption 4 are easily satisfied whenever the loss is obtained by applying a square loss or another smooth, strongly convex loss pointwise to the prediction of the target class; concrete examples are given in Appendix E. \revoneeditFor most of our results, we apply these assumptions with , but the case will prove useful for certain settings in which strong -type estimation guarantees for the target parameter are available (cf. Example 1). In general, Assumptions 1, 2, 4 and 3 do not imply that is uniquely identified unless is a norm. However, if the assumptions are satisfied by two parameters , we must have , meaning convergence in the sense that is equivalent for both representatives.
We now state our main theorem concerning fast rates.
Theorem 1**.**
Suppose there exists such that Assumptions 1, 2, 4 and 3, are satisfied. Then the sample splitting meta-algorithm (Section 1) produces a parameter such that with probability at least ,
[TABLE]
where and C_{2}\leq 2\Big{(}\big{(}\frac{\beta_{2}}{\lambda}\big{)}^{\frac{2}{1+r}}+\frac{\kappa}{\lambda}\Big{)}. In addition,
[TABLE]
\revoneedit
The majority of the results in this paper concern the special case in which . In this case, since , Theorem 1 shows that for Section 1, the impact of the unknown nuisance parameter on the prediction is of second-order, i.e.
[TABLE]
This implies that if the optimal rate without nuisance parameters is of order , it suffices to take to achieve the oracle rate.
Proof of Theorem 1.
We prove Theorem 1 by performing a Taylor expansion to relate the excess risk at to the excess risk at , employing orthogonality and self-bounding arguments to control cross terms. We abbreviate and to simplify notation.
By a second-order Taylor expansion on the risk at , there exists such that
[TABLE]
Next, using the strong convexity assumption (Assumption 4), we have
[TABLE]
Combining these statements, we conclude that
[TABLE]
Using the assumed rate for (Definition 1), this implies the inequality
[TABLE]
We now apply a second-order Taylor expansion (using the assumed derivative continuity from Assumption 3), which implies that there exists such that
[TABLE]
Using orthogonality of the loss (Assumption 1), this is equal to
[TABLE]
We use the second order smoothness assumed in Assumption 3 to upper bound by
[TABLE]
Invoking Young’s inequality and using that , we have that for any constant , this is at most
[TABLE]
Lastly, we use the assumed rate for (Definition 1) to bound by
[TABLE]
Choosing , combining this string of inequalities with Eq. 15, and rearranging, we have:
[TABLE]
Assumption 2 implies that , which establishes the inequality Eq. 13.
To derive the inequality Eq. 14, we use another Taylor expansion, which implies that there exists such that
[TABLE]
We combine Eq. 16 with Eq. 17 to conclude that is bounded by
[TABLE]
The result follows by again using that , along with the fact that without loss of generality. ∎
There is one issue not addressed by Theorem 1: If the nuisance parameter were known, the rate for the target parameters would be , but the bound in Eq. 14 scales instead with . This is addressed in Sections 4 and 5, where—building on Theorem 1—we show that for many standard algorithms, the cost to relate these quantities grows as , and can be absorbed into the second term in Eq. 13 or Eq. 14.
3.2 Beyond Strong Convexity: Slow Rates
The strong convexity assumption used by Theorem 1 requires curvature only in the prediction space, not the parameter space. This is considerably weaker than what is assumed in prior works on double machine learning (e.g., Chernozhukov et al. (2018b)), and is a major advantage of analyzing prediction error rather than parameter recovery. Nonetheless, in some situations even assuming strong convexity on predictions may be unrealistic. A second advantage of studying prediction is that, while parameter recovery is not possible in this case, it is still possible to achieve low prediction error, albeit with slower rates than in the strongly convex case. We now give guarantees under which these (slower) oracle rates for prediction error can be obtained in the presence of nuisance parameters using Section 1. \revoneeditAs in the prequel, we fix a representative throughout this subsection.
The key technical assumption for next result is universal orthogonality, which informally states that the loss is not simply orthogonal around , but rather is orthogonal for all .
Assumption 5** (Universal Orthogonality).**
For all ,
[TABLE]
\revoneedit
Universal orthogonality is a strengthening of Assumption 1, which requires that the cross derivative at vanishes for all , rather than only at . It is satisfied for examples including treatment effect estimation (Section 3.3) and policy learning (Section 3.4), and is used implicitly in previous work in these settings (Nie and Wager, 2021; Athey and Wager, 2017). Beyond orthogonality, we require a mild smoothness assumption for the nuisance class.
Assumption 6**.**
The derivatives and are continuous. Furthermore, there exists a constant such that for all and ,
[TABLE]
Our main theorem for slow rates is as follows.
Theorem 2**.**
Suppose that there is such that Assumption 5 and Assumption 6 are satisfied. Then with probability at least , the target parameter produced by Section 1 enjoys the excess risk bound:
[TABLE]
For generic Lipschitz losses, the optimal rate for parametric classes—in the absence of nuisance parameters—scales with . Without orthogonality, one expects the dependence on nuisance estimation error to scale linearly with , which would require to achieve the oracle rate. Theorem 2 shows that under orthogonality, the impact of nuisance parameter estimation is of lower order, and it suffices that . The proof follows similar reasoning to that of Theorem 1; see Appendix J.
3.3 Example: Treatment Effect Estimation
To make matters concrete, we now walk through a detailed example in which we specialize our general framework to the well-studied problem of treatment effect estimation. We show how the setup falls in our framework, explain what statistical assumptions are required to apply our main theorems, and show how to interpret the resulting excess risk bounds.
Following, e.g., Robinson (1988); Nie and Wager (2021), we receive examples according to the following data generating process:
[TABLE]
where and are covariates, is the treatment variable, and is the target variable. The true target parameter is , but we do not necessarily assume that . The functions and are unknown; we define and take to be the true nuisance parameter. We set and .
3.3.1 Residualized Loss (R-Loss)
Following Robinson (1988); Nie and Wager (2021), we consider the residualized square loss
[TABLE]
Let us take a moment to interpret the meaning of excess risk under this loss. It is simple to verify that if the true nuisance parameters are plugged in, then
[TABLE]
Thus, if a predictor has low risk it, must be good at predicting whenever there is sufficient variation in the treatment . In addition, If the model is not well-specified () but is convex, we can still deduce that
[TABLE]
so in this case low excess risk implies that we predict nearly as well as the best predictor in class (again, assuming sufficient variation in ).
Applying the main results: Fast rates
We now apply Theorem 1 to derive oracle excess risk bounds for the residualized loss. Let us consider the seminorms and and . Establishing the basic orthogonality and first-order conditions required to apply Theorem 1 is a simple exercise (see Appendix J for a full derivation). To establish the smoothness and strong convexity properties Assumption 3 and Assumption 4, we require mild boundedness assumptions, and a lower bound on the coverage parameter
[TABLE]
In particular, we have the following result.
Proposition 1**.**
Consider the treatment effect estimation setting with the residualized loss and norms and . Suppose that and . Then the assumptions of Theorem 1 are satisfied with constants , , , , and . As a result, the sample splitting meta-algorithm (Section 1) with the residualized loss produces a parameter such that with probability at least ,
[TABLE]
More generally, whenever is convex, the same conclusion holds with replaced by , regardless of whether .
Proposition 1 implies that for any class, oracle rates for excess risk are achievable whenever . Interestingly, in the case where the target class is convex, this holds even when the target parameter is arbitrarily misspecified. In addition, the excess risk bound in Proposition 1 has the desirable property that the coverage parameter enters only through the higher-order nuisance error term.
Let us interpret the coverage parameter , which acts as a problem-dependent constant whose value reflects the interaction between the treatment policy and the treatment effect. In general, to lower bound , it suffices to assume that for some , with no further assumptions required on the data distribution or target parameter class. This condition is typically referred to as overlap, since it requires that the treatment is not deterministic for any realization of the covariates, and implies that . On the other hand, even if overlap is not satisfied, one can still lower bound the coverage parameter. To do so, we focus on a special case investigated in Chernozhukov et al. (2017) and Chernozhukov et al. (2018b), where is a class of high-dimensional predictors of the form , where and is a fixed featurization; in general, the dimension may grow with , with . In this case, note that it suffices that the matrix satisfies a restricted minimum eigenvalue condition. Hence, a lower bound on generalizes assumptions used in Chernozhukov et al. (2017, 2018b).
Stronger guarantees for specific target classes
The results in the prequel apply to arbitrary target classes, but require that the nuisance estimation algorithm is close in the norm (i.e., ). For specific target classes (typically, classes with additional structure that facilitates estimation in parameter error), it is possible to provide improved guarantees that scale with weaker estimation error for the nuisance class. To illustrate the flexibility of Theorem 1 in accommodating such cases, we consider a constrained variant of the R-learner of Nie and Wager (2021) and recover the oracle rates from this work.
Example 1** (Constrained R-Learner).**
*The R-learner of Nie and Wager (2021) corresponds to a special case of the treatment effect estimation setup in Eq. 19 in which the target parameter belongs to a kernel class, and is estimated by minimizing the orthogonal loss Eq. 20 with regularization. Specializing the sample splitting meta-algorithm (Section 1) to this setting, we obtain a constrained variant of their method.
In more detail, consider the treatment effect estimation setting with and . Let be a reproducing kernel Hilbert space (RKHS) with norm and kernel . Assume that almost surely and that treatments satisfy overlap, and consider the constrained target parameter class*
[TABLE]
where is a parameter. For the target estimation algorithm, consider the plug-in empirical risk minimizer
[TABLE]
where is the nuisance estimator. Assume that the kernel has eigenvalue decay of the form for some parameter and that a smoothed version of lies in the RKHS for smoothing parameter (refer to proof in Section J.2 for definitions), and choose . If the nuisance estimation algorithm has , then with probability at least , Section 1 has
[TABLE]
where suppresses dependence on regularity parameters and factors. This matches the best known rate for the oracle learner (Nie and Wager, 2021).
This example shows that an rate in -error for the nuisances suffices to achieve the optimal rate in the absence of nuisance parameter. The proof leverages a lemma of Mendelson and Neeman (2010), which states that for all , , where is the eigenvalue decay parameter. This allows us to establish Assumption 3 and Assumption 4 with respect to -error for the nuisance parameter, at the cost of incurring exponent , rather than as in the generic result (Proposition 1). Moreover, as a consequence of the norm comparison inequality of Mendelson and Neeman (2010), the -error bound from our theorem also implies a bound on error.
Slow rates
We mention in passing that some distributions may simply not satisfy the coverage condition in Eq. 21. In this case, we can appeal to Theorem 2 (we show in Section J.2 that the residualized loss satisfies the universal orthogonality property), which does not require any lower bounds in the vein of Eq. 21, but leads to slower rates. In general, whether the fast rate (Theorem 1) or slow rate (Theorem 2) will give better results given finite samples will depend on the behavior of the data distribution and target class.
3.3.2 Doubly-Robust Loss (DR-Loss)
As an alternative to the residualized loss, for the special case of a binary treatment, we can use the doubly-robust approach described in Section 1. Consider the special case of Eq. 24 in which . Recall that that is the treatment propensity, and define . Define
[TABLE]
We take as the nuisance parameter. Then the doubly robust loss takes the form
[TABLE]
One can verify that . As a result, whenever the true nuisance parameters are plugged in, the oracle excess risk satisfies
[TABLE]
and hence is equivalent to -error. It is straightforward to verify that the doubly-robust loss satisfies the preconditions of Theorem 1, which leads to the following result.
Proposition 2**.**
Consider the treatment effect estimation setting with the doubly-robust loss and norms and . Suppose that and that for all . In addition, assume that almost surely, and that for all . Then the assumptions of Theorem 1 are satisfied with constants , , , , and . As a result, the sample splitting meta-algorithm (Section 1) with the doubly-robust loss produces a parameter such that with probability at least ,
[TABLE]
This approach was further developed in the subsequent work of Kennedy (2020), who termed it the DR-Learner, and provided improved oracle estimation rates which are doubly-robust with respect to the estimation errors for the propensities and conditional means. A variant of our two-stage algorithm for the doubly robust loss was also explored in the prior work of Oprescu et al. (2019) for the special case where the target estimation algorithm is a Generalized Random Forest.
\vsedit
We note that the explicit dependence on in the second order term in Proposition 2 can be avoided if one instead re-defines the inverse propensity term as the nuisance function. In this case, the second part of Assumption 3 is satisfied with respect to the product pre-norm: , since we have that:
[TABLE]
Applying Theorem 1 with this definition of nuisance pre-norm yields a doubly robust version of Proposition 2, where only products of nuisance estimation rates arise.
3.4 Example: Policy Learning
As a second example, we show how to apply our framework to the classical problem of policy learning. Compared to our treatment effect estimation example, losses for this setting do not typically satisfy the strong convexity property, meaning that Theorem 2 is the relevant meta-theorem, and slow rates are to be expected.
In policy learning, we receive examples of the form , where is an incurred loss, is a treatment vector and is a vector of covariates. The treatment is chosen based on an unknown, potentially randomized policy which depends on . Specifically, we assume the following data generating process:
[TABLE]
The learner wishes to optimize over a set of treatment policies (i.e., policies take as input covariates and return a treatment). Their goal is to produce a policy that achieves small regret with respect to the population risk:
[TABLE]
This formulation has been extensively studied in statistics (Qian and Murphy, 2011; Zhao et al., 2012; Zhou et al., 2017; Athey and Wager, 2017; Zhou et al., 2023) and machine learning (Beygelzimer and Langford, 2009; Dudík et al., 2011; Swaminathan and Joachims, 2015a; Kallus and Zhou, 2018); in the latter, it is sometimes referred to as counterfactual risk minimization.
The learner does not know the so-called counterfactual outcome function , so it is treated as a nuisance parameter. Typically, orthogonalization of this nuisance parameter is possible by utilizing the secondary treatment equation in Eq. 24 and fitting a parameter for the observational policy , which is also treated as a nuisance parameter. We can then write the expected counterfactual reward as
[TABLE]
for some known loss function that utilizes the treatment parameter . Letting , the learner’s goal can be phrased as minimizing the population risk,
[TABLE]
over . This formulation clearly falls into our orthogonal statistical learning framework, where the target parameter is the policy and the counterfactual outcome and observed treatment policy together form the nuisance parameter . To facilitate the use of estimation for the nuisance components, one typically assumes access to function classes and with and (so that ), and fits the nuisance parameters via regression over these classes.
We make this discussion concrete for the special case of binary treatments , with additional examples in Section G.1. To simplify notation, define , so that if and if . Consider the nuisance parameter . Then the loss function
[TABLE]
has the structure in Eq. 27: it evaluates to the true risk Eq. 25 whenever the true nuisance parameter is plugged in. This formulation leads to the well-known doubly-robust estimator for the counterfactual outcome (Cassel et al., 1976; Robins et al., 1994; Robins and Rotnitzky, 1995; Dudík et al., 2011). It is straightforward to verify that the resulting population risk is orthogonal with respect to . We can also obtain an equivalent loss function by subtracting the loss incurred by choosing treatment [math]. Define
[TABLE]
and set . This formulation leads to a linear population risk:
[TABLE]
This population risk satisfies universal orthogonality, and Theorem 2 can be applied with whenever the nuisance parameters are bounded appropriately. In particular, we have the following corollary of Theorem 2.
Proposition 3**.**
Consider the policy learning setting with binary treatments and norm . Suppose that almost surely, and that all have and for some . Then with probability at least , the target parameter produced by Section 1 enjoys the excess risk bound:
[TABLE]
Note that this bound depends on the overlap parameter only through the nuisance error term. We mention in passing that explicit dependence on this parameter can be avoided entirely by treating the inverse propensity term as nuisance parameter (see, e.g., Chernozhukov et al. (2021)). In this case, note that Assumption 6 is satisfied with respect to the product pre-norm: , since we have that:
[TABLE]
Applying our Theorem 1 with this definition of nuisance pre-norm, yields a doubly robust version of Proposition 3, where only products of nuisance estimation rates arise. Recent follow-up work of Chernozhukov et al. (2021) provides a statistical learning approach for estimating such nuisance functions with respect to mean-squared error, which is based on minimizing an empirical analogue of the risk function .
3.5 Discussion
We close by discussing extensions that build on the results presented in this section, as well as additional connections between our results and existing techniques in the literature on semiparametric inference and double machine learning.
Experiments, additional tools, and applications
Part I of the appendix contains an empirical evaluation of the techniques presented in this section, with applications to treatment effect estimation and policy learning. Part II of the appendix contains supplementary theoretical results that build on the development of this section, including user-friendly variants of the main theorems (Appendix C), construction of orthogonal losses (Appendix D), sufficient conditions to apply the main theorems (Appendix E), and further applications (Appendix G).
Construction of orthogonal losses
While orthogonal losses are already known for many problem settings and statistical models (treatment effect estimation, policy learning, regression with missing or censored data, and so on), for new problems one often begins with a loss that is not necessarily orthogonal. In Appendix D, we give a generic approach to construct orthogonal losses, building on a technique from Chernozhukov et al. (2018b).
One the use of cross-fitting
Section 1 relies on sample splitting. While this strategy is quite general and results in rate-optimal estimates, it can be inefficient, since the target parameter is only estimated using a subset of the data. A more practical alternative is to employ the well-known cross-fitting approach (e.g., Chernozhukov et al. (2018a)), in which we split the data into folds, obtain estimators using complementary folds, and combine the results. Cross-fitting variants of Section 1 are given in Appendix B as Appendices B and B.
One can show that under fairly genreal assumptions, the analysis in this section remains valid when cross-fitting is employed, and we recommend this in practice. However, compared to the setting considered in Chernozhukov et al. (2018a), in which cross-fitting provably enjoys improved efficiency over basic sample splitting, there is no hope of establishing that cross-fitting improves efficiency at the level of generality considered in the present work. This is because our framework permits the use of arbitrary, potentially nonparametric or high-dimensional estimators which may be biased due to the use of regularization or constraints and—for example—may not be asymptotically linear. As a result, even in the absence of nuisance parameters, there is no guarantee that averaging multiple target estimators obtained from independent sample splits will lead to improved efficiency.
One the use of influence functions
A special case of our framework can be phrased in the language of classical semiparametric inference as follows: If the population risk functional is pathwise differentiable, and one estimates the target by minimizing an estimator for the risk based on influence functions, \vseditwhich will typically lead to a Neyman orthogonal loss and the resulting target estimator will have favorable second-order errors dependence on the error of the nuisance estimator; see Curth et al. (2020) for follow-up work which takes this approach explicitly. However, Neyman orthogonality goes beyond pathwise differentiability (for instance, one can construct orthogonal losses assuming only existence of pathwise derivatives locally at ; see Appendix D), and our results apply in settings where influence functions may not exist. Moreover, one can obtain Neyman orthogonal losses without invoking influence functions, hence orthogonal statistical learning is a more flexible framework. See van der Laan and Robins (2003); Tsiatis (2007); Kosorok (2008); Kennedy (2016) for a review of influence functions and semiparametric theory.
4 Instantiating the Main Results: Plug-In Empirical Risk Minimization
\revoneedit
The results in Section 3 are stated at a high level of generality, and concern generic estimation algorithms for the target and nuisance parameters. In this section we shift our focus to specific algorithms, and instantiate our general tools to provide explicit bounds based on intrinsic properties of the function classes under consideration. In particular, we develop algorithms and analysis for orthogonal statistical learning with -estimation losses of the form
[TABLE]
We analyze one of the most natural and widely used estimation algorithms for the target parameter: plug-in empirical risk minimization (plug-in ERM). Specifically, recalling that , we define the empirical risk via
[TABLE]
where we adopt the convention that with to keep notation compact. The plug-in ERM algorithm returns the minimizer plug-in empirical loss obtained by plugging in the first-stage estimate of the nuisance component:
[TABLE]
\revoneedit
We provide oracle excess risk bounds for the plug-in ERM algorithm (and variants) in terms of statistical standard complexity measures for the target class . The main results in this section show that the impact of on the oracle excess risk achieved ERM is of second order, and that classical excess risk bounds carry over up to lower order terms and constant factors. These results are derived by bounding the second-stage using (localized) empirical process tools, then appealing to the main theorems (Theorem 1 and Theorem 2).
In the fast rate regime (i.e., for strongly convex losses) we offer a generalization of the local Rademacher complexity analysis of Bartlett et al. (2005) in the presence of an estimated nuisance component and show that the notion of the critical radius of the class still governs rate . This leads to several applications of our theory to specific target classes, including sparse linear models, neural networks and kernels (Section H.1).
In the slow rate regime (i.e., for generic Lipschitz losses), we offer a novel moment-penalized variant of the plug-in ERM algorithm that achieves a rate whose leading term is equal to the critical radius, multiplied by the variance of the population loss evaluated at the optimal target parameter. This offers an improvement over prior variance-penalized ERM approaches (Maurer and Pontil, 2009), whose leading term depends on the metric entropy of the target function class at single scale, and which typically is larger than the critical radius.
Technical preliminaries
To present our main results, we need to introduce additional tools from empirical process theory and statistical learning. For any real-valued function class , define the localized Rademacher complexity:
[TABLE]
where are independent Rademacher random variables. Let denote the non-localized Rademacher complexity (that is, ). We also make use of the metric entropy of a function class (which is closely related to the Rademacher complexity). \dfeditWe make the mild assumption that and are separable, so as to ensure that associated empirical processes are measurable (cf. Boucheron et al. (2013, pp 314-315)).
Definition 3** (Metric Entropy).**
For any real-valued function class and sample , the empirical metric entropy is the logarithm of the size of the smallest function class , such that for any there exists , with . Moreover will denote the maximal empirical entropy over all possible sample sets .
Finally, for a vector-valued function class , let denote the projection of the class onto the -th coordinate.
4.1 Fast Rates for Plug-In Empirical Risk Minimization
Our first contribution is an extension of the foundational results of Bartlett et al. (2005); Koltchinskii and Panchenko (2000)—which bound the excess risk for empirical risk minimization in terms of local Rademacher complexities—to incorporate misspecification due to nuisance parameter estimation error. A crucial parameter in this approach is the critical radius of a function class , defined as the smallest solution to the inequality
[TABLE]
Classical work shows that in the absence of a nuisance component, if a loss is Lipschitz in its first argument and satisfies standard assumptions required for fast rates (strong convexity in the first argument), then empirical risk minimization achieves an excess risk bound of order . For the case of parametric classes, , leading to the fast rates for strongly convex losses. For more general classes (cf. Wainwright (2019)) the critical radius is—up to constant factors—equal to the solution to an inequality on the metric entropy of the function class (cf. Section H.1.1):
[TABLE]
where ; see Section H.1 for concrete examples.
Our first theorem in this section extends this result in the presence of a nuisance component and bounds the excess risk of the plug-in ERM algorithm by the critical radius of the target function class (more precisely, the worst-case critical radius for each coordinate of the target class, since we deal with vector-valued function classes).
Theorem 3** (Fast Rates for Plug-In ERM).**
Consider a function class with . Let be a solution to the equation:
[TABLE]
where is the projection of onto coordinate . Suppose that is -Lipschitz with respect to the norm and that the population risk satisfies Assumptions 1, 2, 4 and 3 with and arbitrary. Define and B_{2}\vcentcolon=\big{(}\frac{\beta_{2}}{\lambda}\big{)}^{\frac{2}{1+r}}+\frac{\kappa}{\lambda}, and let be the outcome of the plug-in ERM algorithm. Then, with probability at least ,
[TABLE]
and
[TABLE]
\revoneedit
Critically, when , the dependence on the nuisance estimation error scales as due to orthogonality, meaning that we can use a complex function class for nuisance estimation without spoiling the rate for the target class. This result is proven in two steps. First, we show that one can take ; this result uses standard empirical process theory tools, and does not leverage orthogonality. Then, we invoke orthogonality through Theorem 1 to derive the final guarantee. See Appendix L for details.
4.2 Slow Rates and Variance Penalization
We now turn to the slow rate regime from Section 3.2, where the loss is not necessarily strongly convex in the prediction. We prove upper bounds on the generalization error of a variance penalized version of the plug-in ERM algorithm. Our main result gives a slow rate that scales with the variance of the loss rather than the range, and is robust to nuisance estimation error. The basic algorithm we analyze first estimates the nuisance parameter, then estimates the optimal loss value using auxiliary samples, and finally performs plug-in empirical risk minimization with an empirical variance penalty which is centered using the estimate for . \revoneeditSee Appendix B in Appendix B for a full description. To simplify notation, we assume that and is partitioned equal splits . Define the variance of the loss at via
[TABLE]
Theorem 4** (Plug-In ERM with Centered Second Moment Penalization).**
Consider the centered second moment-penalized plugin empirical risk minimizer in Appendix B:
[TABLE]
where is the output of and . Consider the function class , and let and . Let be any solution to the inequality
[TABLE]
Suppose that Assumption 5 holds, is -Lipschitz, and Assumption 6 holds with parameter and . Let . Then with probability at least ,
[TABLE]
\revoneedit
As with the previous result, Theorem 4 is proven by first upper bounding using empirical process tools, then invoking orthogonality through one of the main theorems (in this case, Theorem 2). The only complication is that the result requires the additional step of relating to the function , which entails bounding the variance of the loss at in terms of the variance of the loss at and nuisance estimation error.
Our approach offers an improvement over the rates for empirical variance penalization in Maurer and Pontil (2009), which provides a generalization error bound whose leading term is of the form: . The drawback of such a bound is that it evaluates the metric entropy at a fixed approximation level of , which can be suboptimal compared to the critical radius. \revoneeditIn Section H.2, we show that for classes with bounded VC dimension, this guarantee can be further improved as a consequence of our general machinery, and give a bound which scales with the so-called Alexander capacity function.
Discussion
Due to space constraints, applications to specific target classes (sparse linear models, neural networks, kernel classes) are deferred to Section H.1.
5 Instantiating the Main Results: Sufficient
Conditions for Oracle Rates
The previous section developed guarantees for orthogonal statistical learning with a specific algorithm, plug-in empirical risk minimization. While empirical risk minimization is a workhorse of statistical learning, in general it does not attain minimax excess risk for rich function classes, even in the absence of nuisance parameters. In this section we build on the development so far and, by appealing to aggregation techniques, provide algorithms that always attain minimax excess risk up to second-order dependence on nuisance parameters. Our main results provide sufficient conditions under which oracle rates are achieved which explicitly depend on intrinsic properties of both the target and nuisance parameter classes. In particular, we give sufficient conditions based on the relationship between the metric entropy for the nuisance and target classes.
For any real-valued function class , we say that the complexity of is if for all ,
[TABLE]
where we recall that is the metric entropy defined in Definition 3. When , this corresponds to the case of parametric functions (e.g., linear models and VC-subgraph classes), while for , we recover nonparametric function classes, such as Lipschitz/smooth functions or kernel spaces. We let and denote the maximum complexity of any output coordinate projection for the nuisance and target class, respectively. We provide sufficient conditions on the pair under which the sample splitting meta-algorithm (Section 1)—with an appropriate choice for the target and nuisance estimator—can achieve oracle rates.
We focus on the important special case of square losses of the form
[TABLE]
where and are known functions, and where we recall from Section 2 that , are subsets of the data , and is an arbitrary auxiliary subset of the data. We assume that the nuisance parameters are defined in terms of regression problems, i.e., that for some known random vector . This assumption is standard in semiparametric literature (Bickel et al., 1993; Kosorok, 2008; van der Laan and Rose, 2011), and implies that each coordinate of may be expressed as the minimizer of a squared loss: g_{0,t}=\operatorname*{arg\,min}_{g_{t}\in\mathcal{G}|_{t}}\operatorname{\mathbb{E}}\big{[}\left(g_{t}(w)-u_{t}\right)^{2}\big{]}. In this setting, a sufficient condition for orthogonality is that
[TABLE]
where and denote the gradient of with respect to the first and second argument, respectively.
In the absence of nuisance parameters, minimax optimal rates for excess risk in square loss regression have been characterized for the well-specified setting in which
[TABLE]
for some , and for the misspecified setting where this assumption is removed. In the former setting, the minimax rates are of order (Yang and Barron, 1999), while in the latter setting the optimal rate is (Rakhlin et al., 2017). We show that under orthogonality, the optimal well-specified and misspecified rates can be achieved in the presence of nuisance parameters even when the nuisance class is larger than the target class , provided it is not too much larger. This generalizes the large body of results on semiparametric inference (Levit, 1976; Ibragimov and Has’Minskii, 1981; Pfanzagl, 1982; Bickel, 1982; Klaassen, 1987; Robinson, 1988; Bickel et al., 1993; Newey, 1994; Robins and Rotnitzky, 1995; Ai and Chen, 2003; van der Laan and Dudoit, 2003; van der Laan and Robins, 2003; Ai and Chen, 2007; Tsiatis, 2007; Kosorok, 2008; van der Laan and Rose, 2011; Ai and Chen, 2012; Chernozhukov et al., 2022a; Belloni et al., 2017; Chernozhukov et al., 2018a), which show under various assumptions that if the target class is parametric, one can obtain a -consistent estimator for the target if the nuisance estimator converges at a rate.
Our main workhorse for the results in this section is the “Aggregation of -Nets” or “Skeleton Aggregation” algorithm described in Yang and Barron (1999) and extended to random design in Rakhlin et al. (2017). \revoneeditThe Skeleton Aggregation method operates by splitting the samples in two, building an empirical cover for the function class under consideration using the first split, and then aggregating the elements of the cover using the second split. See Section M.4 for a full description. This approach is related to sieve-based methods (e.g., (Semenova and Chernozhukov, 2021)), which employ parametric methods to learn a linear combination of basis elements that approximate the target, but an important difference is that Skeleton Aggregation builds the basis in a data-dependent fashion. We use Skeleton Aggregation as-is to provide rates for the first stage, and provide an extension in the presence of nuisance parameters for the second stage, which entails relating to .
\revoneedit
We caution that the algorithms in this section are only designed to attain the minimax rates for generic square losses of the type in Eq. 43 (e.g., vanilla square loss regression), and specific special cases may admit better rates. Deriving minimax lower bounds for specific losses of interest (as in Kennedy (2020)) is an interesting direction for future research.
Assumptions
Since our aim is to provide sufficient conditions based on the metric entropy of the classes and , which is already quite technical, we assume that all other problem-dependent parameters are constant. This is only for expository purposes.
Assumption 7**.**
*The classes are bounded in the sense that for all
and \vsedit the following bounds \vsedithold a.s.: a) , b) , c) , d) , \vsedite) , f) almost surely, g) the functions and have -Lipschitz gradients with respect to , h) the strong convexity condition \operatorname{\mathbb{E}}\big{[}\left\langle\Lambda(g_{0}(w),v),\theta(x)-\theta^{\star}(x)\right\rangle^{2}\big{]}\geq\gamma\operatorname{\mathbb{E}}\left\|\theta(x)-\theta^{\star}(x)\right\|_{2}^{2} is satisfied for all for some .*
Assumption 7 implies that Assumption 3 and Assumption 4 are satisfied with respect to the seminorms \left\|\theta\right\|_{\Theta}\vcentcolon=\big{(}\operatorname{\mathbb{E}}\left\langle\Lambda(g_{0}(w),v),\theta(x)\right\rangle^{2}\big{)}^{1/2} and , with . Since typical results on minimax oracle rates provide rates for the nuisance with respect to , we assume control on the ratio between these seminorms.
Assumption 8** (Moment Comparison).**
There is a constant such that
[TABLE]
The moment comparison condition has been used in statistics as a minimal assumption for learning without boundedness (Lecué and Mendelson, 2016; Mendelson, 2014; Liang et al., 2015). For example, suppose that each has the form for . Then if is mean-zero gaussian and if follows any distribution that is independent across all coordinates and symmetric (via the Khintchine inequality). Moment comparison is also implied by the “subgaussian class” assumption used in Mendelson (2011); Lecué and Mendelson (2016).222Suppose is scalar-valued and let . Then the subgaussian class assumption for our setting asserts that for all . We emphasize that the moment constant does not enter the leading term in any of our bounds—only the term in Theorem 1—and so it does not affect the asymptotic rates under conditions on metric entropy growth of that we prescribe in the sequel. We also note that this condition is not required for many classes of interest, where direct estimation rates are available (see discussion in Appendix E). We adopt the condition here because it allows us to develop guarantees for arbitrary classes at the highest possible level of generality.
Main Result
\revoneedit
The main theorem for this section provides sufficient conditions for oracle rates in the well-specified setting Eq. 45. For extensions of this result to misspecified models, as well as non-strongly convex losses, see Appendix F.
Theorem 5** (Oracle Rates, Well-Specified Case).**
Suppose that we are in the well-specified setting Eq. 45, and that Assumptions 1 and 2 and Assumptions 7 and 8 are satisfied for the class defined below. Suppose that the following relationship holds:
[TABLE]
Then for appropriate choice of sub-algorithms, the sample splitting meta-algorithm Section 1 produces a predictor that guarantees that with probability at least ,
[TABLE]
\revoneedit
where hides problem-dependent parameters and terms. This result matches the minimax rate in the absence of nuisance parameters. In particular, when it suffices to take and use plug-in ERM for stage two, and when it suffices to take and use Skeleton Aggregation for stage two; in both cases, it suffices to use Skeleton Aggregation for stage one.
\revoneedit
Theorem 5 is proven by combining the main theorem (Theorem 1) with algorithm-specific upper bounds on and . Fig. 1 summarizes the sufficient conditions under which Theorem 5 leads to the oracle rate (Yang and Barron, 1999). In particular, whenever is a parametric class (i.e. ), it suffices to take , which recovers the usual setup for semiparametric inference.
6 Discussion
This paper initiates the systematic study of prediction error and excess risk guarantees in the presence of nuisance parameters and Neyman orthogonality. Our results highlight that orthogonality is beneficial for learning with nuisance parameters even in the presence of possible model misspecification, and even when the target parameters belong to large nonparametric classes. We also show that many of the typical assumptions used to analyze estimation in the presence of nuisance parameters can be relaxed when excess risk is the focus. There are many promising future directions, including weakening assumptions, obtaining sharper guarantees for specific settings and losses of interest (e.g., doubly-robust guarantees), and analyzing further algorithms for general function classes (along the lines of Sections 4 and 5). \revoneeditWe refer to the appendix for additional results, as well as empirical results.
Acknowledgements
We are grateful to the anonymous COLT reviewers and to Xiaohong Chen for pointing out additional related work. Part of this work was completed while DF was an intern at Microsoft Research, New England. DF acknowledges support from the Facebook PhD fellowship and NSF Tripods grant #1740751.
Organization of Appendix
The appendix is organized as follows.
- •
Part I contains experimental results evaluating the performance of orthogonal risk minimization methods for examples considered in Section 3.
- •
Part II contains supplemental theoretical results: Algorithms omitted from the main body for space (Appendix B), user-friendly variants of the main theorems (Appendix C), construction of orthogonal losses (Appendix D), sufficient conditions to apply the main theorems, (Appendix E), sufficient conditions for oracle rates that extend Section 5 (Appendix F), applications of the main results (Appendix G), further results regarding plug-in empirical risk minimization (generalization guarantees for specific function classes (Section H.1), and an application of the guarantees for plug-in variance-penalized ERM developed in Section 4 to VC classes (Section H.2).
- •
Part III contains proofs for the results presented in the main body of the paper: Appendix I contains preliminaries, Appendix J proves the main theorems for the sample-splitting meta-algorithm, Appendix L proves the main results for plug-in empirical risk minimization (building on technical tools developed in Appendix K), and Appendix M proves our results concerning oracle rates.
Part I Experiments
Appendix A Experiments
We conducted experiments to evaluate of the benefit of risk minimization with orthogonal loss functions as compared to non-orthogonal losses, focusing on the main applications considered in Section 3: conditional average treatment effect (CATE) estimation under conditional exogeneity (Section 3.3) and policy learning (Section 3.4).
For both applications, we use synthetic data. Each dataset contains i.i.d. examples, where each example is a tuple consisting of a -dimensional random vector of confounders (), a random binary treatment (), and a random scalar outcome (). These random variables are drawn from a data generating process of the form:
[TABLE]
We refer to as the propensity function, as the base response function, and as the treatment effect function. The dimension and noise level are parameters that vary across experiments.
Inspired by the set of experiments in the prior work of Nie and Wager (2021); Athey and Wager (2017) we consider six variants of the data-generating process in Eq. 48 that capture different qualitative challenges for CATE estimation and policy learning.
- •
In Setup we consider , , , and , where denotes the operator that clips to the interval . This setup has a complicated propensity function and base response, but a relatively simple treatment effect function .
- •
In Setup , we consider , , , and . This setup is a randomized trial with a complex base response and treatment effect function.
- •
In Setup , we consider , , , and . This setup has a complex propensity and base response function, but a constant treatment effect.
- •
In Setup , we consider , , , and . In this setup, the response under treatment is independent of the response under control, and hence there is no statistical benefit of jointly learning the two responses.
- •
In Setup , we consider , , , and . In this setup, we have a very wide range of responses that is largely explained by the confounder. Hence, failing to appropriately center the response around the predictable part can lead to large variance in the learning objective. Moreover, the treatment effect function is discontinuous.
- •
In Setup , we consider , , , . This setup is similar to setup , but with a smooth treatment effect function.
For each setup, we consider three values for the sample size (), two values for the dimension of the confounders (), and three values for the scale of the response noise (). For each such parameter setting, we run experiments and report average metrics and standard errors for the average metric (for the appropriate metric for each task).
Reproducibility
Code for reproducing the results is available at https://github.com/vsyrgkanis/orthogonal_learning. We note that because the FLAML package used for nuisance model estimation does not allow for control of random seeds for fully reproducible execution, results will have slight variation across multiple executions, albeit not to an extent that affects the qualitative take-aways.
A.1 Conditional Average Treatment Effect Estimation
We first consider the task of estimating the conditional average treatment effect (i.e., we have ) with respect to the mean squared error metric \mathbb{E}\big{[}\big{(}\widehat{\theta}(X)-\theta_{0}(X)\big{)}^{2}\big{]}. We use the two orthogonal losses considered in Section 3.3. The first is the residual-on-residual loss:
[TABLE]
where , with the true values of the nuisance parameters given by and . Second, we consider the doubly-robust loss:
[TABLE]
where , with the true values of the nuisance parameters given by and . For each loss, we consider two risk minimization approaches: a semi-crossfitting approach that allows for some sample re-use, and a pure sample splitting approach.
- •
In the semi-crossfitting approach, we perform model selection using the FLAML (Wang et al., 2021) automated machine learning package, which performs hyperparameter tuning and selection using a broad class of forest based models. Using the model class and hyperparameters chosen by FLAML, we perform cross-fitting: we train a set of the nuisance models (e.g. ) on half of the data and compute their values (e.g. ) on the other half, and vice-versa. We then minimize the loss function over using all the samples with their computed nuisance values (e.g. for the residual-on-residual loss); this phase is also performed using FLAML.
- •
In the pure sample-splitting approach, we split the dataset in half and perform hyperparameter tuning, model selection, and model fitting with FLAML on the first half. We then compute nuisance values on the second half. Finally, we estimate the target function , also using the second half of the dataset. This approach corresponds to the meta-algorithm presented in the main body (Section 1).
We note that while the theoretical results in this paper only concern the pure sample-splitting approach, we expect that the semi-crossfitting approach will also benefit from the orthogonality of the loss, and it has been analyzed in prior literature for estimation of finite-dimensional target parameters (Chernozhukov et al., 2018a).
We refer to the estimation approach that uses the residual-on-residual loss with semi-crossfitting as dml, and refer to the variant with pure sample splitting as dml_split. Similarly, we refer to the two variants that use the doubly robust loss as dr and dr_split. Finally, as a benchmark we also consider the non-orthogonal loss:
[TABLE]
which we apply without any sample splitting or cross-fitting. We refer to this method as the slearner.
For each method, to perform risk minimization for the target parameter , we applied the FLAML package, which performs automated machine learning and hyperparameter tuning across many candidate forest models. In order to make the target parameter inherently simpler than then nuisance parameters, we constrained the final stage model to depend only on the first coordinates of the variables . For all setups except , we used , while for setup we used to maintain well-specification of the target model.
Results
The average mean-squared error for the estimated target parameters across the experiments and across the different setups are depicted in Figures 2, 3, 4, 5, 6, 7. We find that the methods that use orthogonal losses either significantly out-perform or are comparable to the non-orthogonal loss in most domains. The superior performance vanishes only in the presence of small samples and high noise level , potentially due to large errors in the nuisance functions. Moreover, we find that for relatively small sample size, the orthogonal losses perform comparably to the oracle method, which uses the true models and minimizes the residual-on-residual loss (denoted by oracle). Moreover, we find that sample-splitting is sufficient to get a substantial boost in performance, and that cross-fitting tends to give small further improvement over sample-splitting.
A.2 Policy Learning
Our second set of experiments concern the task of policy learning, where the goal is to minimize the loss (or equivalently, maximize the reward ), where is the treatment policy (the target parameter) and is a pre-defined treatment cost; for all experiments, we use the mean of the heterogeneous treatment effect as the cost (i.e. ). We implemented the orthogonal doubly-robust loss (denoted dr) described in Section 3.4 (Eq. Eq. 29):
[TABLE]
where is the nuisance parameter, with and . We consider two non-orthogonal losses as benchmarks. The first benchmark loss (denoted direct) is based solely on the regression model :
[TABLE]
while the second benchmark loss (denoted ips), is based solely on the propensity model :
[TABLE]
We consider the same data generating process setups (A-F) as in the CATE estimation setting (in particular, the true nuisance parameters and are chosen in the same fashion). As in the CATE estimation experiments, we used FLAML to fit the nuisance and target parameters. We performed hyperparameter tuning, model selection and model fitting on half of each dataset, and performed loss minimization over on the other half. For the target parameter (the treatment policy) we searched over binary decision trees of depth at most , with a minimum leaf size of at least samples. These trees where constructed using a greedy method, where—starting at the root node—we recursively choose at each node a split that greedily leads to the largest improvement in the target criterion.
Results
We present our results in Figures 8, 9, 10, 11, 12, which display the mean reward (negative of loss) for the policies learned by the approaches above; we omit setup C because the treatment effect is constant, which renders the policy learning problem trivial. Similar to our CATE results, we find that in most settings, the orthogonal loss approach gives comparable results to the non-orthogonal losses, but for setups and , the orthogonal loss method significantly outperforms the non-orthogonal losses. As a secondary result, we compare to
- •
The unrestricted optimal policy (denoted opt).
- •
An oracle method (denoted as or) that uses the true nuisance functions and optimizes the oracle objective over the same space of target policies as in dr, direct, and ips.
We find that when the sample size is large and the variance of the outcome is relatively small, the learned policies perform comparably to the oracle policy.
Part II Additional Results
Appendix B Additional Algorithms
Appendix B contains pseudocode for the variance-penalized plug-in empirical risk minimization method described in Section 4.2, which is omitted from the main body due to space constraints.
Appendix B and Appendix B present variants of Section 1 that employ cross-fitting rather than sample splitting, and serve as statistical learning counterparts to the DML1 and DML2 methods described in Chernozhukov et al. (2018a). Appendix B is specialized to M-estimation losses of the type considered in Section 4, in which for a point-wise loss function .
We note that most the natural instantiation of the DML2 meta-algorithm is to use a second-stage algorithm that minimizes the average empirical loss across the folds. In particular, assuming the empirical loss takes the form , one can apply plug-in ERM to the average empirical risk across the folds
[TABLE]
The localized Rademacher complexity techniques we develop in this paper can be adapted to this method to provide average excess risk bounds of order , where is as described in Theorem 3. This guarantee can then be combined with our main theorems to achieve oracle excess risk bounds with second order dependence. For example, the follow-up work of Dao et al. (2020) invokes such an analysis in the context of knowledge distillation. Appendix B is stated in a more general form to allow for second stage algorithms that go beyond plug-in ERM (e.g., penalized ERM variants or aggregation methods).
Appendix C Orthogonal Statistical Learning: User-Friendly Tools
Our main results, Theorem 1 and Theorem 2, give excess risk bounds for Section 1 (with generic nuisance and target estimators) under Neyman orthogonality. In this section we provide some additional consequences and variants of these results which will prove useful in deriving guarantees for specific estimators.
The first result gives a consequence of Theorem 1 for the case where the target estimator satisfies a certain self-bounding property. Such is the case for plug-in empirical risk minimization.
Lemma 1**.**
Suppose that the conditions of Theorem 1 hold, and that for all ,
[TABLE]
for functions and . Then the sample splitting meta-algorithm (Section 1) produces an estimate such that with probability at least ,
[TABLE]
and
[TABLE]
where and are defined as in Theorem 1.
Proof of Lemma 1.
Theorem 1 implies that with probability at least ,
[TABLE]
where and are as in Theorem 1. By the AM-GM inequality, we can upper bound this by
[TABLE]
Rearranging yields Eq. 57. To prove Eq. 58, we begin by applying, Assumption 3, which gives that
[TABLE]
Applying Lemma 2, we have
[TABLE]
where C\leq{}\frac{\lambda}{2}\bigg{(}\bigg{(}\frac{\beta_{2}}{\lambda}\bigg{)}^{\frac{2}{1+r}}+\frac{\kappa}{\lambda}\bigg{)}. We further bound this by
[TABLE]
where we have applied the AM-GM inequality. It follows that
[TABLE]
where we have applied Eq. 57 and then used that to simplify. ∎
The next lemma we provide is a variant of Theorem 1 which gives a bound on the first derivative of the oracle risk for Section 1.
Lemma 2** (Variant of Theorem 1).**
Suppose there exists such that Assumptions 1, 2, 4 and 3, are satisfied. Then the sample splitting meta-algorithm (Section 1) produces a parameter such that with probability at least ,
[TABLE]
where C\leq{}\frac{\lambda}{2}\bigg{(}\bigg{(}\frac{\beta_{2}}{\lambda}\bigg{)}^{\frac{2}{1+r}}+\frac{\kappa}{\lambda}\bigg{)}.
Proof of Lemma 2.
See Eq. 16 within the proof of Theorem 1. ∎
The next lemma (Lemma 3) uses orthogonality to provide a bound on the plug-in excess risk in terms of the target error and nuisance error. Recall that our main theorems give a bound on the oracle excess risk as a function of the plug-in excess risk. Lemma 3 is—in some sense—a converse, upper bounding the plug-in excess risk with the estimation error under orthogonality. This type of guarantee can be useful as an intermediate result when analyzing the plug-in excess risk for second stage algorithms. In this paper, we use it in the context of Skeleton Aggregation (Appendix M), to control misspecification error for the pseudo-excess risk.
Lemma 3**.**
Suppose that Assumption 1 holds, that
[TABLE]
and that the following derivative bounds hold:
- •
:
[TABLE]
- •
:
[TABLE]
Then for all ,
[TABLE]
Proof of Lemma 3.
Let be fixed. Using a second-order Taylor expansion, there exists such that
[TABLE]
Using another second-order Taylor expansion, there exists for which
[TABLE]
where the second equality uses the first-order condition and orthogonality. The result now follows from the assumed derivative bounds.
∎
Appendix D Construction of Orthogonal Losses
While orthogonal losses are already known for many problem settings and statistical models (treatment effect estimation, policy learning, regression with missing/censored data, and so on), for new problems we often begin with a loss which is not necessarily orthogonal. A natural question, which we address now, is whether one can modify the loss to satisfy orthogonality so that our main theorems can be applied.
Suppose we begin with a loss such that the nuisance and target parameter are specified by the moment equations
[TABLE]
where is a random variable, , and denotes the derivative with respect to the first argument. If is not orthogonal, we can construct an orthogonal loss using a generalization of a construction in Chernozhukov et al. (2018b). For simplicity, we sketch the approach for the special case where is scalar-valued.
To begin, assume that there exists a function such that for all , we have
[TABLE]
Under this assumption, we can expand our nuisance parameters to include —that is, define —and construct a new orthogonal loss:
[TABLE]
Letting \tilde{L}_{\mathcal{D}}(\theta,\tilde{g})=\operatorname{\mathbb{E}}\big{[}\tilde{\ell}(\theta(x),\tilde{g};z)\big{]} be the new population risk, we have the following claim.
Lemma 4**.**
The population risk satisfies Assumption 1 and Assumption 2.
Proof of Lemma 4.
The first-order condition (Assumption 2) follows immediately by using that , and by the assumption that . To establish orthogonality with respect to the nuisance parameter , we compute
[TABLE]
where the second equality follows from the definition of and the final inequality is the law of total expectation. For orthogonality with respect to , we have
[TABLE]
where the final inequality uses that and . ∎
As a first example, in the special case where the loss depends on only through its evaluation at (i.e., Eq. 61 simplifies to ), then we can take
[TABLE]
Of course, to make use of the lemma, we must be able to estimate the new nuisance parameter . This can be accomplished through an additional plug-in estimation step based on sample splitting: Split into folds , , , and of equal size. Estimate on , then obtain an initial estimate for by solving , where denotes the empirical loss over . Next, use the initial estimator to compute a plug-in estimator for by regressing onto the “targets” on ; \revoneeditthis requires access to an additional function class containing , and the resulting guarantee will have (second-order) dependence on the complexity of this class. Finally, produce the main estimator for the target parameter by solving . \revoneeditA full description is given in Appendix D.
The key idea behind this scheme is that the initial estimator will not be able to take advantage of orthogonality, but its estimation error will only enter the final bound through the error of , and thus will only have higher-order impact on the rate. \revoneeditWe omit details, which are beyond the scope of the present paper.
This approach is applicable for the problem of estimating utility functions in models of strategic competition, as used in Chernozhukov et al. (2018b); see Section D.1 for a detailed example. For some models—including utility function estimation— is a known function of and , so that the extra regression to estimate given the initial estimators is not required.
A more general setting where the loss has the form in Eq. 62 is as follows. Suppose that all functions are conditionally square-integrable in the sense that for all , , and suppose there exist functions , such that we can write
[TABLE]
where is a linear operator on with uniformly bounded operator norm:
[TABLE]
By the Riesz-Frechet representation theorem, we can express the operator as
[TABLE]
where we have used that to simplify. Hence, we have
[TABLE]
so that Eq. 62 is satisfied for induced by the family of Riesz representers for operators . This is a variant of the Riesz representer approach presented in Chernozhukov et al. (2022b). In Section D.1, we show that this construction recovers the treatment effect estimation example presented in the introduction.
We mention in passing that that another, perhaps more standard, approach to constructing an orthogonal loss is to derive the influence function for the risk function using standard calculations from semiparametric theory (van der Laan and Robins, 2003; Tsiatis, 2007; Kosorok, 2008; Kennedy, 2016).
D.1 Orthogonal Loss Construction: Examples
We now walk through concrete examples of the orthogonal loss construction approach outlined in Appendix D. Both examples use the loss structure in Eq. 61.
Estimating utility functions in models of strategic competition
In this setting, we have
[TABLE]
where is the logistic function. The motivation of this problem stems from estimating games of incomplete information, where is the entry decision of one player, is the entry decision of the opponent, is a featurized state of the world, is the non-strategic part of the utility of the player and is the competitive part of the utility, i.e. the effect of the opponent’s entry decision on the player’s utility. For this setting, we can take the auxiliary nuisance variable to be
[TABLE]
Thus, is a known function of and , and to estimate it suffices to construct preliminary estimates for and (e.g., using plug-in estimation with the original non-orthogonal loss). We can simplify the final orthogonal loss based our generic construction to
[TABLE]
where and is a preliminary estimate for .
Treatment effect estimation
In this setting, we denote the treatment as and , for some vector of extra control variables . In this setting, we take and , where is a vector of extra control variables and is a binary treatment. We begin from the moment equation
[TABLE]
with identified by the local moment equation . In this case, the Riesz representer takes the form , and the resulting orthogonal loss created using the generic construction takes the form
[TABLE]
where we recall . Minimizing this over is equivalent to minimizing the loss
[TABLE]
which is precisely the loss presented in the introduction of the paper.
Appendix E Sufficient Conditions for Theorems
and : Single Index Losses
Our setup and main results in Section 3 (Theorem 1 and Theorem 2) are stated at a high level of generality, with abstract assumptions on the structure of the population risk. In this section of the appendix we provide conditions under which these assumptions follow from concrete structural assumptions on the risk. We give sufficient guarantees for general families of loss functions. In particular, the conditions we give here suffice to derive guarantees for the applications considered in Appendix G.
E.1 Fast Rates
In this section we give a broad class of losses under which the conditions for fast rates in Section 3.1 are satisfied. The population loss is defined as the expectation of a point-wise loss acting on the predictions of the nuisance and target parameters. We assume existence of functions and such that the loss has the structure
[TABLE]
so that
[TABLE]
Here we recall from Section 2 that , are subsets of the data , and let be an auxiliary subset of the data. We also assume existence of functions and such that the partial derivative of may be written as
[TABLE]
where is non-decreasing and Lipschitz. A simple example is square loss regression, where .
To provide fast rates of the type in Section 3 for losses with this structure, we let be given, define , and consider the norms
[TABLE]
Taking large leads to a more stringent notion of distance for the nuisance parameter, but weakens the regularity conditions we consider, which depend on the dual parameter . In particular, we make the following assumptions.
Assumption 9** (Sufficient Conditions for Fast Rates: Single Index Losses).**
The loss satisfies the following conditions for parameters :
[TABLE]
Let us highlight the condition Eq. 81, which involves the dual exponent . Taking large strengthens the notion of distance for the nuisance parameter, but weakens the condition Eq. 81. For example, with , when we take , Eq. 81 becomes an average-case eigenvalue-type condition, but we require distance on the nuisance parameter. On the other hand, with , Eq. 81 essentially requires that parameter error () and error are equivalent for the target, but leads to distance for the nuisance parameter, which is more permissive than . As a concrete example, consider the logistic loss, where , where the target class is a subset of the data and is the logistic function, so that . Observe that the gradient of the loss with respect to the target index value can be written as
[TABLE]
Moreover, whenever the arguments to the loss are bounded, the Hessian can be bounded above and below via
[TABLE]
and thus, when , the ratio condition in Eq. 81 is implied by a minimum eigenvalue assumption on the conditional covariance matrix .
We now show that these conditions are sufficient to satisfy the assumptions of Theorem 1, and thus guarantee higher order impact from the nuisance parameters. Our main result is as follows.
Lemma 5**.**
If Assumption 9 holds, then the Assumptions 1, 2, 4 and 3 required by Theorem 1 are satisfied with constants , , , and , and with respect to the norms and defined in Eqs. 75 and 76.
Combining this lemma with the guarantee from Theorem 1 directly yields an oracle excess risk bound.
Corollary 1**.**
Suppose that there is some such that Assumption 9 is satisfied. The sample splitting meta-algorithm Section 1 produces a predictor that guarantees that with probability at least ,
[TABLE]
where and C_{2}=2\Big{(}\left(\frac{16\mu_{\mathsf{si}}^{2}K_{2}}{\tau_{\mathsf{si}}^{2}\lambda_{\mathsf{si}}}\right)^{\frac{1}{1+r_{\mathsf{si}}}}+32\left(\frac{L_{\mathsf{si}}^{4}R_{\mathsf{si}}^{2}}{\lambda_{\mathsf{si}}}\right)^{\frac{1}{1+r_{\mathsf{si}}}}\Big{)}. Furthermore, with as in Lemma 5, the following prediction error guarantee is satisfied with probability at least ,
[TABLE]
Observe that for both of the bounds in Corollary 1, the only problem-dependent parameters that multiply the target estimation rate are and . Importantly, this implies that if the more restrictive parameters and so forth are held constant as grows, they have negligible impact asymptotically, so long as the nuisance parameter can be estimated quickly enough. For the square loss this is particularly desirable, since .
En route to proving Lemma 7, we also prove the following, slightly stronger, smoothness bound, which is used for several results.
Lemma 6**.**
Suppose Assumption 9 holds with . Then for all functions , , and ,
[TABLE]
Discussion: Estimation for the first stage
When we work with error for the target parameter (that is, ), Corollary 1 (with ) provides guarantees in terms of the estimation rate for the nuisance parameters, i.e.
[TABLE]
Since error rates are somewhat less common than (i.e., square loss) estimation rates, let us briefly discuss conditions under which out-of-the box algorithms can be used to give guarantees on the error.
First, for many nonparametric classes of interest, minimax error rates have been characterized and can be applied directly. This includes smooth classes (Stone, 1980, 1982), Hölder classes (Lepski, 1992; Kerkyacharian et al., 2001, 2008), Besov classes (Delyon and Juditsky, 1996), Sobolev classes (Tsybakov, 1998), and convex regression (Guntuboyina and Sen, 2015)
Second, whenever the is a linear class or more broadly a parametric class, classical statistical theory (Lehmann and Casella, 2006) guarantees parameter recovery. Up to problem-dependent constants, this implies a bound on the error as soon as the fourth moment is bounded. This approach also extends to the high-dimesional setting (Hastie et al., 2015).
Last, if the class has well-behaved moments in the sense that for all , we can directly appeal to square loss regression algorithms for the first stage; this is the approach taken in Section 5. This condition is related to the so-called “subgaussian class” assumption, and both have been explored in recent works (Lecué and Mendelson, 2016; Mendelson, 2014; Liang et al., 2015).
Of course, whenever guarantees are available for the target class itself (e.g., for parametric models), we can instead take , which permits the use of the more standard distance for the nuisance parameter.
E.2 Slow Rates
In the single-index setup, assumptions much weaker than Assumption 9 suffice to obtain slow rates via Theorem 2. In particular, the following conditions are sufficient.
Assumption 10** (Sufficient Slow Rate Conditions for Single Index Losses).**
[TABLE]
Compared to Assumption 9, the most important difference is that since we require universal orthogonality, the first condition is required to hold for all , not just at . Assumption 10 has the following immediate consequences.
Lemma 7**.**
If Assumption 10 holds, then Assumption 5 is satisfied and Assumption 6 is satisfied with constant and with respect to .
Corollary 2**.**
Suppose Assumption 10 holds. Then with probability at least , the target predictor produced by Section 1 enjoys the excess risk bound
[TABLE]
E.3 Proofs
Proof of Lemma 5.
We prove that the assumptions required by Theorem 1 are implied by our conditions one by one.
Assumption 1
From the definition of the directional derivative, the law of iterated expectations, and the fact that , we have
[TABLE]
Assumption 2
This likewise follows immediately by expanding the directional derivative and applying the law of iterated expectation:
[TABLE]
We now argue about the remaining assumptions. We will repeatedly invoke the following expression for the second derivative of the population risk.
[TABLE]
Let us introduce some additional notation. Let and be the free variables in the statements of Assumption 3 and Assumption 4. We define the following vector- and matrix-valued random variables:
[TABLE]
To prove the lemma, it suffices to verify that Assumption 4 and Assumption 3 hold for the norms \left\|\theta-\theta^{\prime}\right\|_{\Theta}^{2}=\operatorname{\mathbb{E}}\big{[}\left\langle W_{0},\theta(x)-\theta^{\prime}(x)\right\rangle^{2}\big{]} and .
Assumption 3(a)
Using Eq. 84 and Eq. 86 we have:
[TABLE]
since for all . It follows that Assumption 3(a) is satisfied with .
Assumption 3(b)
Define random vectors , , , and . Then, invoking the assumed structure for the loss function, we have
[TABLE]
Using that , we have
[TABLE]
All that remains is to relate these norms to the norms appearing in the lemma statement.
[TABLE]
Thus, we have
[TABLE]
so that Assumption 3(b) is satisfied with .
Assumption 4
[TABLE]
Using Hölder’s inequality, we have that
[TABLE]
Hence, by Young’s inequality, we have that for any
[TABLE]
Choosing yields
[TABLE]
Thus, Assumption 4 is satisfied with and .
∎
Proof of Lemma 6.
We adopt the same notation as in the proof of Lemma 5. This proof is almost the same as the proof that Assumption 4 is satisfied in that lemma.
[TABLE]
Using the AM-GM inequality, this implies the inequality
[TABLE]
∎
Proof of Lemma 7.
Immediate. ∎
Appendix F Sufficient Conditions for Oracle Rates: Further Results
The results in Section 5 provide sufficient conditions for fast oracle rates under the assumption of strong convexity and a well-specified model, which is not satisfied for all losses used in practice. For example, linear losses used in policy learning (cf. Section 3.4) do not satisfy strong convexity property. In this section we extend the \revoneeditsufficient conditions for oracle rates given in Section 5. First, we give guarantees for strongly convex losses in the presence of misspecification, and then give guarantees for any (potentially non-strongly convex) loss that is Lipschitz in the target prediction . We follow the same notation as in Section 5.
F.1 Oracle Rates for Square Losses with Misspecification
Here we consider strongly convex losses with the same structure as in Section 5, but consider the case in which the target parameter is misspecified. This setting has been relatively unexplored in recent results on double machine learning (Chernozhukov et al., 2018a; Mackey et al., 2018; Chernozhukov et al., 2018b); this is perhaps not surprising since for many settings, assuming a well-specified model is critical to establish orthogonality. However, for certain settings including treatment effect estimation (Section 3.3), orthogonality can indeed hold even without model correctness. The following theorem shows that we can obtain oracle rates in the presence of both misspecification and nuisance parameters as long as the nuisance class has moderate complexity.
Theorem 6** (Oracle Rates, Misspecified Case).**
Suppose that the target class is convex. Suppose that Assumptions 1 and 2 and Assumptions 7 and 8 hold. If the relationship
[TABLE]
holds, then for appropriate choice of sub-algorithms, the sample splitting meta-algorithm Section 1 produces a predictor such that with probability at least ,
[TABLE]
\revoneedit
thereby matching the minimax rate in the absence of nuisance parameters. In particular, it suffices to use Skeleton Aggregation for the first stage and plug-in ERM for the second stage.
\revoneedit
Like the previous result, Theorem 6 is proven by combining the main theorem (Theorem 1) with algorithm-specific upper bounds on and . Fig. 13 summarizes sufficient conditions under which Theorem 6 matches the oracle rate (Rakhlin et al., 2017). Comparing to the well-specified case (Theorem 5/Fig. 1), we see that in the misspecified case, the condition on the metric entropy for the nuisance parameter is significantly more permissive when the target parameter class is large (i.e. ). For example, if then we require for oracle rates in the well-specified case, but only require in the misspecified case. On the other hand, when the conditions on the nuisance metric entropy match the well-specified case. In particular, whenever is a parametric class it again suffices to take so that the first stage has an -rate.
F.2 Oracle Rates for Generic Lipschitz Losses
For arbitrary Lipschitz losses (in particular, for the linear loss), the optimal rate in the absence of nuisance parameters is under the metric entropy growth assumed in Section 5 (cf. Section 12.8/12.9 of Rakhlin and Sridharan (2012)). Our main theorem for this section shows that this rate is still obtained in the presence of nuisance parameters when the nuisance metric entropy parameter is not too much larger than . We make the following regularity assumption on the loss.
Assumption 11**.**
The loss has absolute value bounded by , the mapping is -Lipschitz with respect to , and the mapping has -Lipschitz gradients with respect to . Assumption 5 holds for all when , and for all if .
Compared to the oracle rates for square losses, the assumptions required for our main Lipschitz loss theorem are relaxed significantly: We no longer require strong convexity, nor do we require any type of moment comparison for the nuisance class. On the other hand, we do require the additional universal orthogonality condition from Section 3.2.
Theorem 7**.**
Suppose that Assumption 11 is satisfied. If the relationship
[TABLE]
holds, then for appropriate choice of sub-algorithms, the sample splitting meta-algorithm produces a predictor that guarantees that, with probability at least ,
[TABLE]
\revoneedit
thereby matching the minimax rate in the absence of nuisance parameters.
\revoneedit
Theorem 7 is proven as a corollary of Theorem 2, by proving upper bounds on and for plug-in empirical risk minimization and skeleton aggregation, respectively; see Appendix M for the proof. The theorem is summarized in Fig. 14. In particular, \revoneeditsuppose that our aim is to match the minimax optimal rate in the absence of nuisance parameters, which is (Rakhlin and Sridharan, 2012). Whenever is a parametric class (i.e. ), it suffices to take , as in the well-specified and misspecified square loss setups in Section 5, and as in standard semiparametric results (Newey, 1994; van der Vaart, 2000; Robins et al., 2008; Zheng and van der Laan, 2010; Chernozhukov et al., 2018a). That the condition matches is somewhat interesting given that the final rate in this case is slower than in the square loss setup. Note however that we cannot tolerate until , as compared to our results strongly convex losses, where the admissible value of is growing for all values of . This result generalizes the condition given in Athey and Wager (2017) for the specific case of policy learning, which applies only in the parametric setting, and only for a specific loss.
Appendix G Additional Applications
In this section of the appendix we show how three additional families of applications—offline policy learning/optimization, domain adaptation/sample bias correction, and learning with missing data—fall into our general orthogonal statistical learning framework. We also sketch some statistical consequences based on our general algorithmic tools, and show how these results generalize and extend previous work.
G.1 Policy Learning: Further Examples
In this section we show how to view some additional policy learning models in our framework. As in Section 3.4, we consider the data-generating process in Eq. 24, but here we go beyond binary treatments.
Multiple finite treatments
The binary setting above can easily be extended to the case of possible treatments, analyzed in Zhou et al. (2023). Formally, let , where is the -th standard basis vector. We still follow the data generating process Eq. 24, but now and . To simplify notation, let so that . Then the following loss function is an unbiased estimate of the counterfactual loss:
[TABLE]
This formulation leads to the standard extension of the doubly-robust estimator to multiple outcomes Dudík et al. (2011); Zhou et al. (2023). Define an -dimensional vector-valued function to have the -th coordinate is equal to . Then, as in the binary case, we can equivalently optimize a population risk that is linear in the target parameter:
[TABLE]
This population risk is easily shown to satisfy universal orthogonality.
Counterfactual risk minimization and general continuous treatments
Counterfactual risk minimization (CRM) is a learning framework introduced by Swaminathan and Joachims (2015a). It is mathematically equivalent to the policy learning setup with arbitrary treatment and outcome spaces, but is motivated by a different set real-world learning scenarios and was developed in a parallel line of research in the machine learning literature. To highlight the relationship with policy learning and the applicability of our results to this setting we will present the CRM framework using the notation of policy learning.
In counterfactual risk minimization we receive data from the policy learning data generating process Eq. 24. The goal is to choose a hypothesis (i.e., the policy takes as input covariates and returns a distribution over treatments) that minimizes the population risk:
[TABLE]
As in policy learning, we construct an unbiased estimate of this counterfactual loss via inverse propensity scoring. Let denote the probability density of treatment conditional on covariates and (overloading notation) let denote the density that hypothesis assigns to treatment . Then we can formulate a new risk function that provides an unbiased estimate of the target risk Eq. 98:
[TABLE]
In the CRM framework the propensity is assumed to be known (Swaminathan and Joachims, 2015a; Swaminathan and Joachims, 2015b). When the propensity is not known, we can treat it as a nuisance parameter to be estimated from data. However, the loss Eq. 99 is not orthogonal to . We can orthogonalize the population risk by also constructing an estimate of (see Eq. 24) by regressing on . This leads to an analogue of the doubly robust formulation from the finite treatment setup:
[TABLE]
For finite treatments this formulation is mathematically equivalent to the population risk for multiple finite treatments presented in the prequel.
For continuous treatments, the empirical version of the problem Eq. 100 may be ill-posed, even if we assume that the propensity has density ower bounded by some constant (the analogue of the overlap condition). Swaminathan and Joachims (2015a) proposed to regularize the empirical risk via variance penalization. A similar variance penalization approach is also proposed in recent work of Bertsimas and McCord (2018), who consider policy learning over arbitrary treatment spaces. The variance-penalized empirical risk minimization algorithm—in the context of Section 1—can be seen as a second stage algorithm that achieves a rate whose leading term scales with the variance of the optimal policy rather than some worst-case upper bound on the risk. Hence, it can be used in our framework to achieve variance-dependent excess risk bounds.
Kallus and Zhou (2018) develop alternative algorithms for policy learning with continuous treatments via a kernel smoothing approach. This approach is equivalent to adding noise to a deterministic hypothesis space , e.g. for each , where . In our framework is the space of density functions induced by this construction. The value of a deterministic policy (or equivalently the value of its corresponding density ) is equal to
[TABLE]
where is the pdf of a normal distribution with standard deviation . This is equivalent to the formulation in Kallus and Zhou (2018), since the empirical version of this risk is the kernel-weighted loss:
[TABLE]
though we note that Kallus and Zhou (2018) do not restrict themselves to only the gaussian kernel.
This idea falls into our framework by simply defining to be this space of randomly perturbed policy functions. The resulting analysis in our framework is slightly different than that of Kallus and Zhou (2018), where kernel weighting is invoked to show consistency of the empirical risk, and subsequently optimization of the empirical risk over deterministic policies is analyzed. With our framework, we directly calculate the regret with respect to randomized policies by applying our general theorem. This implies that we enjoy robustness to errors in estimating and .
Observe that the rate for the second stage will depend on the amount of randomization , since the variance of the empirical risk is governed by . Consequently, if one is interested in regret against deterministic strategies, we can invoke Lipschitzness of the reward function to control the regret added by the extra randomness we are injecting, which would typically be of order . We can then choose an optimal as a function of the number of samples to trade-off the bias and variance of the regret. If we wish to further optimize dependence on problem-dependent parameters in the resulting rates one can use variance penalization in the kernel-based framework to achieve a regret rate whose leading term scales with the variance of the optimal policy.
G.2 Domain Adaptation and Sample Bias Correction
Domain adaptation is a widely studied topic in machine learning (Daume III and Marcu, 2006; Jiang and Zhai, 2007; Ben-David et al., 2007; Blitzer et al., 2008; Mansour et al., 2009). The goal is to choose a hypothesis that minimizes a given loss in expectation over a target data distribution, where the target distribution may be different from the distribution of data that is already collected.
We consider a particular instance of domain adaptation called covariate shift, encountered in supervised learning (Shimodaira, 2000). We assume that we have data , where are co-variates drawn from some distribution with density and are labels, drawn from some distribution conditional on . Our goal is to choose a hypothesis from some hypothesis space , so as to minimize a loss function in expectation over a different distribution of co-variates with density . Both of the densities are unknown, and we solve this issue in the orthogonal statistical learning framework by treating their ratio as a nuisance parameter for an importance-weighted loss function. Let and , so that
[TABLE]
We assume the hypothesis space satisfies a realizability condition, i.e. there exists such that:
[TABLE]
where corresponds to the gradient with respect to the first input of . For instance, for the case of the square loss , then this condition has the natural interpretation that there exists such that:
[TABLE]
Observe that when we treat the density ratio as a nuisance function, the loss function is orthogonal. Indeed,
[TABLE]
Also, note that this setup fits in to the single index structure from Appendix E by writing as the expectation of a new loss . Focusing on the square loss for concreteness, it is simple to show that all of the conditions of Assumption 9 are satisfied with as long as we have for all . Corollary 1 then implies that with probability at least , Section 1 enjoys the bound
[TABLE]
Note that whenever the dependence on is negligible asymptotically. Of course, it is also important to develop algorithms for which the rate of the target class does not depend on . As one example, we can employ the variance-penalized ERM guarantee from Theorem 8. When has VC dimension , and the variance of the loss at and the capacity function are bounded, this gives , with entering only lower-order terms. The final result is that if , we get an excess risk bound for which the dominant term is , with no dependence on .
Related work
Cortes et al. (2010) gave generalization error guarantees for the important weighted loss Eq. 103 in the case where the densities and are known. At the other extreme, Ben-David and Urner (2012) showed strong impossibility results in the regime where the densities are unknown. Our results lie in the middle, and show that learning with unknown densities is possible in the regime where the weights belong to a nonparametric class that is not much more complex than the target predictor class . We remark in passing that algorithms based on discrepancy minimization Ben-David et al. (2007); Mansour et al. (2009) offer another approach to domain adaptation that does not require importance weights, but these results are not directly comparable to our own.
G.3 Missing Data
As a final application, we apply our tools to the well-studied problem of regression with missing/censored data (Robins and Rotnitzky, 1995; van der Laan and Robins, 2003; Rubin and van der Laan, 2005; Tsiatis, 2007; Wang et al., 2010; Graham, 2011). In this setting we receive data is generated through standard regression model, but label/target variables are sometimes “missing” or unobserved. The learner observes whether or not the target is missing for each example, and the conditional probability that the target is missing is treated as an unknown nuisance parameter. As usual, the target is the unknown regression function.
To proceed, we formalize the setting through the following data-generating process for the observed variables :
[TABLE]
Here is an auxiliary variable (observed by the learner) that indicates whether the target variable is missing, and is the unknown propensity for . The parameter is the true regression function. We make the standard unobserved confounders assumption that and . We define , take , and use the loss
[TABLE]
Observe that this loss has the property that
[TABLE]
so that the excess risk relative to precisely corresponds to prediction accuracy whenever the true nuisance parameter is plugged in.
Proposition 4**.**
This model satisfies Assumption 1 and Assumption 2 whenever , i.e. we have , , and .
Note that the extra nuisance parameter is only required here because we consider the general setting in which . Whenever this is unnecessary (and indeed ). This parameter can generally be estimated at a rate no worse than the rate for and (absent nuisance parameters); see Chernozhukov et al. (2018b) for discussion.
As to rates and algorithms, the situation here is essentially the same as that of the domain adaptation example, so we discuss it only briefly. The setup has the single index structure from Appendix E, and all of the sufficient conditions for fast rates from Assumption 9 are satisfied with as long as we have for all in the nuisance class. Thus, with probability at least , Section 1 enjoys the bound
[TABLE]
As with the previous example, the variance-penalized ERM guarantees from Section 4 can be applied here to provide bounds on for which the dominant term in the excess risk does not scale with the inverse propensity range.
Proof of Proposition 4.
We first show that the gradient vanishes in the sense of Assumption 2 when evaluated at . In particular, for any choice of we have
[TABLE]
To establish orthogonality with respect to , we have
[TABLE]
To establish orthogonality with respect to , we have
[TABLE]
Using that and , the expression above is seen to be equal to zero. ∎
Appendix H Plug-in Empirical Risk Minimization: Further Results
H.1 Plug-in Empirical Risk Minimization: Examples
In this section of the appendix, we instantiate the general plug-in ERM framework from Section 4 to give concrete guarantees for some concrete classes of interest. In all examples we use to hide dependence on problem-dependent constants, factors, and factors.
High-dimensional linear classes
For our first set of examples, we focus on high-dimensional linear predictors. Chernozhukov et al. (2018b) gave orthogonal/debiased estimation guarantees for high-dimensional predictors using Lasso-type algorithms. Our first example shows how to recover the type of guarantee they gave, and our second example shows that we can give similar guarantees under weaker assumptions by exploiting that we work in the excess risk / statistical learning (rather than parameter estimation) framework.
Example 2** (High-Dimensional Linear Predictors with Constraint).**
Suppose that is an -sparse linear function with support set and that and almost surely under . Define the target class via
[TABLE]
Given , define the restricted eigenvalue for the target class as
[TABLE]
where has as rows. Then under the assumptions of Theorem 3, the empirical risk minimizer guarantees that with probability at least ,
[TABLE]
For parameter estimation it is well known that restricted eigenvalue or related conditions are required to ensure parameter consistency. For prediction however, such as assumptions are not needed if we are willing to consider inefficient algorithms. The next example shows that ERM over predictors with a hard sparsity constraint obtains the optimal high-dimensional rate for prediction in the presence of nuisance parameters with no restricted eigenvalue assumption.
Example 3** (High-Dimensional Linear Predictors with Hard Sparsity).**
Suppose that is a class of high-dimensional linear predictors obeying exact or “hard” sparsity:
[TABLE]
and suppose almost surely under . Then under the assumptions of Theorem 3, the empirical risk minimizer guarantees that with probability at least ,
[TABLE]
Neural networks
We now move beyond the classical linear setting to the case to the case where the target parameters belong to a class of neural networks, a considerably more expressive class of models. Let be the logistic link function and let be the ReLU function.333For vector-valued inputs we overload and to denote element-wise application.
Our first neural network example is inspired by Chen and White (1999); Farrell et al. (2021), who analyzed neural networks for nuisance parameter estimation with parametric target parameters. We depart from their approach by using neural networks to estimate target parameters.
Example 4**.**
Suppose that the target parameters are a class of neural networks , where
[TABLE]
and and . Let denote the total number of weights in the network. Under the assumptions of Theorem 3, the empirical risk minimizer guarantees that with probability at least ,
[TABLE]
The target class in this example is well-suited to estimation of binary treatment effects. Note that in this example our only quantitative assumption on the network weights is that they guarantee boundedness of the output. However, the bound scales linearly with the number of parameters , and thus may be vacuous for modern overparameterized neural networks. Our next example, which is based on neural networks covering bounds from Bartlett et al. (2017), shows that by making stronger assumptions on the weight matrices we can obtain weaker dependence on the number of parameters. This comes at the price of a slower rate— vs. .
Example 5**.**
Suppose that the target parameters are a class of neural networks , where
[TABLE]
and denotes the sum of row-wise -norms. Suppose almost surely under . Under the assumptions of Theorem 3, the empirical risk minimizer guarantees that with probability at least ,
[TABLE]
Let us give a concrete example where the neural network guarantees above enable oracle rates for the target class while using a more flexible parameter class for the nuisance. Suppose the target parameters belong to the class in Example 4 with layers and weights, and suppose the nuisance parameters also belong to a neural network class, but with layers and weights. In the next section we will show that under certain assumptions one can guarantee for such a class. In this case, Example 5 shows that we obtain oracle rates whenever , meaning the number of parameters in the nuisance network can be significantly larger than for the target network. Similar guarantees can be derived for Example 4.
Deriving tight generalization bounds for neural networks is an active area of research and there are many more results that can be used as-is to give guarantees for the second stage in our general framework (Golowich et al., 2018; Arora et al., 2019).
Kernels
For our final example, we give rates for some basic kernel classes. These examples were chosen only for concreteness, and the machinery in this section and the subsequent sections can be invoked to give guarantees for more rich and general nonparametric classes.
Example 6** (Gaussian Kernels).**
Suppose that is unit ball in the reproducing kernel Hilbert space with the Gaussian kernel . Suppose is drawn from the uniform distribution over . Under the assumptions of Theorem 3, the empirical risk minimizer guarantees that with probability at least ,
[TABLE]
Example 7** (Sobolev Spaces).**
Suppose the target class is the Sobolev space
[TABLE]
and suppose that is drawn from the uniform distribution on . Under the assumptions of Theorem 3, the empirical risk minimizer guarantees that with probability at least ,
[TABLE]
H.1.1 Proofs
Throughout this section we adopt the shorthand . We first recall some basic technical lemmas which will be used in the proofs for the examples.
Lemma 8** (Mendelson (2002), Lemma 4.5).**
For any real-valued function class with for all and any with ,
[TABLE]
Lemma 9** (Wainwright (2019), Proposition 14.1).**
Let be the minimal solution to
[TABLE]
where is a star-shaped set with . Then with probability at least over the draw of data ,
[TABLE]
where is the minimal solution to
[TABLE]
The following result is an immediate consequence of Lemma 15.
Lemma 10**.**
Define
[TABLE]
Then any minimal solution to
[TABLE]
is an upper bound for the fixed point for Eq. 110.
Proof for Example 2.
Let data be fixed. Let . Under our assumption that , Zhang (2002), Theorem 3, implies that
[TABLE]
We will now establish that
[TABLE]
for an appropriate choice of using the restricted eigenvalue bound. Let
[TABLE]
We first claim . Indeed, fix and let . Then we have
[TABLE]
Rearranging, we get as desired. Now observe that for any , we have
[TABLE]
This implies that for , and as a consequence
[TABLE]
We plug this bound into Eq. 112 and derive an upper bound of
[TABLE]
where we have used that . Using Lemma 10, we may now take in Eq. 37, then combine with Theorem 3 and Theorem 1 to get the result.
∎
Proof for Example 3.
Since and , the standard covering number bound for linear classes states that the covering number at scale for any fixed sparsity pattern is at most . We take the union over all such covers for all sparsity patterns, which implies . Lemma 8 further implies that
[TABLE]
It is now a standard calculation to show that
[TABLE]
Thus, via Lemma 9 and Lemma 10, we may take in Eq. 37. The final result follows by combining Theorem 3 and Theorem 1.
∎
Proof for Example 4.
Since our the target class is bounded, Lemma 8 implies
[TABLE]
Recall
[TABLE]
Then, since is -Lipschitz and positive-homogeneous, we have . Recall that for -valued classes of regressors we can relate the empirical metric to an empirical metric for a closely related VC class as follows. Let , let , and write
[TABLE]
Consequently, we see that the covering number for on the distribution at scale , is at most the size of the cover of the class on distribution at scale . Thus, invoking Haussler’s covering number bound for VC classes (Haussler, 1995), we have
[TABLE]
where denotes the VC dimension and denotes the pseudodimension. Using Theorem 14.1 from Anthony and Bartlett (1999) and Theorem 6 from Bartlett et al. (2017), we have
[TABLE]
With this bound on the metric entropy we have
[TABLE]
Thus, it suffices to take in Eq. 37 and appeal to Theorem 3 and Theorem 1. ∎
Proof for Example 5.
As in Example 4, we have . Theorem 3.3 of Bartlett et al. (2017) implies that under our assumptions,
[TABLE]
The result follows by plugging this bound into Lemma 10 and proceeding exactly as in the previous examples. ∎
Proof for Example 6 and Example 7.
Note that each target class has range bounded by . By examples 14.4 and 14.3 in Wainwright (2019), we may take and in Eq. 37 for the gaussian and Sobolev classes respectively. We combine with this with Theorem 3 and Theorem 1. ∎
H.2 Plug-in Empirical Risk Minimization: Refined Guarantees for VC Classes
In this section of the appendix we use the general tools developed in Section 4 to provide efficient/variance-dependent oracle rates for VC classes with general Lipschitz losses. Our main result shows that for VC classes with dimension , the excess risk enjoyed by variance penalization grows exactly as (where , as before, is the variance of the loss at the pair ) so long as the nuisance estimator converges at a rate of . The key to our approach is to assume boundedness of the so-called Alexander capacity function, a classical quantity that arises in the study of ratio type empirical processes (Giné and Koltchinskii, 2006).
To be more precise, for this example we assume that is a class of binary predictors with VC dimension , and let have the following policy learning structure:
[TABLE]
where is a known function. Our goal is to derive a bound for which the leading term only scales with rather than the loss range. Our results depend on a variant of the Alexander capacity function (Giné and Koltchinskii, 2006; Hanneke, 2014). Letting
[TABLE]
the capacity function is defined as
[TABLE]
When is the unweighted classification loss, this definition recovers the classical definition of the capacity function (Giné and Koltchinskii, 2006; Hanneke, 2014). Beyond boundedness of the capacity function, we make the following assumption.
Assumption 12**.**
Assumption 5* holds along, with the following bounds:*
- •
* almost surely for all , for some .*
- •
* almost surely.*
- •
* for all , for some seminorm .*
- •
The first stage algorithm provides an estimation error bound with respect to , i.e. with probability at least over the draw of .
- •
Assumption 6* holds with respect to with constant .*
Theorem 8**.**
Suppose that Assumption 12 holds, and define . Then variance-penalized empirical risk minimizer guarantees that with probability at least ,
[TABLE]
Note that whenever the asymptotic rate depends only on the variance at and , not on the problem-dependent parameters . Furthermore, whenever the capacity function is constant the asymptotic rate is exactly .
Variance-dependent bounds that obtain the efficient rate have been the subject of much recent investigation, and there is much interest in understanding when the rate obtained by naive approaches can be improved. To give a brief survey, the seminal empirical variance bound due to Maurer and Pontil (2009) when applied directly to this setting gives a suboptimal rate. Recent work of Athey and Wager (2017) shows that for a specific loss and nuisance parameter setup arising in policy learning, the can be replaced with a certain worst-case variance parameter. Our result, Theorem 8 is complementary, and shows that the can be replaced by the capacity function for general losses. It appears unlikely that the factor can be removed without at least some type of assumption. Indeed the results in Rakhlin et al. (2017) imply that there are indeed VC classes for which the critical radius grows as in the worst case.
The proof of Theorem 8 can be broken into three parts: First, we apply the previous results of this section to show that the excess risk obtained by variance penalization depends on the critical radius of the class . Second, we show that in the absence of first-stage estimation error, the capacity function controls the critical radius. Finally, we show that the impact of nuisance estimation error on the capacity function is of second order.
H.2.1 Proof of Theorem 8
Define the function class . We assume for now that for all , that is -Lipschitz (i.e. in the theorem statement) and that ; the general case will be handled by rescaling at the end of the proof..
Our starting point is to appeal to Theorem 9 . In particular, let be any solution to the inequality:
[TABLE]
where . Then if is the outcome of variance-penalized ERM, we have that by Theorem 4, with probability at least ,
[TABLE]
Moving to capacity function at
Per the discussion in the prequel, we focus on bounding the critical radius in the case where is bounded by and is -Lipschitz. We wish to make use of the capacity function, which is defined at , but the local Rademacher complexity we need to bound is that of , which evaluates the weight function at . To make progress, we show how to use the capacity function defined in the theorem statement to bound the following “plug-in” variant:
[TABLE]
We first show how to relate the norm at to the norm at . Define . Then for any we have
[TABLE]
Using AM-GM and boundedness of , for any this is lower bounded by
[TABLE]
Using the Lipschitz assumption and conditional lower bound on , we further lower bound by
[TABLE]
Hence, by choosing and rearranging, we get
[TABLE]
We proceed to bound the capacity function . Let . Let be fixed and let be any policy with . Then equation Eq. 115 implies that that , and so
[TABLE]
To handle the term in the numerator we proceed similar to the proof of Eq. 115. We have
[TABLE]
Fix any . We use AM-GM and boundedness of policies to upper bound the second term as
[TABLE]
Putting everything together, we get
[TABLE]
Thus, for all we have
[TABLE]
Bounding the critical radius
For any class we define . We work with the following empirical version of the local Rademacher complexity
[TABLE]
which has . Let the draw of be fixed. Invoking Lemma 15, we have
[TABLE]
Using that any can be written as , where and , a simple discretization argument (cf. proof of Lemma 8) shows that
[TABLE]
Let us adopt the shorthand . It follows from the usual symmetrization argument that . Letting be fixed, we can summarize our argument so far as
[TABLE]
Furthermore, using a change of variables we have
[TABLE]
We now handle the covering integral for . Let
[TABLE]
Our approach is to upper bound the empirical covering number for by the covering number of the class with respect to Hamming error. Let the Hamming error on a set be defined via
[TABLE]
We claim that there is a choice for the dataset such that for all , the empirical error on is upper bounded by the empirical Hamming error of associated policies on .
Let and be fixed elements of , and let be such that and likewise for and . Define , and take to contain of copies of example for each . With this choice, we have
[TABLE]
Thus, if we let , then any -cover in Hamming error is an -cover in . Now define
[TABLE]
and note that by definition. We invoke the following facts.
- •
.
- •
Haussler’s bound (Haussler, 1995) implies that any class with VC dimension admits a –Hamming error cover of size .
Putting everything together, we have
[TABLE]
It follows from the usual symmetrization argument that . Furthermore, using the concentration bound in Eq. 122 (we use the assumed boundedness of elements of to simplify Eq. 122 to the form that appears on this page), there exists a constant such that for any , with probability at least over the draw of ,
[TABLE]
Thus, conditioning on this event, we have
[TABLE]
where the second inequality uses a change of variables and that .
Now, to summarize our developments so far, we have shown that with probability at least ,
[TABLE]
Using Markov’s inequality, we have that with probability at least , . Thus, by union bound, with probability at least ,
[TABLE]
Integrating out this tail bound, we get that
[TABLE]
Using AM-GM and that , then rearranging, this implies
[TABLE]
We now bound the ratio . We have
[TABLE]
Thus, using the relationship between and established in the previous section of the proof, if we have
[TABLE]
and so for all ,
[TABLE]
In particular, we can see from this expression that taking
[TABLE]
yields a valid upper bound on the critical radius.
Final bound
Putting together the excess risk bound and the critical radius bound, we have
[TABLE]
To handle the square Rademacher complexity term, we recall that since , the main result of Haussler (1995) implies that ; since this term is squared in the final bound, its contribution is of lower order.
To deduce the final bound in the general -bounded -Lipschitz case we divide the class by , then rescale the final bound (observing that , , and all vary appropriately under the rescaling).
∎
Part III Proofs for Main Results
Appendix I Preliminaries
We invoke the following version of Taylor’s theorem and its directional derivative generalization repeatedly.
Proposition 5** (Taylor expansion).**
Let be fixed and let , where is an open interval containing . If is -times differentiable, then there exists such that
[TABLE]
Let , where is a vector space of functions. For any , if is -times differentiable over an open interval containing , then there exists such that
[TABLE]
Appendix J Proofs from Section 3
J.1 Omitted Proofs for Main Results
Proof of Theorem 2.
To begin, we use the guarantee for the second stage from Definition 1 and perform straightforward manipulation to show
[TABLE]
Using continuity guaranteed by Assumption 6, we perform a second-order Taylor expansion with respect to for each pair of loss terms in the preceding expression to conclude that there exist such that
[TABLE]
To relate the two derivative terms, we apply another second-order Taylor expansion (which is possible due to Assumption 6), this time with respect to the target predictor.
[TABLE]
where . Universal orthogonality immediately implies that
[TABLE]
Furthermore, observe that
[TABLE]
Since for all , including , universal orthogonality (Assumption 5) implies that both terms in the numerator are zero, and hence . We conclude that . Using this identity in the excess risk upper bound, we arrive at
[TABLE]
∎
J.2 Proofs for Examples
Proof of Proposition 1.
We first verify that the conditions of Theorem 1 are satisfied. To establish orthogonality (Assumption 1) for the propensities , let be fixed. Then we have
[TABLE]
To handle the first term, we use that for any ,
[TABLE]
Similarly, the second term is handled by using that
[TABLE]
To establish orthogonality for the expected value parameter , for any we have
[TABLE]
which follows from the assumption . Note that both of these orthogonality proofs held for any choice of , not just , and hence Assumption 1 is satisfied for all .
Next, we show that Assumption 2 holds whenever the second stage is well-specified (i.e. ); the fact that Assumption 2 whenever is convex is immediate. We have
[TABLE]
In particular, for any we have
[TABLE]
and so .
It remains to verify Assumptions 3 and 4. We do so appealing to Lemma 5. Following the notation of Appendix E, we set , , and . With this choice, we can take , and . To bound the parameter , for , , and , we write
[TABLE]
which has
[TABLE]
It follows that whenever . As a result Lemma 5 implies that Assumptions 4 and 3 are satisfied with constants , , and . The result now follows from Theorem 1.
∎
Proof and Details for Example 1.
Formally, we consider the following setup (Nie and Wager, 2021). Let is an RKHS with norm and kernel . We assume that is a compact metric space, and that is a kernel with respect to . We define via
[TABLE]
Using Mercer’s theorem (Cucker and Smale, 2002), there exist eigenfunctions and eigenvalues such that
[TABLE]
The main assumptions are as follows.
Eigenvalue decay. There exists such that
[TABLE] 2. 2.
Approximation. There exists such that the function has
[TABLE]
In addition, we assume that , that (note that ), and that almost surely. We also assume overlap, i.e. . Throughout the proof, we use to suppress dependence on , , , , and . Recall that the estimator is obtained via plug-in empirical risk minimization with respect to the constrained function class
[TABLE]
for a parameter . As noted in Eq. (19) of Nie and Wager (2021), if we define (note that , since is constrained), we have
[TABLE]
In particular, by choosing , the approximation error scales as to . In light of this approximation result, is remains to derive an oracle excess risk bound against for fixed .
To proceed, we verify that the assumptions required by Theorem 1 (in particular, smoothness and strong convexity) are satisfied. We do so appealing to Lemma 5. Following the notation of Appendix E, we set , , and . With this choice, we can take . It remains to bound and .
- •
To bound the parameter , for , , and , we write
[TABLE]
which has
[TABLE]
It follows that whenever .
- •
To bound , we recall Lemma 5.1 of Mendelson and Neeman (2010), which states that for all ,
[TABLE]
where is a constant depending on , , and . In addition, we have and . Hence, we may take and .
As a result, Lemma 5 implies that Assumptions 4 and 3 are satisfied with constants , , , and . Theorem 3 now implies that with probability at least , the plug-in empirical risk minimizer satisfies
[TABLE]
where is the solution to the fixed point equation in Eq. 37. From Lemma 5 of Nie and Wager (2021), we have
[TABLE]
This implies that is a valid fixed point, so we have
[TABLE]
Whenever , this simplifies to
[TABLE]
and combining with the approximation result in Eq. 118 yields
[TABLE]
Choosing gives excess risk .
∎
Proof of Proposition 2.
We verify that the conditions of Theorem 1 are satisfied. We do so by showing that Assumption 9 is satisfied and appealing to Lemma 5. In particular, following the notation of Appendix E, we observe that the doubly-robust loss has the structure required by Assumption 9, since we may take , , and . Furthermore, it can be written as a square loss of the form
[TABLE]
where, for , we overload notation and write
[TABLE]
To establish orthogonality (Assumption 1), it suffices to show that
[TABLE]
We have
[TABLE]
and it is straightforward to verify that as a consequence of the double robustness property
Next, since , it is immediate that Assumption 2 is satisfied.
Finally, we verify the regularity conditions required to establish Assumptions 3 and 4. With , it is immediate that we may take , , and for Assumption 9 whenever . To bound the parameter , for , and we observe that
[TABLE]
Since by assumption, it suffices to bound . It is straightforward to verify that whenever , , and , we have , so we may take . As a result, Lemma 5 implies that Assumptions 4 and 3 are satisfied with constants , , , and . The result now follows from Theorem 1. ∎
Proof of Proposition 3.
We first verify that Assumption 5 is satisfied. Let , , and be arbitrary. Abbreviate . We have
[TABLE]
and
[TABLE]
Similarly, we have
[TABLE]
This establishes the universal orthogonality property.
We now verify that Assumption 6 is satisfied with . Consider the function given by
[TABLE]
Assumption 6 holds whenever is bounded for all and all in the range of . In particular, one can verify by inspection that whenever , all have and .
With both assumptions satisfied, the result now follows from Theorem 2. ∎
Appendix K Technical Lemmas for Constrained -Estimators
In this section of the appendix we give self-contained technical results for -estimation over general function classes, in the absence of nuisance parameters. The results here serve as a building block for the results of Section 4.
Let be the function class, and let be the loss. We receive a sample set drawn from distribution independently. Let denote the random variable and let
[TABLE]
denote the population risk and empirical risk over samples. The constrained empirical risk minimizer is given by
[TABLE]
Additional Notation
To keep notation compact, we adopt the abbreviations and . We drop the subscript for real-valued function classes. We recall that denotes the th coordinate projection of and, likewise, for any , the th coordinate projection is .
The following lemmas provide a vector-valued extension of the analysis of constrained ERM based on local Rademacher complexities given in Wainwright (2019). The key idea behind the extension is to invoke a vector-valued contraction theorem for Rademacher complexity due to Maurer (2016). For completeness we give a include proofs for these lemmas, even though some parts are straightforward adaptations of lemmas in Wainwright (2019).
Lemma 11**.**
Consider a function class , with and pick any . Assume that the loss is -Lipschitz in its first argument with respect to the -norm and let
[TABLE]
Then there are universal constants such that
[TABLE]
Moreover, if is any solution to the inequalities
[TABLE]
then for each ,
[TABLE]
Lemma 12**.**
Consider a vector valued function class with , and pick any . Let be any solution to the system of inequalities
[TABLE]
Moreover, assume that the loss is -Lipschitz in its first argument with respect to the norm. Consider the following event:
[TABLE]
Then for some universal constants , .
Lemma 13**.**
Consider a vector valued function class and pick any . Let be any solution to the inequalities
[TABLE]
Suppose . Moreover, assume that the loss is -Lipschitz in its first argument with respect to the norm and also linear, i.e. and . Consider the following event:
[TABLE]
Then for some universal constants , .
Lemma 14**.**
Consider a function class , with , and pick any . Let be any solution to the inequalities:
[TABLE]
Moreover, assume that the loss is -Lipschitz in its first argument with respect to the norm. Then for some universal constants , with probability ,
[TABLE]
Hence, the outcome of constrained ERM satisfies that with the same probability,
[TABLE]
If the loss is also linear in , i.e. and , then the lower bound on is not required.
K.1 Proofs of Lemmas for Constrained -Estimators
Proof of Lemma 11.
By the Lipschitz condition on the loss and the boundedness of the functions, we have . Moreover,
[TABLE]
\dfedit
Thus, by Bousquet’s concentration inequality (see Theorem 7.3 of Bousquet (2003) or Theorem 12.5 of Boucheron et al. (2013)) we have that for all ,
[TABLE]
for absolute constants . Moreover, by a standard symmetrization argument,
[TABLE]
where the second inequality follows from the fact that each summand is non-negative, since we can always choose . By invoking the multivariate contraction inequality of Maurer (2016), letting be independent Rademacher random variables, we have
[TABLE]
where we also used the fact that for any fixed , . This completes the proof of the first part of the lemma. For the second part, observe that: . Moreover, for any star shaped function class , the function is monotone non-increasing. Thus for any ,
[TABLE]
Rearranging yields that . Hence, . This completes the proof of the second part of the lemma. ∎
Proof of Lemma 12.
We invoke a peeling argument. Consider the events
[TABLE]
for . Since , it must be that any with belongs to some for , where . Thus by a union bound we have
[TABLE]
Moreover, observe that if the event occurs then there exists a with , such that
[TABLE]
Thus, by the definition of , we have
[TABLE]
Applying Lemma 11 with and , yields that the latter probability is at most , where we used the fact that in the last inequality. Subsequently, taking a union bound over the events, we have
[TABLE]
Since, by assumption on the lower bound on we have , we get
[TABLE]
∎
Proof of Lemma 13.
For simplicity, let . Suppose that there exists a function , with , such that
[TABLE]
Then we will show that there exists a function , with , such that
[TABLE]
To do so, we simply choose to satisfy
[TABLE]
Since and by the definition of the star hull, we know that . Moreover, by the definition of , we also have that . Moreover, by the linearity of the loss with respect to , we have:
[TABLE]
Thus we have that the probability of event is upper bounded by the probability of the event
[TABLE]
Invoking Lemma 11 with and , we conclude that the probability of the second event is also at most , for some universal constants . ∎
Proof of Lemma 14.
Consider the events:
[TABLE]
with as defined in Lemma 11. Observe that if Eq. 130 is violated, then one of these events must occur. Applying Lemma 11 with and yields, that event happens with probability at most , where . Moreover, applying Lemma 12 we get that . Thus by a union bound with probability , neither events occur. If the loss is linear then we apply Lemma 13 instead of Lemma 12, which does not require a lower bound on . ∎
Appendix L Proofs from Section 4
Let be shorthand for the random variable , and recall that
[TABLE]
denote, respectively, the population risk and empirical risk over the samples in . We consider the two-stage plugin ERM algorithm
[TABLE]
As in Appendix K, we adopt the abbreviations and ; we drop the subscript for real-valued function classes.
Throughout this section, we repeatedly make use of the following fact: If solves a fixed point equation such as Eq. 37, then does as well. By expanding the radius in this fashion, we can replace success probabilities of the form (e.g., in Theorem 3 below) by , as long as is replaced by in the final excess risk bound.
L.1 Proof of Theorem 3
We prove Theorem 3 for the case where . The general case follows by observing that ERM over is equivalent to ERM over the class with the loss , with the problem parameters remapped as , , , , , and .
Proof of Theorem 3.
Since is the outcome of the Plug-In ERM and since , we have
[TABLE]
Applying Lemma 12, with , and , we know that with probability at least , where are numerical constants,
[TABLE]
Since Assumptions 1, 2, 4 and 3 are satisfied, Lemma 1 (a corollary of Theorem 1) with and implies that
[TABLE]
and
[TABLE]
where and C_{2}\leq 2\Big{(}\big{(}\frac{\beta_{2}}{\lambda}\big{)}^{\frac{2}{1+r}}+\frac{\kappa}{\lambda}\Big{)} are as in Theorem 1; note that we have used that and to simplify. This establishes the result.
∎
L.2 Proof of Theorem 4
We first prove a theorem about non-centered -moment penalization, and then show that this implies Theorem 4.
Theorem 9** (Moment-Penalized Plug-In ERM).**
Consider the function class , with and . Let be any solution to the inequality
[TABLE]
and be the second moment-penalized empirical risk minimizer,
[TABLE]
Then with probability at least ,
[TABLE]
Proof of Theorem 9.
We first consider the case where . We apply Lemma 14 with (i.e., the identity loss). Observe that with probability (via Theorem 4.1 of Wainwright (2019)), and that is linear. As a result, we have that for any that satisfies the conditions of Theorem 9, with probability ,
[TABLE]
This implies that with probability at least ,
[TABLE]
where the second inequality follows by the definition of the moment-penalized algorithm, since
[TABLE]
Now, for general values of , we may apply the reasoning above to the normalized class . In particular, the condition Eq. 136 implies that
[TABLE]
so we may take as the critical radius for the normalized class. This, combined with Eq. 137, implies that probability at least ,
[TABLE]
which establishes the result. ∎
Proof of Theorem 4.
We first sketch the intuition behind how to deduce the variance-based bound from Theorem 4 from the second moment-based bound from Theorem 9, then give a formal proof. Observe that if the optimal loss is zero then the two are equivalent. This motivates the following approach: if one has access to a good preliminary estimate of the value , then using moment penalization one can always attain a bound that depends on . The latter is achieved by simply redefining the function class in Theorem 9 to be the centered class of losses . This leads to the algorithm in the theorem statement, which penalizes the centered second moment:
[TABLE]
As long as the error in the preliminary estimate is vanishing, i.e. , then the impact of this error on the regret is only of second order, since the final regret bound takes the form.
[TABLE]
In more detail, using the bound from Theorem 9, we have
[TABLE]
which, using the AM-GM inequality, we can bound by
[TABLE]
We are nearly ready to apply Theorem 2, but first we must bound the error term and relate the variance on the right-hand side above to the true variance . We bound the error for the estimate using vanilla (non-localized) Rademacher complexity and two-sided uniform convergence arguments over the function class . In particular, using standard arguments, we can guarantee that with probability at least over the draw of , we have
[TABLE]
Finally, we observe that since is -Lipschitz for all and , we have for all . Hence, we have , and
[TABLE]
Finally, we observe that
[TABLE]
After another application of the AM-GM inequality, this yields
[TABLE]
Note that we have not used orthogonality up to this point. Applying Theorem 2 (using Assumption 5 and Assumption 6) gives the final result. ∎
Appendix M Proofs from Section
5 and Appendix F
M.1 Notation
We state the guarantees in this section using slightly more refined notation than in Section 5 and Appendix F. In particular, we consider both the case where the target and nuisance classes are parametric and where they are nonparametric. For the nuisance parameter class , the two cases we consider are:
- a)
Parametric case. There exists a constant such that
[TABLE] 2. b)
Nonparametric case. There exists a constant such that
[TABLE]
Likewise, for the target class, the two cases we consider are:
- a)
Parametric case. There exists a constant such that
[TABLE] 2. b)
Nonparametric case. There exists a constant such that
[TABLE]
M.2 Preliminaries
Definition 4** (Empirical Rademacher Complexity).**
For a real-valued function class and sample set , the Rademacher complexity is defined via
[TABLE]
where are i.i.d. Rademacher random variables. We define .
We require the following technical lemmas. First is the Dudley entropy integral bound; we use the following form from Srebro et al. (2010) (Lemma A.3). In all results that follow, we use to denote an absolute constant whose value may change from line to line.
Lemma 15**.**
For any real-valued function class , we have
[TABLE]
As a consequence, whenever takes values in the following bounds hold:
- •
If , then , where satisfies
[TABLE]
- •
If , then .
We also require the following lemma, which controls the rate at which the empirical metric converges to the population metric in terms of metric entropy behavior.
Lemma 16**.**
Let , and let be a collection of samples in drawn i.i.d. from .
- •
If for some , then with probability at least , for all ,
[TABLE]
where
[TABLE]
- •
If , then with probability at least , for all ,
[TABLE]
Proof of Lemma 16.
Using Lemma 8 and Lemma 9 from Rakhlin et al. (2017), it also holds that with probability at least , for all , we have
[TABLE]
where , and where in the parametric case, and in the general case. The final result follows by applying the Rademacher complexity bounds from Lemma 15.
∎
Remark 2**.**
Technically, the result in Rakhlin et al. (2017) we appeal to in the proof above is stated for -valued classes, but it may be applied to our -valued setting by shifting and rescaling the class (i.e., invoking with ). We appeal to the same reasoning throughout this section, shifting regression targets in the same fashion when necessary.
M.3 Overview of Proofs
We now sketch the high-level approach behind the main results in Section 5 and Appendix F. The idea is to use out-of-the-box learning algorithms for both the nuisance and target stage. However, which algorithm gives an optimal rate will depend on the complexity of and . Moreover, some of the algorithms we employ for the target class require new analyses based on orthogonality to bound the error due to nuisance parameter estimation.
First stage
Base on our assumptions on the metric entropy, we can obtain rates by appealing to the following generic algorithms.
- •
Global ERM: For each , select
[TABLE]
- •
Skeleton Aggregation/Aggregation of -nets (Yang and Barron (1999); Rakhlin et al. (2017); see Section M.4 for a formal description): For each , run the Skeleton Aggregation algorithm with the class on the dataset of instance-target pairs . Let be the result.
Proposition 6** (Rates for first stage, informal).**
Suppose that Assumption 7 or Assumption 11 holds. Then Global ERM guarantees that with probability at least ,
[TABLE]
Skeleton Aggregation guarantees that with probability at least ,
[TABLE]
Here the notation suppresses , , and factors.
A precise version of Proposition 6 and detailed description of the algorithms are given in Section M.5. Note that the minimax rate is (Yang and Barron, 1999), and so Skeleton Aggregation is optimal for all values of , while Global ERM is optimal only for . While these are not the only algorithms in the literature for which we have generic guarantees based on metric entropy (other choices include Star Aggregation (Liang et al., 2015) and Aggregation-of-Leaders (Rakhlin et al., 2017)), they suffice for our goal in this section, which is to characterize the spectrum of admissible rates.
In all applications we study, the dimension is constant. Nevertheless, studying procedures that jointly learn all output dimensions of and, in particular, deriving the correct statistical complexity when is large is an interesting direction for future research and may be practically useful.
Second stage
The idea behind the second-stage rates we provide is that the problem of obtaining a target predictor for the second stage can be solved by reducing to the classical square loss regression setting. We map our setting onto square loss regression by defining auxiliary variables and , and by defining auxiliary predictor classes
[TABLE]
With these definitions, our goal to bound the excess risk in, e.g., Theorem 5 can be equivalently stated as producing a predictor that enjoys a bound on
[TABLE]
which is the standard notion of square loss excess risk used in, e.g., Liang et al. (2015); Rakhlin et al. (2017). Defining, , we can apply any standard algorithm for the class to the dataset . Note however that, due to the use of as a plug-in estimate, predictors produced via Section 1 will—invoking Definition 1—give a guarantee of the form
[TABLE]
where and the benchmark belong to instead of . The machinery developed in Section 3 relates the left-hand-side of this expression to the oracle excess risk Eq. 143. Depending on the setting, more work is required to show that the right-hand-side of Eq. 144 is controlled. This challenge is only present in the well-specified setting. The difficulty is that while the original problem Eq. 143 is well-specified in this case, the presence of the plug-in estimator in Eq. 144 introduces additional “misspecification”. We show for global ERM and Skeleton Aggregation the right-hand-side of the expression is controlled as well, meaning that is not much larger than the rate that would have been achieved if the true value for the nuisance parameter was known. This achieved by exploiting orthogonality once again.
In the misspecified setting, we can simply upper bound the right-hand side of Eq. 144 by the worst-case bound and get the desired growth. Since the model is misspecified to begin with, any extra misspecification introduced by using the plugin estimate here is irrelevant. To be precise, the algorithm configuration is as follows.
- •
For stage one, use Skeleton Aggregation (Yang and Barron, 1999; Rakhlin et al., 2017). If , global ERM can be used instead.
- •
For stage two, in the misspecified setting with convex, use global ERM.
- •
For stage two, in the well-specified setting, we use Skeleton Aggregation, with a new analysis to account for the small amount of “model misspecification” introduced by the plug-in nuisance estimate . If , global ERM can be used instead; this is because skeleton ERM and global ERM are both optimal for , even in the presence of nuisance parameters.
M.4 Skeleton Aggregation
Here we briefly describe the Skeleton Aggregation meta-algorithm for real-valued regression (Yang and Barron, 1999; Rakhlin et al., 2017). The setting is as follows: we receive examples i.i.d. from a distribution . For a function class , we define . Our goal is to produce a predictor for which the excess risk is small.
We call a sharp model selection aggregate any algorithm that, given a finite collection of functions and i.i.d. samples, returns a convex combination for which
[TABLE]
with probability at least . One such model selection aggregate is the star aggregation algorithm of Audibert (2008), which produces a -sparse convex combination with the property Eq. 145 whenever almost surely and the functions in take values in .
We use the following variant of skeleton aggregation, following Rakhlin et al. (2017). Given a dataset , we split it into two equal-sized parts and .
- •
Fix a scale , and let . Let \big{\{}\widehat{f}_{i}\big{\}}_{i\in\left[N\right]} be a collection of functions that realize the cover, and assume the cover is proper without loss of generality.444Any improper -cover can be made into a proper -cover.
- •
Let be the output of the star aggregation algorithm run with the collection \big{\{}\widehat{f}_{i}\big{\}}_{i\in\left[N\right]} on the dataset .
For a simple analysis of this algorithm, see Section 6 of Rakhlin et al. (2017). In general, the algorithm is optimal only in the well-specified setting in which for some . We give a more refined analysis in the presence of nuisance parameters in the sequel. As final remark, note that since we use a proper cover and the star aggregate is -sparse, the final predictor lies in the class
M.5 Rates for Specific Algorithms
Given an example , we define an auxiliary example via , . For the remainder of this section we make use of the auxiliary second-stage dataset defined via
[TABLE]
We make use of the following auxiliary predictor classes:
[TABLE]
Finally, we define and , where are sampled from the distribution introduced by drawing , and taking . With these definitions, observe that for any of the form , we have
[TABLE]
We relate the metric entropy of the auxiliary class to that of as follows.
Proposition 7**.**
Under Assumption 7, it holds that
[TABLE]
Lemma 17** (Rates for first stage).**
Global ERM guarantees that with probability at least ,
[TABLE]
where is a constant that depends only on . Skeleton Aggregation guarantees that with probability at least ,
[TABLE]
Proof of Lemma 17.
In what follows we analyze the algorithms under consideration for the class for a fixed coordinate . The final result follows by union bounding over coordinates and summing the coordinate-wise error bounds we establish.
Global ERM. When we are either in the parametric case or the nonparametric case with , the result is given by Theorem 5.2 of Koltchinskii (2011). See Example 3 and Example 4 that follow the theorem for precise calculations under these assumtions. See also Remark 2.
On the other hand, when we apply to the standard Rademacher complexity bound for ERM (e.g. Shalev-Shwartz and Ben-David (2014)), which states that with probability at least ,
[TABLE]
where . The result follows by applying Lipschitz contraction to the Rademacher complexity (using that the class is bounded) and appealing to the Rademacher complexity bound from Lemma 15.
Skeleton Aggregation. We appeal to Section 6 of Rakhlin et al. (2017). See Remark 2.
∎
Lemma 18**.**
Consider the plug-in ERM algorithm for the setting in Section 5, i.e.
[TABLE]
Under the assumptions of Theorem 5 and Theorem 6, global ERM guarantees
[TABLE]
Proof of Lemma 18.
To begin, let \widehat{f}(\widetilde{X})\vcentcolon={}\big{\langle}\Lambda(\widehat{g}(w),w),\widehat{\theta}(x)\big{\rangle} and observe that we can write as the global ERM for the auxiliary dataset :
[TABLE]
Case . In the misspecified case we appeal to Theorem 5.1 in Koltchinskii (2011), using that is the global ERM for the class . To invoke the theorem, we verify that a) takes values in under Assumption 7 (see Remark 2), b) inherits convexity from , and c) , following Proposition 7. The theorem (see also the following discussion in Example 3 and Example 4) therefore guarantees that with probability at least ,
[TABLE]
The result now follows from Eq. 149, in particular that .
Case . We apply the standard Rademacher complexity bound for ERM (e.g. Shalev-Shwartz and Ben-David (2014)), which states that with probability at least ,
[TABLE]
where we have applied Lipschitz contraction to the Rademacher complexity (using that the class is bounded). To complete the result, we use that and appeal to the Rademacher complexity bound from Lemma 15. ∎
Lemma 19**.**
Consider the following variant of the Skeleton Aggregation algorithm:555See Section M.4 for background.
- •
Split into equal-sized subsets and .
- •
Fix a scale , and let . Let be a collection of functions that realize the cover, and assume the cover is proper without loss of generality. Define for each .
- •
Let realize the output of the star aggregation algorithm using the function class \big{\{}\widehat{f}_{i}\big{\}}_{i\in\left[N\right]} on the dataset \big{\{}(\widetilde{X}(z),\widetilde{Y}(z))\big{\}}_{z\in{}S^{\prime\prime}}.
Under the assumptions of Theorem 5, when the model is well-specified, Skeleton Aggregation guarantees that with probability at least ,
[TABLE]
so long as .
Proof of Lemma 19.
Let . The Skeleton Aggregation algorithm as described outputs a predictor (see Section M.4) such that
[TABLE]
Translating back into the language of the lemma statement, recall that we can express each via , with since we have assumed a proper cover. Since this parameterization is linear in , there must be some that realizes . Using the expression for the risk in Eq. 149, this implies
[TABLE]
Adding and subtracting from both sides, we rewrite the inequality as
[TABLE]
Let . Observe that by Lemma 6, we have that for all ,
[TABLE]
and by Lemma 5, we have that for all and ,
[TABLE]
Hence, since we have assumed orthogonality, Lemma 3 implies that for all ,
[TABLE]
Furthermore, Assumption 7 implies that . Plugging this bound back into Eq. 151, we have that
[TABLE]
for constants . We now invoke Lemma 16 for each of the output coordinates of the target space separately and union bound, which implies that with probability at least over the draw of ,
[TABLE]
where
[TABLE]
in the nonparametric case, and
[TABLE]
in the parametric case. Returning to the excess risk, this implies
[TABLE]
The cover property implies that . So we are left with
[TABLE]
Solving for the balance , leads the first two terms to be of order in the parametric case and in the nonparametric case. Thus, in the parametric case, the term dominates and the final bound is . In the nonparametric case, our assumption on the growth of implies that is of lower order.
∎
M.6 Proofs for Oracle Rates
Proof of Theorem 5.
First we invoke Lemma 17, which along with Assumption 8 implies that depending on whether , one of either global ERM or skeleton aggregation guarantees that with probability at least ,
[TABLE]
Observe that the assumption that the model is well-specified at implies that Assumption 2 is satisfied. We invoke Assumption 9 (implied by Assumption 7) and Corollary 1 to get
[TABLE]
We now invoke Lemma 19 using that the model is assumed to be well-specified, which implies that with probability at least , Skeleton Aggregation enjoys
[TABLE]
Combining these results, we get
[TABLE]
The final result follows by setting to guarantee that the first term dominates this expression.
We mention in passing that to show that global ERM achieves the desired rate for stage two when , one can appeal to the rates in Appendix L. ∎
Proof of Theorem 6.
As in the proof of Theorem 5, we invoke Lemma 17, which implies that Skeleton Aggregation666Alternatively, global ERM can be applied when . guarantees that with probability at least ,
[TABLE]
Observe that since is convex, Assumption 2 is satisfied, and we can invoke Assumption 9 (implied by Assumption 7) and Corollary 1 to get
[TABLE]
We use global ERM for the second stage. Lemma 18 implies that since the class is convex, with probability at least , global ERM guarantees
[TABLE]
This leads to a final guarantee of
[TABLE]
and the stated result follows by setting to guarantee that the first term dominates this expression.
∎
Proof of Theorem 7.
Lemma 17 implies that either Skeleton Aggregation or global ERM (for ) guarantees that with probability at least ,
[TABLE]
Theorem 2 guarantees
[TABLE]
We use global ERM for the second stage. The standard Rademacher complexity bound for ERM (e.g. Shalev-Shwartz and Ben-David (2014)), states that with probability at least , the excess risk is bounded by the Rademacher complexity of the target class composed with the loss class as follows:
[TABLE]
Using Lemma 15 and boundedness of the loss, we have
[TABLE]
Since the loss is -Lipschitz with respect to , we have . Under the assumed growth of , this gives
[TABLE]
This leads to a final guarantee of
[TABLE]
The theorem statement follows by setting so that the first term dominates.
∎
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