The importance of better models in stochastic optimization

TL;DR

This paper introduces the aProx family of models for stochastic optimization, enhancing stability, convergence, and robustness across diverse problem classes, including non-convex objectives, with theoretical guarantees and empirical validation.

Contribution

It proposes the aProx modeling framework that improves robustness and convergence in stochastic optimization, extending to weakly convex and non-convex problems with theoretical and experimental support.

Findings

aProx models improve stability and convergence

Robustness extends to non-convex and weakly convex objectives

Experimental results show reduced sensitivity and faster convergence

Abstract

Standard stochastic optimization methods are brittle, sensitive to stepsize choices and other algorithmic parameters, and they exhibit instability outside of well-behaved families of objectives. To address these challenges, we investigate models for stochastic minimization and learning problems that exhibit better robustness to problem families and algorithmic parameters. With appropriately accurate models---which we call the aProx family---stochastic methods can be made stable, provably convergent and asymptotically optimal; even modeling that the objective is nonnegative is sufficient for this stability. We extend these results beyond convexity to weakly convex objectives, which include compositions of convex losses with smooth functions common in modern machine learning applications. We highlight the importance of robustness and accurate modeling with a careful experimental evaluation of convergence time and algorithm sensitivity.

Figures

Equations

$$\begin{array}{rl}\mathrm{minimize}& F(x):={\mathbb{E}}_P[f(x;S)]={\int }_{\mathcal{S}}f(x;s)dP(s)\\ \mathrm{subjectto}& x\in \mathcal{X}.\end{array}$$

$$x\mapsto \left\{f(x;s)+\frac{\rho (s)}{2}{\Vert x\Vert }_2^2\right\}\text{mbox}isconvex.$$

$$x_{k+1}:={\mathrm{argmin}}_{x\in \mathcal{X}}\left\{f_{x_k}(x;S_k)+\frac{1}{2{\alpha }_k}{\Vert x-x_k\Vert }_2^2\right\}.$$

$$f_x(y;s)\le f(y;s)+\frac{\rho (s)}{2}{\Vert y-x\Vert }_2^2\text{mbox}forally\in \mathcal{X}.$$

$${{\partial }_y{f}_x(y;s)|}_{y=x}\subset {\partial }_{x}f(x;s).$$

$$f_x(y;s):=f(x;s)+⟨f^{\prime }(x;s),y-x⟩.$$

$$f_x(y;s):=f(y;s)+\frac{\rho (s)}{2}{\Vert y-x\Vert }_2^2.$$

$$f_x(y;s):=h(c(x;s)+\nabla {c(x;s)}^T(y-x);s).$$

$${\mathrm{minimize}}_{x\in {\mathbb{C}}^n}\frac{1}{m}\sum _{i=1}^{m}f(x;(a_i,b_i))\text{mbox}wheref(x;(a_i,b_i)):=\left||⟨a_i,x⟩{|}^2-b_i\right|.$$

$${\mathrm{minimize}}_{X\in {\mathbb{R}}^{m\times r},Y\in {\mathbb{R}}^{n\times r}}\frac{1}{|\Omega |}\sum _{(i,j)\in \Omega }f(x_i,y_j;M_{i,j})\text{mbox}wheref(x,y;z):=|⟨x,y⟩-z|$$

$$f_x(y;s):=\max \left\{f(x;s)+⟨f^{\prime }(x;s),y-x⟩,\underset{z\in \mathcal{X}}{\inf }f(z;s)\right\}.$$

$$f_x(y;s)\ge \underset{z\in \mathcal{X}}{\inf }f(z;s).$$

$$\underset{k}{\sup }\mathrm{dist}(x_k,{\mathcal{X}}^{\star })<\infty \text{mbox}withprobability1.$$

$$\mathbb{E}\left[{\Vert f^{\prime }(x;S)\Vert }_2^2\right]\le {\mathsf{G}}_{\mathrm{big}}({\Vert x\Vert }_2).$$

$$\sqrt{k}(\overline{x}_{k}-x^{\star})\mathop{\stackrel{{\scriptstyle d}}{{\rightarrow}}}\mathsf{N}\big{(}0,\nabla^{2}F(x^{\star})^{-1}\mathop{\rm Cov}(\nabla f(x^{\star};S))\nabla^{2}F(x^{\star})^{-1}\big{)}.$$

$$\mathbb{E}\left[{\Vert f^{\prime }(x;S)\Vert }_2^2\right]\le C(1+\mathrm{dist}(x,{\mathcal{X}}^{\star }{)}^p),$$

$$\underset{k\in \mathbb{N}}{\sup }\mathrm{dist}(x_k,{\mathcal{X}}^{\star })<\infty \text{with probability 1.}$$

$$\mathrm{Var}(f^{\prime }(x;S))\le C_1{\Vert F^{\prime }(x)\Vert }_2^2+C_2.$$

$$F_{\lambda}(x):=\inf_{y\in\mathcal{X}}\Big{\{}F(y)+\frac{\lambda}{2}\left\|{y-x}\right\|_{2}^{2}\Big{\}}~{}~{}\mbox{and}~{}~{}{\rm prox}_{F/\lambda}(x):=\mathop{\rm argmin}_{y\in\mathcal{X}}\Big{\{}F(y)+\frac{\lambda}{2}\left\|{y-x}\right\|_{2}^{2}\Big{\}}.$$

$$\nabla F_{\lambda }(x)=\lambda (x-x^{\lambda }),F(x^{\lambda })\le F(x),\mathrm{dist}(0,\partial F(x^{\lambda }))\le {\Vert \nabla F_{\lambda }(x)\Vert }_2.$$

$$F_{\lambda }(x_k)\stackrel{a.s.}{\to }{G}_{\lambda }$$

$$\underset{k\in \mathbb{N}}{\sup }\mathrm{dist}(x_k,{\mathcal{X}}^{\star })<\infty \text{with probability 1.}$$

$$\sum _k{\alpha }_k{\Vert \nabla F_{\lambda }(x_k)\Vert }_2^2<\infty \text{mbox}and{F}_{\lambda }(x_k)\to G_{\lambda }\text{mbox}withprobability1.$$

$$\underset{k}{\mathrm{limsup}}{k}^{1-\beta }\mathbb{E}\left[{\Vert \nabla F_{\lambda }(x_{I_k})\Vert }_2^2\mid {\mathcal{F}}_k\right]<\infty \text{mbox}withprobability1.$$

$$\underset{x\in \mathcal{X}}{\inf }f(x;s)=f(x^{\star };s).$$

$${\Vert x_{k+1}-x^{\star }\Vert }_2^2\le (1+{\alpha }_k\rho (S_k)){\Vert x_k-x^{\star }\Vert }_2^2-[f(x_k;S_k)-f(x^{\star };S_k)]\min \left\{{\alpha }_k,\frac{f(x_k;S_k)-f(x^{\star };S_k)}{{\Vert f^{\prime }(x_k;S_k)\Vert }_2^2}\right\}.$$

$$\mathbb{E}\left[\min \left\{\alpha ,\frac{f(x;S)-f(x^{\star };S)}{\Vert f^{\prime }(x;S){\Vert }_2^2}\right\}(f(x;S)-f(x^{\star };S))\right]\ge \mathrm{dist}(x,{\mathcal{X}}^{\star })\min \left\{{\lambda }_0\alpha ,{\lambda }_1\mathrm{dist}(x,{\mathcal{X}}^{\star })\right\}.$$

$$E_{1,i}(x)$$

$$E_{2,i}(x)$$

$$\underset{x\in {\mathbb{R}}^n}{\inf }\frac{1}{m}\sum _{i=1}^m{E}_i(x)\ge \frac{3}{4}-2ϵ-C\sqrt{\frac{n+t}{m}}.$$

$$\frac{1}{m}\sum _{i=1}^m\min \left\{\alpha ,\frac{f(x;(a_i,b_i))-f(x^{\star };(a_i,b_i))}{{\Vert f^{\prime }(x;(a_i,b_i))\Vert }_2^2}\right\}\left[f(x;(a_i,b_i))-f(x^{\star };(a_i,b_i))\right]$$

Full text

1 Introduction

A major challenge in stochastic optimization—the algorithmic workhorse for much of modern statistical and machine learning applications—is in setting algorithm parameters (or hyper-parameter tuning). The challenge arises because most algorithms are sensitive to their parameters, and different applications require different tuning. This sensitivity causes multiple issues. It results in thousands to millions of wasted engineer and computational hours. It also leads to a lack of clarity in research and development of algorithms—in claiming that one algorithm is better than another, it is unclear if this is due to judicious choice of dataset, judicious parameter settings, or if indeed the algorithm does exhibit new desirable behavior. Consequently, in this paper we have two main thrusts: first, by using better models than naive first-order models common in stochastic gradient methods, we develop families of stochastic optimization algorithms that are provably more robust to input parameter choices, with several corresponding optimality properties. Second, we argue for a different type of experimental evidence in evaluating stochastic optimization methods, where one jointly evaluates convergence speed and sensitivity of the methods.

The wasted computational and engineering energy is especially pronounced in deep learning, where engineers use models with millions of parameters, requiring a days to weeks to train single models. To get a sense of this energy use, we consider a few recent papers that we view as exemplars of this broader trend: in searching for optimal neural network architectures and hyperparameters, the papers [37, 30, 36] used approximately 2000, 3150, 22000 GPU-days of computation, respectively. The paper [7] uses approximately 750000 CPU days in its parameter search. To put this in perspective, assuming standard CPU energy use of between 60-100 Watts, the energy (ignoring network interconnect, monitors, etc.) for the paper [7] is roughly between $4$ and $6\cdot 10^{12}$ Joules. At $10^{9}$ Joules per tank of gas, this is sufficient to drive 4000 Toyota Camrys the $380$ miles between San Francisco and Los Angeles.

To address these challenges, we develop stochastic optimization procedures that exhibit similar convergence to classical approaches—when the classical approaches are provided good tuning parameters—but they enjoy better robustness and achieve this performance over a range of parameters. We additionally argue for evaluation of optimization algorithms based not only on convergence time, but also on robustness to input choices. Briefly, a fast algorithm that converges for a small range of stepsizes is too brittle; we argue instead for (potentially slightly slower) algorithms that converge for broad ranges of stepsizes and other parameters. Our theory and experiments demonstrate the effectiveness of our methods for many applications, including phase retrieval, matrix completion, and deep learning.

1.1 Problem setting and approach

We begin by making our setting concrete. We study the stochastic optimization problem

[TABLE]

In problem (1), the set $\mathcal{S}$ is a sample space, $\mathcal{X}\subset\mathbb{R}^{n}$ is a closed convex, and $f(x;s)$ is the instantaneous loss parameter $x$ suffers on sample $s$. In this paper, we move beyond convex optimization by considering $\rho(s)$-weakly convex functions $f$, meaning [cf. 33, 12] that

[TABLE]

We recover the convex case when $\rho(s)\leq 0$. Examples in this framework include linear regression, where $f(x;(a,b))=(\langle a,x\rangle-b)^{2}$, robust phase retrieval [34, 14] where $f(x;(a,b))=|\langle a,x\rangle^{2}-b|$, which is $2\left\|{a}\right\|_{2}^{2}$-weakly convex, or bilinear prediction, $f(x,y;b)=|\langle x,y\rangle-b|$, which is $1$-weakly convex.

Most optimization methods iterate by making an approximation—a model—of the objective near the current iterate, then minimizing this model and re-approximating. Stochastic (sub)gradient methods [31, 27] instantiate this approach using a linear approximation; following initial work of our own and others [2, 15, 8], we study the modeling approach in more depth for stochastic optimization. Thus, the aProx algorithms we develop [2] iterate as follows: for $k=1,2,\ldots$, we draw a random $S_{k}\sim P$, then update the iterate $x_{k}$ by minimizing a regularized approximation to $f(\cdot;S_{k})$, setting

[TABLE]

We call $f_{x}(\cdot;s)$ the model of $f$ at $x$, where $f_{x}$ satisfies three conditions [cf. 2, 15, 8]:

1. (C.i)

[Model convexity] The function $y\mapsto f_{x}(y;s)$ is convex and subdifferentiable. 2. (C.ii)

[Weak lower bound] The model $f_{x}$ satisfies

[TABLE] 3. (C.iii)

[Local accuracy] We have $f_{x}(x;s)=f(x;s)$ and the containment

[TABLE]

We provide examples in Section 2.

We show that by using just slightly more accurate models than the first order model used by the stochastic gradient method—sometimes as simple as recognizing that if the function loss $f$ is non-negative, we should truncate our approximation at zero—we can achieve substantially better theoretical guarantees and practical performance. While the iterates of gradient methods can (super-exponentially) diverge as soon as we have mis-specified stepsizes, our methods guarantee that the iterates never diverge. Even more, this stability guarantees convergence of the methods, and in convex cases, optimal asymptotic normality of the averaged iterates. Finally, we evaluate the performance of our methods, reaffirming our theoretical findings on convergence and robustness for a range of problems, including matrix completion, phase retrieval, and classification with neural networks. We defer all proofs to the appendices.

Notation and basic assumptions

For a weakly convex function $f$, we let $\partial f(x)$ denote its Fréchet subdifferential at the point $x$, and $f^{\prime}(x)\in\partial f(x)$ denotes an arbitrary element of the subdifferential. Throughout, we let $x^{\star}$ denote a minimizer of problem (1) and $\mathcal{X}^{\star}=\mathop{\rm argmin}_{x\in\mathcal{X}}F(x)$ denote the optimal set for problem (1). We let $\mathcal{F}_{k}:=\sigma(S_{1},\ldots,S_{k})$ denote the $\sigma$-field generated by the first $k$ random variables $S_{i}$. Note that $x_{k}\in\mathcal{F}_{k-1}$ for all $k$. Unless stated otherwise, we assume that the function $f(x;s)$ is $\rho(s)$-weakly convex for each $s\in\mathcal{S}$. Finally, the following assumption will implicity hold throughout the paper.

Assumption A1**.**

The set $\mathcal{X}^{\star}:=\mathop{\rm argmin}_{x\in\mathcal{X}}\{F(x)\}$ is non-empty, and there exists $\sigma^{2}<\infty$ such that for each $x^{\star}\in\mathcal{X}^{\star}$ and selection $f^{\prime}(x^{\star};s)\in\partial f(x^{\star};s)$, we have $\mathbb{E}[\left\|{f^{\prime}(x^{\star};S)}\right\|_{2}^{2}]\leq\sigma^{2}$.

2 Methods

To make our approach a bit more concrete, we identify several models that fit into our framework. These have appeared [15, 8, 2], but we believe a self-contained presentation beneficial. Each of these models satisfies our conditions (C.i)–(C.iii). The most widely used model in stochastic optimization, is the simple first-order model:

Stochastic subgradient methods:

The stochastic subgradient method uses the model

[TABLE]

Proximal point methods:

In the convex setting [4, 28, 2], the stochastic proximal point method uses the model $f_{x}(y;s):=f(y;s)$; in the weakly convex setting, we regularize and use

[TABLE]

We now turn to models that require less knowledge than proximal model (4), but preserve important structural properties in the original function.

Prox-linear model:

Let the function $f$ have the composite structure $f(x;s)=h(c(x;s);s)$ where $h(\cdot;s)$ is convex and $c(\cdot;s)$ is smooth. The stochastic prox-linear method applies $h$ to a first-order approximation of $c$, using

[TABLE]

In the non-stochastic setting, these models are classical [18, 19], while in stochastic setting, recent work establishes convergence and convergence rates in restrictive settings [15, 8]. See Figure 1 for illustration of this model. When $h$ is $L_{h}$-Lipschitz and $c$ has $L_{c}$-Lipschitz gradient, then $f$ is $\rho=L_{h}\cdot L_{c}$-weakly convex.

Examples help highlight the applicability of this composite structure:

Example 1 (Phase Retrieval): 

In phase retrieval [34], we wish to recover an object $x^{\star}\in\mathbb{C}^{n}$ from a diffraction pattern $Ax^{\star}$, where $A\in\mathbb{C}^{m\times n}$, but physical sensor limitations mean we only observe the amplitudes $b=|Ax^{\star}|^{2}$. A natural objective is thus

[TABLE]

This is evidently the composition of $h(z)=|z|$ and $c(x;(a_{i},b_{i}))=|\langle a_{i},x\rangle|^{2}-b_{i}$. Moreover, a calculation [14] shows that $f(\cdot;(a_{i},b_{i}))$ is $2\left\|{a_{i}}\right\|_{2}^{2}$-weakly convex. $\Diamond$

Example 2 (Matrix Completion): 

In the matrix completion problem [5], which arises (for example) in the design of recommendation systems, we have a matrix $M\in\mathbb{R}^{m\times n}$ with decomposition $M=X_{\star}Y_{\star}^{T}$ for $X_{\star}\in\mathbb{R}^{m\times r}$ and $Y_{\star}\in\mathbb{R}^{n\times r}$. Based on the incomplete set of known entries $\Omega\subset[m]\times[n]$, our goal is to recover the matrix $M$, giving rise to the objective

[TABLE]

and $X_{i}$ and $Y_{j}$ are the $i$th and $j$th rows of $X$ and $Y$. This is the composition of $h(z)=|z|$ and $c(x,y,z)=\langle x,y\rangle-z$, so that $f=h\circ c$ is $1$-weakly convex. $\Diamond$

Truncated models:

The prox-linear model (5) may be challenging to implement for complex compositions (e.g., deep learning), and it requires a common but potentially restrictive structure. If instead we know a lower bound on $f$, we may incorporate this to model

[TABLE]

In most of our examples—linear and logistic regression, phase retrieval, matrix completion (and more generally, all typical loss functions in machine learning)—we have $\inf_{z}f(z;s)=0$. The assumption that we have a lower bound is thus rarely restrictive. This model satisfies the conditions (C.i)–(C.iii), also satisfying the additional condition

1. (C.iv)

[Lower optimality] For all $s\in\mathcal{S}$, the models $f_{x}(\cdot;s)$ satisfy

[TABLE]

As we show, Condition (C.iv) is sufficient to derive several optimality and stability properties.

3 Stability and its consequences

In our initial study of stability in optimization [2], we defined an algorithm as stable if its iterates remain bounded, then showed several consequences of this in convex optimization (which we review presently). A weakness, however, was that the only algorithms whose stability we were able to demonstrate were very close approximations to stochastic proximal point methods. Here, we develop two important extensions. First, we show that any model satisfying Condition (C.iv) has stable iterates under mild assumptions, in strong contrast to models (e.g. linear) that fail the condition. Second, we develop an analogous stability theory for weakly convex functions, proving that accurate enough models are stable. In parallel to the convex case, stability suffices for more: it implies convergence (with an asymptotic rate) to stationary points for any model-based method on weakly convex functions.

Let us formalize stability. A pair $(\mathcal{F},\mathcal{P})$ a collection of problems if $\mathcal{P}$ consists of probability measures on a sample space $\mathcal{S}$ and $\mathcal{F}$ of functions $f:\mathcal{X}\times\mathcal{S}\to\mathbb{R}$.

Definition 3.1**.**

An algorithm generating iterates $x_{k}$ according to the model-based update (2) is stable in probability for the problems $(\mathcal{F},\mathcal{P})$ if for all $f\in\mathcal{F}$, square-summable positive stepsize sequences $\{\alpha_{k}\}$, and $P\in\mathcal{P}$ defining $F(x)=\mathbb{E}_{P}[f(x;S)]$ and $\mathcal{X}^{\star}=\mathop{\rm argmin}_{x\in\mathcal{X}}F(x)$,

[TABLE]

Standard models, such as the linear model (3) and consequent subgradient method, are unstable [2, Sec. 3]. They may even cause super-exponential divergence:

Example 3 (Divergence): 

Let $F(x)=e^{x}+e^{-x}$, $p<\infty$, and $\alpha_{0}>0$, and let $\alpha_{k}$ be any sequence satisfying $\alpha_{k}\geq\alpha_{0}k^{-p}$. Let $x_{k+1}=x_{k}-\alpha_{k}F^{\prime}(x_{k})=x_{k}-\alpha_{k}(e^{x_{k}}-e^{-x_{k}})$ be generated by the gradient method. Whenever $x_{1}$ is large enough, then $\log\frac{x_{k+1}}{x_{k}}\geq 2^{k}$ for all $k$. $\Diamond$

3.1 The importance of stability in stochastic convex optimization

To set the stage for what follows, we begin by motivating the importance of stable procedures. Briefly, any stable aProx model converges for any convex function under weak assumptions, which we now elucidate. First, we make an

Assumption A2**.**

There exists an increasing function $\mathsf{G}_{\rm big}:\mathbb{R}_{+}\to\left[{0},{\infty}\right)$ such that for all $x\in\mathcal{X}$ and each selection $f^{\prime}(x;s)\in\partial f(x;s)$,

[TABLE]

Assumption A2 is relatively weak, and is equivalent to assuming that $\mathbb{E}[\|{f^{\prime}(x;S)}\|_{2}^{2}]$ is bounded on compacta; it allows arbitrary growth—exponential, super-exponential—in the norm of the subgradients, just requiring that the second moment exists.

Corollary 3.1** (Asi and Duchi [2], Prop. 1).**

Assume that $f(\cdot;s)$ is convex for each $s\in\mathcal{S}$ and let Assumption A2 hold. Let the iterates $x_{k}$ be generated by any method satisfying Conditions (C.i)–(C.iii), and additionally assume that with probability 1, $\sup_{k}\left\|{x_{k}}\right\|<\infty$. Then $\sum_{k}\alpha_{k}(F(x_{k})-F(x^{\star}))<\infty$. If in addition $\inf_{x\in\mathcal{X}}\left\{F(x)-F(x^{\star})\mid\mathop{\rm dist}(x,\mathcal{X}^{\star})\geq\epsilon\right\}>0$ for all $\epsilon>0$, then $\mathop{\rm dist}(x_{k},\mathcal{X}^{\star})\mathop{\stackrel{{\scriptstyle a.s.}}{{\rightarrow}}}0$.

Corollary 3.1 establishes convergence of stable procedures, and also (via Jensen’s inequality) provides asymptotic rates of convergence for weighted averages $\sum_{k}\alpha_{k}x_{k}/\sum_{k}\alpha_{k}$.

Additionally, when the functions $f$ are smooth, any aProx method achieves asymptotically optimal convergence whenever the iterates remain bounded. In particular, let us assume that $F$ is $\mathcal{C}^{2}$ near $x^{\star}=\mathop{\rm argmin}_{\mathcal{X}}F(x)$ with $\nabla^{2}F(x^{\star})\succ 0$, and that on some neighborhood of $x^{\star}$ the functions $f(\cdot;s)$ have $L(s)$-Lipschitz gradient with $\mathbb{E}[L(S)^{2}]<\infty$. We have

Corollary 3.2** (Asi and Duchi [2], Theorem 2).**

In addition to the conditions of Corollary 3.1, let the conditions of the previous paragraph hold. Then $\overline{x}_{k}=\frac{1}{k}\sum_{i=1}^{k}x_{i}$ satisfies

[TABLE]

This convergence is optimal for any method given samples $S_{1},\ldots,S_{k}\stackrel{{\scriptstyle\rm iid}}{{\sim}}P$ (see [16]), and Corollary 3.2 highlights the importance of stability: if any aProx method is stable, it enjoys (asymptotically) optimal convergence.

3.2 Stability of lower-bounded models for convex functions

With these consequences of stability in hand—convergence and asymptotic optimality—it behooves us to provide conditions sufficient to guarantee stability. To that end, we show that lower bounded models satisfying Condition (C.iv) are stable in probability (Def. 3.1) for functions whose (sub)gradients grow at most polynomially. We begin by stating our polynomial growth assumption.

Assumption A3**.**

There exist $C,p<\infty$ such that for every $x\in{\mathcal{X}}$,

[TABLE]

and $\mathbb{E}[(f(x^{\star};S)-\inf_{z\in\mathcal{X}}f(z;S))^{p/2}]\leq C$ for all $x^{\star}\in\mathcal{X}^{\star}$.

The analogous condition [29] for stochastic gradient methods holds for $p=2$, or quadratic growth, without which the method may diverge. In contrast, Assumption A3 allows polynomial growth; for example, the function $f(x)=x^{4}$ is permissible, while the gradient method may exponentially diverge even for stepsizes $\alpha_{k}=1/k$. The key consequence of Assumption A3 is that if it holds, truncated models are stable:

Theorem 1**.**

Assume the function $f(\cdot;s)$ is convex for each $s\in\mathcal{S}$. Let Assumption A3 hold and $\alpha_{k}=\alpha_{0}k^{-\beta}$ with $\frac{p+2}{p+4}<\beta<1$. Let $x_{k}$ be generated by the iteration (2) with a model satisfying Conditions (C.i)–(C.iv). Then

[TABLE]

Theorem 1 shows that truncated methods enjoy all the benefits of stability we outline in Corollaries 3.1 and 3.2 above. Thus, these models, whose updates are typically as cheap to compute as a stochastic gradient step (especially in the common case that $\inf_{z}f(z;s)=0$) provide substantial advantage over methods using only (sub)gradient approximations.

3.3 Stability and its consequences for weakly convex functions

We continue our argument that—if possible—it is beneficial to use more accurate models, even in situations beyond convexity, investigating the stability of proximal models (4) for weakly convex functions. Establishing stability in the weakly convex case requires a different approach to the convex case, as the iterates may not make progress toward a fixed optimal set. In this case, to show stability, we require an assumption bounding the size of $f^{\prime}(x;S)$ relative to the population subgradient $F^{\prime}$.

Assumption A4**.**

There exist $C_{1},C_{2}<\infty$ such that for all measurable selections $f^{\prime}(x;s)\in\partial f(x;s)$ and $F^{\prime}(x)\in\partial F(x)$,

[TABLE]

By providing a “relative” noise condition on $f^{\prime}$, Assumption A4 allows for a broader class of functions without global Lipschitz properties (as are typically assumed [8]), such as the phase retrieval and matrix completion objectives (Examples 2 and 2). It can allow exponential growth, addressing the challenges in Ex. 3.1. For example, let $f(x;1)=e^{x}$ and $f(x;2)=e^{-x}$, where $S$ is uniform in $\{1,2\}$ so that $F(x)=\frac{1}{2}(e^{x}+e^{-x})$; then $\mathbb{E}[f^{\prime}(x;S)^{2}]=2F^{\prime}(x)^{2}+\frac{1}{2}$.

To describe convergence and stability guarantees in non-convex (even non-smooth) settings, we require appropriate definitions. Finding global minima of non-convex functions is computationally infeasible [26], so we follow established practice and consider convergence to stationary points, specifically using the convergence of the Moreau envelope [8, 13]. To formalize, for $x\in\mathbb{R}^{n}$ and $\lambda\geq 0$, the Moreau envelope and associated proximal map are

[TABLE]

For large enough $\lambda$, the minimizer $x^{\lambda}:={\rm prox}_{F/\lambda}(x)$ is unique whenever $F$ is weakly convex. Adopting the techniques pioneered by Davis and Drusvyatskiy [8] for convergence of stochastic methods on weakly convex problems, our convergence machinery relies on the Moreau envelope’s connections to (near) stationarity:

[TABLE]

The three properties (8) imply that any nearly stationary point $x$ of $F_{\lambda}(x)$—when $\|{\nabla F_{\lambda}(x)}\|_{2}$ is small—is close to a nearly stationary point $x^{\lambda}$ of the original function $F(\cdot)$. To prove convergence for weakly convex methods, then, it is sufficient to show that $\nabla F_{\lambda}(x_{k})\to 0$.

Using full proximal models, it turns out, guarantees convergence.

Theorem 2**.**

Let Assumption A4 hold and $\lambda$ be large enough that $\mathbb{E}[\rho(S)]<\lambda$, and assume that $\inf_{x\in\mathcal{X}}F(x)>-\infty$ and $\mathbb{E}[\rho(S)^{2}]<\infty$. Let $x_{k}$ be generated by the iteration (2) with the proximal model (4). Then there exists a random variable $G_{\lambda}<\infty$ such that

[TABLE]

and $\sum_{k=1}^{\infty}\alpha_{k}\left\|{\nabla F_{\lambda}(x_{k})}\right\|_{2}^{2}<\infty$ with probability one.

The theorem shows that $F_{\lambda}(x_{k})$ is bounded almost surely. Thus, if the function $F(\cdot)$ is coercive, meaning $F(x)\uparrow\infty$ as $\left\|{x}\right\|\to\infty$, then the Moreau envelope $F_{\lambda}(\cdot)$ is also coercive, so that we have the following corollary.

Corollary 3.3**.**

Let the conditions of Theorem 2 hold and let $F$ be coercive. Then

[TABLE]

In parallel with our devlopment of the convex case, stability is sufficient to develop convergence results for any model-based aProx method, highlighting its importance. Indeed, we can show that stable methods guarantee various types of convergence to stationary points, though for probability one convergence of the iterates, we require a slightly elaborate assumption [cf. 15, 10], which rules out some pathological limiting cases.

Assumption A5** (Weak Sard).**

Let $\mathcal{X}_{\rm stat}=\{x\mid 0\in\partial F(x)\}$ be the collection of stationary points of $F$ over $\mathcal{X}$. The Lebesgue measure of the image $F(\mathcal{X}_{\rm stat})$ is zero.

Under this assumption, aProx methods converge to stationary points whenever the iterates are stable.

Proposition 1**.**

Let Assumption A2 hold and the iterates $x_{k}$ be generated by any method satisfying Conditions (C.i)–(C.iii). Assume that $\lambda$ is large enough that $\mathbb{E}[\rho(S)]<\lambda$. There exists a finite random variable $G_{\lambda}$ such that on the event that $\sup_{k}\left\|{x_{k}}\right\|_{2}<\infty$, we have

[TABLE]

If additionally Assumption A5 holds, then $\left\|{\nabla F_{\lambda}(x_{k})}\right\|_{2}\mathop{\stackrel{{\scriptstyle a.s.}}{{\rightarrow}}}0$ and $\mathop{\rm dist}(x_{k},\mathcal{X}_{\rm stat})\mathop{\stackrel{{\scriptstyle a.s.}}{{\rightarrow}}}0$.

In passing, we note that the finite sum condition (9) is enough to develop a type of conditional $\ell_{2}$-convergence, which is similar to the non-asymptotic rates of convergence that stochastic (sub)gradient methods achieve to stationary points [8, 20]. Indeed, assume $\alpha_{k}=\alpha_{0}k^{-\beta}$ for some $\beta\in(\frac{1}{2},1)$ and that the iterates $x_{k}$ are stable. Now let $I_{k}$ be an index chosen from $\{1,\ldots,k\}$ with probabilities $P(I_{k}=i)=\alpha_{i}/\sum_{j=1}^{k}\alpha_{j}$. Then inequality (9) shows that

[TABLE]

This provides an asymptotic analogue of the convergence rates that stochastic model-based methods achieve on Lipschitzian functions [8].

4 Fast convergence for easy problems

In many engineering and learning applications, solutions interpolate the data. Consider, for example, signal recovery problems with $b=Ax^{\star}$, or modern machine learning applications, where frequently training error is zero [25, 3]. We consider such problems here, showing how models that satisfy the lower bound condition (C.iv) enjoy linear convergence, extending our earlier results [2] beyond convex optimization. We begin with a

Definition 4.1**.**

Let $F(x):=\mathbb{E}_{P}[f(x;S)]$. Then $F$ is easy to optimize if for each $x^{\star}\in\mathcal{X}^{\star}:=\mathop{\rm argmin}_{x\in\mathcal{X}}F(x)$ and $P$-almost all $s\in\mathcal{S}$ we have

[TABLE]

For such problems, we can guarantee progress toward minimizers as long as $f$ grows quickly enough away from $x^{\star}$, as the following lemma (generalizing our result [2, Lemma 4.1]) shows.

Lemma 4.1**.**

Let $F$ be easy to optimize (Definition 4.1). Let $x_{k}$ be generated by the updates (2) using a model satisfying Conditions (C.i)–(C.iv). Then for any $x^{\star}\in\mathcal{X}^{\star}$,

[TABLE]

Lemma 4.1 allows us to prove fast convergence so long as $f$ grows quickly enough away from $x^{\star}$; a sufficient condition for us is a so-called sharp growth condition away from the optimal set $\mathcal{X}^{\star}$. To meld with the progress guarantee in Lemma 4.1, we consider the following assumption.

Assumption A6** (Expected sharp growth).**

There exist constants $\lambda_{0},\lambda_{1}>0$ such that for all $\alpha\in\mathbb{R}_{+}$ and $x\in\mathcal{X}$ and $x^{\star}\in\mathcal{X}^{\star}$,

[TABLE]

Assumption A6 is perhaps too-tailored to Lemma 4.1 for obvious use, so we discuss a few situations where it holds. One simple sufficient condition is the small-ball-type condition that there exists $C<\infty$ such that $\mathbb{P}(f(x;S)-f(x^{\star};S)\geq\epsilon\mathop{\rm dist}(x,\mathcal{X}^{\star}))\geq 1-C\epsilon$ for $\epsilon>0$ and that $\mathbb{E}[\left\|{f^{\prime}(x;S)}\right\|_{2}^{2}]\leq C(1+\mathop{\rm dist}(x,\mathcal{X}^{\star})^{2})$. We can also be more explicit:

Example 4 (Phase retrieval, example 2 continued): 

Consider the (real-valued) phase retrieval problem with objective $f(x;(a,b))=|\langle a,x\rangle^{2}-b|$. Let us assume the vectors $a_{i}\in\mathbb{R}^{n}$ are drawn from a distribution satisfying the small ball condition $P(|\langle a_{i},u\rangle|\geq\epsilon\left\|{u}\right\|_{2})\geq 1-\epsilon$ for $\epsilon>0$ and any $u\in\mathbb{R}^{n}$, and additionally that $\mathbb{E}[\left\|{a_{i}}\right\|_{2}^{2}]\leq M^{2}n$ and $\mathbb{E}[\langle a_{i},x\rangle^{2}]\leq M^{2}\left\|{x}\right\|_{2}^{2}$ for some $M<\infty$. For each $i$, define the events

[TABLE]

The small ball condition implies $P(E_{1,i}(x))\geq 1-2\epsilon$, and Markov’s inequality gives $P(E_{2,i}(x))\geq\frac{7}{8}$ and $P(E_{3,i}(x))\geq\frac{7}{8}$ for all $x\in\mathbb{R}^{n}$. Letting $E_{i}(x)=1$ if $E_{1,i}(x),E_{2,i}(x),E_{3,i}(x)$ each occur and $E_{i}(x)=0$ otherwise, a VC-dimension calculation (see, for example, [14, Appendix A]) thus implies that with probability at least $1-e^{-t}$, we have

[TABLE]

Let $\epsilon=\frac{1}{8}$ for simplicity. Then evidently with high probability over the draw of a random data matrix $A\in\mathbb{R}^{m\times n}$ and $b=|Ax^{\star}|^{2}$, the rows $a_{i}$ satisfy condition (10), so we have

[TABLE]

for a numerical constant $c>0$. On this finite sample of size $m$, Assumption A3 holds for the empirical objective $F(x)=\frac{1}{m}\sum_{i=1}^{m}f(x;(a_{i},b_{i}))$ with $\lambda_{0}=c\left\|{x^{\star}}\right\|_{2}$, and $\lambda_{1}=\frac{c}{16M^{4}n}$. $\Diamond$

The following proposition is our main result in this section, showing that lower bounded models enjoy linear convergence on easy problems.

Proposition 2**.**

Let Assumption A6 hold and $x_{k}$ be generated by the stochastic iteration (2) using any model satisfying Conditions (C.i)–(C.iv), where the stepsizes $\alpha_{k}$ satisfy $\alpha_{k}=\alpha_{0}k^{-\beta}$ for some $\beta\in(0,1)$. If $f(\cdot;S_{k})$ is $\rho(S_{k})$-weakly convex with $\mathbb{E}[\rho(S_{k})]=\overline{\rho}$, then for any $m\in\mathbb{N}$ and $\epsilon>0$, there exists a finite random variable $V_{\infty,m}<\infty$ such that

[TABLE]

When the functions $f$ are convex, we have $\overline{\rho}=0$, so that Proposition 2 guarantees linear convergence for easy problems. In the case that $\overline{\rho}>0$, the result is conditional: if an aProx method converges to one of the sharp minimizers of $f$, then this convergence is linear (i.e. geometrically fast). In the case of phase retrieval, we can guarantee convergence:

Example (Phase retrieval, Example A6 continued): 

With $F(x)=\frac{1}{m}\left\|{|Ax|^{2}-|Ax^{\star}|^{2}}\right\|_{1}$, the conditions in Example A6 guarantee that with high probability $F(x)\geq\lambda\left\|{x-x^{\star}}\right\|_{2}\left\|{x+x^{\star}}\right\|_{2}$ for a numerical constant $\lambda>0$ and that $F$ is $\rho$-weakly convex for $\rho$ a numerical constant [14]. In this case, an integration argument [9] guarantees that

[TABLE]

If $\left\|{x-x^{\star}}\right\|_{2}\leq\left\|{x^{\star}}\right\|_{2}$, then $\left\|{x+x^{\star}}\right\|_{2}\geq\left\|{x^{\star}}\right\|_{2}$, so $\langle F^{\prime}(x),x-x^{\star}\rangle>0$ whenever $\left\|{x-x^{\star}}\right\|_{2}<\frac{2\lambda}{\rho}\left\|{x^{\star}}\right\|_{2}$. Thus $F$ has no stationary points in the set $\{x:\left\|{x-x^{\star}}\right\|_{2}<\frac{2\lambda}{\rho}\left\|{x^{\star}}\right\|_{2}\}$. This set is unavailable; however, it is possible to use a spectral algorithm to construct an initializer $x_{0}$ satisfying $\left\|{x_{0}-x^{\star}}\right\|_{2}<\frac{\lambda}{\rho}\left\|{x^{\star}}\right\|_{2}$ and estimate $\left\|{x^{\star}}\right\|_{2}$ to high accuracy as soon as the number of measurements $m\gtrsim n$ [14]. By defining $\mathcal{X}:=\{x:\left\|{x-x_{0}}\right\|_{2}\leq\frac{\lambda}{\rho}\left\|{x^{\star}}\right\|_{2}\}$, we then guarantee (via Proposition 1) that the iterates $x_{k}$ converge to $x^{\star}$, and Proposition 2 implies that for a numerical constant $c>0$, there is a (random) $B<\infty$ such that $\left\|{x_{k}-x^{\star}}\right\|_{2}^{2}\leq B\cdot(1-c/n)^{k}$ eventually. To achieve an $\epsilon$-accurate solution to the (robust) phase retrieval objective, the truncated stochastic model (6) requires $O(n)$ operations per iteration and at most $O(n\log\frac{1}{\epsilon})$ steps. Combined with the spectral initialization, which requires time $O(mn\log\frac{1}{\epsilon})$, the overall computation time is at most $O(mn\log\frac{1}{\epsilon})$, which is the best known for phase retrieval. $\Diamond$

5 Experiments

An important question in the development of any optimization method is its sensitivity to algorithm parameters. Consequently, we conclude by experimentally examining convergence time and robustness of each of our optimization methods. We consider each of the models we investigate in the paper: the stochastic gradient method (i.e. the linear model (3)), the proximal model (4), the prox-linear model (5), and the (lower) truncated model (6).

We test both convergence time and, dovetailing with our focus in this paper, robustness to stepsize specification for several problems: phase retrieval (Section 5.1), matrix completion (Section 5.2), and two classification problems using deep learning (Section 5.3). We consider stepsize sequences of the form $\alpha_{k}=\alpha_{0}k^{-\beta}$, where $\beta\in(1/2,1)$, and perform $K$ iterations over a wide range of different initial stepsizes $\alpha_{0}$. (For brevity, we present results only for the stepsize $\beta=.6$; our other experiments with varied $\beta$ were similar.) For a fixed accuracy $\epsilon>0$, we record the number of steps $k$ required to achieve $F(x_{k})-F(x^{\star})\leq\epsilon$, reporting these times (where we terminate each run at iteration $K$). We perform $T$ experiments for each initial stepsize choice, reporting the median time to $\epsilon$-accuracy and 90% confidence intervals.

5.1 Phase Retrieval

We start our experiments with the phase retrieval problem in Examples 2 and A6, focusing on the real case for simplicity, where we are given $A\in\mathbb{R}^{m\times n}$ with rows $a_{i}\in\mathbb{R}^{n}$ and $b=(Ax^{\star})^{2}\in\mathbb{R}_{+}^{m}$ for some $x^{\star}\in\mathbb{R}^{n}$. Our objective is the non-convex and non-smooth function

[TABLE]

In our experiments, we sample the entries of the vectors $a_{i}$ and $x^{\star}$ i.i.d. $\mathsf{N}(0,I_{n})$.

We present the results of this experiment in Figure 2, comparing the stochastic gradient method (3), proximal method (4), and truncated method (6) (which yields updates identical to the prox-linear model (5) in this case). The plots demonstrate the expected result that the stochastic gradient method has good performance in a narrow range of stepsizes, $\alpha_{1}\approx 10$ in this case, while better approximations for aProx yield better convergence over a large range of stepsizes. The truncated model (6) exhibits some oscillation for large stepsizes, in contrast to the exact model (4), which is robust to all stepsizes $\alpha_{0}\geq 10$.

5.2 Matrix Completion

For our second experiment, we investigate the performance of aProx procedures for the matrix completion problem of Example 2. In this setting, we are given a matrix $MX_{\star}Y_{\star}^{T}$, for $X_{\star}\in\mathbb{R}^{m\times r}$ and $Y_{\star}\in\mathbb{R}^{n\times r}$. We are also given a set of indices $\Omega\subset[m]\times[n]$ where the value of matrix $M$ is known. We aim to recover $M$ while our access to $M$ is restricted only to indices in the set $\Omega$, so our goal is to minimize

[TABLE]

Here the matrices $X\in\mathbb{R}^{m\times\hat{r}}$ and $Y\in\mathbb{R}^{n\times\hat{r}}$, where the estimated rank $\hat{r}\geq r$. We generate the data by drawing the entries of $X_{\star}$ and $Y_{\star}$ i.i.d. $\mathsf{N}(0,1)$ and choosing $\Omega$ uniformly at random of size $|\Omega|=5(nr+mr)$. We present the timing results in Figure 3, which tells a similar story to Figure 2: better approximations, such as the truncated models (which again yield identical updates to the prox-linear models (5)), are significantly more robust to stepsize specification. In this case, the full proximal update requires solving a nontrivial quartic, so we omit it.

5.3 Neural Networks

As one of our main motivations is to address the extraordinary effort—in computational and engineering hours—carefully tuning optimization methods, we would be remiss to avoid experiments on deep neural networks. Therefore, in our last set of experiments, we test the performance of our models for training neural networks for classification tasks over the CIFAR10 dataset [24] and the fine-grained 128-class Stanford dog multiclass recognition task [23]. For our CIFAR10 experiment, we use the Resnet18 architecture [21]; we replace the Rectified Linear Unit (RELU) activations internal to the architecture with Exponentiated Linear Units (ELUs [6]) so that the loss is of composite form $f=h\circ c$ for $h$ convex and $c$ smooth. For Stanford dogs we use the VGG16 architecture [35] pretrained on Imagenet [11], again substituting ELUs for RELU activations. For this experiment, we also test a modified version of the truncated method, TruncAdaGrad, that uses the truncated model in iteration (2) but uses a diagonally scaled Euclidean distance [17], updating at iteration $k$ by

[TABLE]

where $H_{k}=\mathop{\rm diag}(\sum_{i=1}^{k}g_{i}g_{i}^{T})^{1/2}$ for $g_{i}=f^{\prime}(x_{i};S_{i})$. This update requires no more out of standard deep learning software than computing a gradient (backpropagation) and loss. We also compare our optimization methods to Adam, the default optimizer in TensorFlow [1].

Figures 4 and 5 show our results for the CIFAR10 and Stanford dogs datasets, respectively. Plot (a) of each figure gives the number of iterations required to achieve $\epsilon$ test-classification error (on the highest or “top-1” predicted class), while plot (b) shows the maximal accuracy each procedure achieves for a given initial stepsize $\alpha_{0}$. The plots demonstrate the sensitivity of the standard stochastic gradient method to stepsize choice, which converges only for a small range of stepsizes, in both experiments. Adam exhibits better robustness for CIFAR10, while it is extremely sensitive in the second experiment (Fig. 5), converging only for a small range of stepsizes—this difference in sensitivities highlights the importance of robustness. In contrast, our procedures using the truncated model are apparently robust for all large enough stepsizes. Plot (b) of the figures shows additionally that the maximal accuracy the two truncated methods achieve changes only slightly for $\alpha_{0}\geq 10^{-1}$, again in strong contrast to the other methods, which achieve their best accuracy only for a small range of stepsizes.

These results reaffirm the insights from our theoretical results and experiments: it is important and possible to develop methods that enjoy good convergence guarantees and are robust to algorithm parameters.

Appendix A Technical lemmas

Before beginning our technical appendices, we collect a few results that will be useful for our derivations to come. The first three provide a guarantee on the progress and convergence that single iterates of the updates (2) make.

Lemma A.1** (Asi and Duchi [2], Lemma 3.2).**

Let $g$ be convex and subdifferentiable on a closed convex set $\mathcal{X}$ and let $\beta>0$. Then for all $x_{0},x_{1},y\in\mathcal{X}$, and $g^{\prime}(y)\in\partial g(y)$,

[TABLE]

Lemma A.2** (Asi and Duchi [2], Lemma 3.3).**

Let Condition (C.i) hold. In each step of the method (2), for any $x\in\mathcal{X}$,

[TABLE]

Lemma A.3**.**

Let Conditions (C.i)–(C.iii) hold and let $x_{k}$ be generated by the updates (2). Then for any $x\in\mathcal{X}$,

[TABLE]

Proof    Let $\alpha>0$ and $x_{0},x_{1},x\in\mathcal{X}$ and $s\in\mathcal{S}$ be otherwise arbitrary. Then for any $g\in\partial f(x_{0};s)$, we have

[TABLE]

where inequality $(i)$ used the convexity of $f_{x_{0}}$ and inequality $(ii)$ used Condtition (C.iii) to obtain $f_{x_{0}}(x_{0};s)=f(x_{0};s)$ and Condition (C.ii) on $f_{x_{0}}(x;s)$. Making the obvious substitutions for $x_{0}\mapsto x_{k}$ and others in Lemma A.2 then gives the result. ∎

We also need the Robbins-Siegmund almost supermartingale convergence theorem.

Lemma A.4** ([32]).**

Let $A_{k},B_{k},C_{k},D_{k}\geq 0$ be non-negative random variables adapted to the filtration $\mathcal{F}_{k}$ and satisfying $\mathbb{E}[A_{k+1}\mid\mathcal{F}_{k}]\leq(1+B_{k})A_{k}+C_{k}-D_{k}$. Then on the event $\{\sum_{k}B_{k}<\infty,\sum_{k}C_{k}<\infty\}$, there is a random $A_{\infty}<\infty$ such that $A_{k}\mathop{\stackrel{{\scriptstyle a.s.}}{{\rightarrow}}}A_{\infty}$ and $\sum_{k}D_{k}<\infty$.

Appendix B Proof of Theorem 1

We begin with two lemmas useful for the proof. The first lemma provides a recursive inequality on the iterates for any model satisfying Conditions (C.i)–(C.iv).

Lemma B.1**.**

Let $x_{k}$ be generated by the iteration (2) with a model satisfying Conditions (C.i)–(C.iv). Then for any $x^{\star}\in\mathcal{X^{\star}}$

[TABLE]

Proof    Lemma A.2 implies that

[TABLE]

where inequality $(\star)$ uses that $f_{x_{k}}(x^{\star};S_{k})\leq f(x^{\star};S_{k})$ and $f_{x_{k}}(x_{k+1};S_{k})\geq\inf_{z\in\mathcal{X}}f(z;S_{k})$. ∎

The preceding recursion allows us to bound the expected norm of the gradients of the iterates.

Lemma B.2**.**

Let Assumption A3 hold and let $x_{k}$ be generated by the iteration (2) with a model satisfying Conditions (C.i)–(C.iv) and stepsizes $\alpha_{k}=\alpha_{0}k^{-\beta}$. Then there exist $C_{1},C_{2}<\infty$ such that

[TABLE]

Proof    Assumption A3 implies that

[TABLE]

Let $f^{\star}_{k}=f(x^{\star};S_{k})-\inf_{z\in\mathcal{X}}f(z;S_{k})$ for shorthand. Lemma B.1 implies that for any $x^{\star}\in\mathcal{X^{\star}}$,

[TABLE]

where inequality $(i)$ is a consequence of the inequality $(a+b)^{\ell}\leq 2^{\ell}(a^{\ell}+b^{\ell})$, which holds for any $a,b,\ell\geq 0$, inequality $(ii)$ follows from Jensen’s inequality, and inequality $(iii)$ follows from Assumption A3 and that $\alpha_{i}\propto i^{-\beta}$ so that $\sum_{i=1}^{k}\alpha_{i}\leq Ck^{1-\beta}$. ∎

We use the preceding lemmas to prove Theorem 1. Lemma A.3, with the weak convexity constant $\rho\equiv 0$, implies that if $x^{\star}$ is the projection of $x_{k}$ onto $\mathcal{X}^{\star}$, then

[TABLE]

Let us denote $B_{k}=\alpha_{k}^{2}\mathbb{E}[\left\|{{f^{\prime}(x_{k};S_{k})}}\right\|_{2}^{2}\mid\mathcal{F}_{k-1}]$. By Lemma A.4, Theorem 1—that the $x_{k}$ are bounded—will follow if we can prove that $\sum_{k=1}^{\infty}B_{k}<\infty$ almost surely. As $\mathbb{E}[B_{k}]=\alpha_{k}^{2}\mathbb{E}[\left\|{f^{\prime}(x_{k};S_{k})}\right\|_{2}^{2}]$, Lemma B.2 implies that for some $C_{1},C_{2}<\infty$,

[TABLE]

where $\gamma=-p/2+\beta(p/2+2)$. Whenever $\frac{p+2}{p+4}<\beta<1$, we get that $\gamma>1$, implying that $\lim_{k\to\infty}\mathbb{E}[\sum_{i=1}^{k}B_{i}]<\infty$. The monotone convergence theorem implies that $\mathbb{E}[\sum_{k=1}^{\infty}B_{k}]<\infty$. We conclude that $\sum_{k=1}^{\infty}B_{k}<\infty$ almost surely, completing the proof.

Appendix C Proof of Theorem 2

We assume without loss of generality that $\inf_{x\in\mathcal{X}}F(x)=0$. It is enough to prove that the recursion in the next lemma holds, because Eq. (8) implies that $\left\|{x^{\lambda}_{k}-x_{k}}\right\|_{2}=\left\|{\nabla F_{\lambda}(x_{k})}\right\|_{2}/\lambda$. The theorem will then follow from the Robbins-Siegmund almost supermartingale convergence lemma (Lemma A.4).

Lemma C.1**.**

Let the conditions of Theorem 2 hold and let $A_{\lambda}=2(C_{1}\lambda^{2}+\mathbb{E}[(\rho(S)-\lambda)^{2}])$. Then

[TABLE]

Proof    First, we have

[TABLE]

Now we bound the term $\|{x^{\lambda}_{k}-x_{k+1}}\|_{2}^{2}$. Since $x_{k+1}$ solves the update (2) with the model $f_{x_{k}}$, we have by the standard optimality conditions for strongly convex minimization that

[TABLE]

Now, we take $y=x^{\lambda}_{k}$ in the last inequality and apply Lemma A.1 with $x_{1}=x_{k+1}$, $x_{0}=x_{k}$, $y=x^{\lambda}_{k}$, and $\beta=\alpha_{k}$ to find

[TABLE]

for all $f^{\prime}(x^{\lambda}_{k};S_{k})\in\partial f(x^{\lambda}_{k};S_{k})$. But since $f_{x_{k}}(x;S_{k})=f(x;S_{k})+\frac{\rho(S_{k})}{2}\left\|{x-x_{k}}\right\|_{2}^{2}$, we have that

[TABLE]

Taking expectations, the definition of $x_{k}^{\lambda}$ as the minimizer of $F(x)+\frac{\lambda}{2}\left\|{x-x_{k}}\right\|_{2}^{2}$ over $\mathcal{X}$ implies that w.l.o.g. we have $\mathbb{E}[f^{\prime}(x_{k}^{\lambda};S_{k})\mid\mathcal{F}_{k-1}]=F^{\prime}(x_{k}^{\lambda})$ for the element $F^{\prime}(x_{k}^{\lambda})\in\partial F(x_{k}^{\lambda})$ satisfying $\langle F^{\prime}(x_{k}^{\lambda})+\lambda(x_{k}^{\lambda}-x_{k}),y-x_{k}^{\lambda}\rangle\geq 0$ for all $y\in\mathcal{X}$. Thus we obtain that

[TABLE]

Moreover, we have

[TABLE]

Combining Assumption A4 with the fact that $\mathbb{E}[f^{\prime}(x_{k}^{\lambda};S_{k})\mid\mathcal{F}_{k-1}]=F^{\prime}(x_{k}^{\lambda})=-\lambda(x_{k}^{\lambda}-x_{k})$, we have that

[TABLE]

Substituting this bound into inequality (14) gives

[TABLE]

Now we derive our final recursion. Taking expectations in the bound (12) and applying inequalities (13) and (15) with the choice $A=2(C_{1}\lambda^{2}+\mathbb{E}[(\rho(S)-\lambda)^{2}])$, we get

[TABLE]

Using the convexity bound (11), we thus have

[TABLE]

Using that $F_{\lambda}(x)=F(x)+\frac{\lambda}{2}\|{x^{\lambda}-x}\|_{2}^{2}$ gives the lemma. ∎

Appendix D Proof of

Proposition 1

Throughout the proof, let $\overline{\rho}=\mathbb{E}[\rho(S)]$ be the expected weak convexity constant.

We prove the first claim of the proposition. By Lemma A.3, we have

[TABLE]

Now, taking expectations conditional on $\mathcal{F}_{k-1}$, we have

[TABLE]

where we have used Assumption A2 and the definition of $\mathsf{G}_{\rm big}$. Adding and subtracting $\frac{\lambda\alpha_{k}}{2}\|{x_{k}-x_{k}^{\lambda}}\|_{2}^{2}=\frac{\alpha_{k}}{2\lambda}\|{\nabla F_{\lambda}(x_{k})}\|_{2}^{2}$, we have

[TABLE]

As $\sup_{k}\left\|{x_{k}}\right\|_{2}<\infty$ by assumption, the final term above satisfies $\mathsf{G}_{\rm big}(\left\|{x_{k}}\right\|_{2})\leq B<\infty$ for some (random) $B$ for all $k$. The martingale convergence Lemma A.4 then gives that $\sum_{k}\alpha_{k}\left\|{\nabla F_{\lambda}(x_{k})}\right\|_{2}^{2}<\infty$ with probability 1, establishing the convergence guarantees (9).

Now for the second claim of Proposition 1, which relies on Assumption A5. We collect a few useful results. First, we note that $F_{\lambda}(\mathcal{X}_{\rm stat})=F(\mathcal{X}_{\rm stat})$ for all $\lambda\geq\overline{\rho}$, as any $x\in\mathcal{X}_{\rm stat}$ is a minimizer of $y\mapsto F(y)+\frac{\lambda}{2}\left\|{y-x}\right\|_{2}^{2}$. Second, we claim that

[TABLE]

For the sake of contradiction, assume that this fails, noting that we have $F_{\lambda}(x_{k})\mathop{\stackrel{{\scriptstyle a.s.}}{{\rightarrow}}}G_{\lambda}$ for some $G_{\lambda}<\infty$. We know that $x\mapsto\nabla F_{\lambda}(x)$ is a continuous function, and so over compacta, $\left\|{\nabla F_{\lambda}(x)}\right\|$ achieves its minimum. Let $B<\infty$ be such that $\left\|{x_{k}}\right\|<B$ for all $k$ ($B$ may be random, but it exists). Then $\mathop{\rm dist}(F_{\lambda}(x_{k}),F(\mathcal{X}_{\rm stat}))\to\delta>0$. As $y\mapsto\mathop{\rm dist}(F_{\lambda}(y),F(\mathcal{X}_{\rm stat}))$ is continuous, the pre-image $Y_{\delta}:=\{y\in\mathcal{X}\mid\left\|{y}\right\|\leq B,\mathop{\rm dist}(F_{\lambda}(y),F(\mathcal{X}_{\rm stat}))\geq\delta/2\}$ is compact. Then $\left\|{\nabla F_{\lambda}(y)}\right\|$ achieves its infimum over $Y_{\delta}$, which must be positive (as otherwise there would exist a stationary point in $Y_{\delta}$, a contradiction). But then for $K$ large enough that $\mathop{\rm dist}(F_{\lambda}(x_{k}),F(\mathcal{X}_{\rm stat}))>\delta/2$ for $k\geq K$, we have

[TABLE]

The first sum is finite by the first part of the proposition, a contradiction to the fact that $\sum_{k}\alpha_{k}=\infty$ and $\inf_{y\in Y_{\delta}}\|{\nabla F_{\lambda}(y)}\|_{2}^{2}>0$.

Now, consider the limiting value $G(\lambda)=\lim_{k}F_{\lambda}(x_{k})$. Then for any limit point $x_{\infty}$ of $x_{k}$, we have $G(\lambda)=F_{\lambda}(x_{\infty})$, and thus $G(\lambda)\in F(\mathcal{X}_{\rm stat})$. As by assumption $F(\mathcal{X}_{\rm stat})$ has measure zero, it must be the case that $G(\lambda)$ is constant in $\lambda$. That is, $x_{\infty}$ satisfies

[TABLE]

for some fixed $G$. We now make the following claim.

Lemma D.1**.**

Let $h$ be convex and assume that $h_{\lambda}(x_{0})=\inf_{x}\{h(x)+\frac{\lambda}{2}\left\|{x-x_{0}}\right\|_{2}^{2}\}$ is constant in $\lambda$. Then $x_{0}$ minimizes $h$.

Deferring the proof of Lemma D.1 temporarily, note that $y\mapsto F(y)+\frac{\lambda}{2}\left\|{y-x_{\infty}}\right\|_{2}^{2}$ is convex for large enough $\lambda$. But of course $F_{\lambda}(x_{\infty})$ is constant in $\lambda$, and thus $x_{\infty}$ must minimize $F(x)+\frac{\lambda}{2}\left\|{x-x_{\infty}}\right\|_{2}^{2}$, so that $x_{\infty}$ is stationary.

Summarizing, we have shown that all subsequences of $x_{k}$ have a further subsequence that converges to some $x_{\infty}\in\mathcal{X}_{\rm stat}$, and thus $\mathop{\rm dist}(x_{k},\mathcal{X}_{\rm stat})\mathop{\stackrel{{\scriptstyle a.s.}}{{\rightarrow}}}0$.

Proof of Lemma D.1     Let $h^{\prime}(x_{0};d)=\lim_{t\downarrow 0}\frac{h(x_{0}+td)-h(x_{0})}{t}$ be the directional derivative of $h$, which exists and satisfies $h(x_{0}+td)-h(x_{0})-th^{\prime}(x_{0};d)=o(t)$ as $t\downarrow 0$ (cf. [22, Lemma VI.2.2.1]). Moreover, $x_{0}$ minimizes $h$ if and only if $h^{\prime}(x_{0};d)\geq 0$ for all $d\in\mathbb{R}^{n}$, and $\lim_{\lambda\uparrow\infty}h_{\lambda}(x)=h(x)$ for all $x$ (cf. [22, Proposition XV.4.1.5]). Now, let $d$ satisfy $\left\|{d}\right\|_{2}=1$, and assume for the sake of contradiction that $h^{\prime}(x_{0};d)<0$. Then

[TABLE]

and taking $t=-\frac{1}{\lambda}h^{\prime}(x_{0};d)$ and letting $\lambda\uparrow\infty$, we have

[TABLE]

which contradicts that $h_{\lambda}(x_{0})$ is constant in $\lambda$. ∎

Appendix E Proofs of fast convergence on easy problems

E.1 Proof of Lemma 4.1

We assume without loss of generality that $f(x^{\star};s)=0$ for all $x^{\star}\in\mathcal{X}^{\star}$, as we may replace $f$ with $f-\inf f$. Lemma 3.3 from [2] implies

[TABLE]

For shorthand, let $g_{k}=f^{\prime}(x_{k};S_{k})$ and $f_{k}=f(x_{k};S_{k})$. Now, note that $f_{x_{k}}(x^{\star};S_{k})\leq f(x^{\star};S_{k})+\frac{\rho(S_{k})}{2}\left\|{x_{k}-x^{\star}}\right\|_{2}^{2}$, and by Condition (C.iv) we have

[TABLE]

Thus we have

[TABLE]

If we let $\widetilde{x}_{k+1}$ denote the unconstrained minimizer

[TABLE]

then because $x_{k+1}\in\mathcal{X}$ we have

[TABLE]

By inspection, the guarded stepsize $\lambda_{k}$ guarantees that $\left[{f_{k}+\langle g_{k},\widetilde{x}_{k+1}-x_{k}\rangle}\right]_{+}=f_{k}-\lambda_{k}\left\|{g_{k}}\right\|_{2}^{2}$, and thus inequality (17) (setting $f(x^{\star};S_{k})=0$) yields

[TABLE]

We have two possible cases: whether $f_{k}/\left\|{g_{k}}\right\|_{2}^{2}\lessgtr\alpha_{k}$. In the case that $f_{k}/\left\|{g_{k}}\right\|_{2}^{2}\leq\alpha_{k}$, we have $\lambda_{k}=f_{k}/\left\|{g_{k}}\right\|_{2}^{2}$ and so $-\lambda_{k}f_{k}+\lambda_{k}^{2}\left\|{g_{k}}\right\|_{2}^{2}/2=-f_{k}^{2}/(2\left\|{g_{k}}\right\|_{2}^{2})$. In the alternative case that $f_{k}/\left\|{g_{k}}\right\|_{2}^{2}>\alpha_{k}$, we have $\lambda_{k}=\alpha_{k}$ and $\alpha_{k}^{2}\left\|{g_{k}}\right\|_{2}^{2}/2\leq\alpha_{k}f_{k}/2$. Combining these cases gives

[TABLE]

which is the desired result.

E.2 Proof of

Proposition 2

Let $D_{k}=\mathop{\rm dist}(x_{k},\mathcal{X}^{\star})$ for shorthand throughout this proof. Lemma 4.1 implies that

[TABLE]

Let $\epsilon>0$ be arbitrary. Then

[TABLE]

In particular, taking $K_{0}\geq m$ to be the smallest integer such that $k\geq K_{0}$ implies $\epsilon\overline{\rho}\alpha_{k}\leq\lambda_{1}$,

[TABLE]

we obtain that

[TABLE]

Now, we give an argument analogous to our proof of linear convergence for convex problems [2, Proposition 2]. We have for any $\epsilon>0$, $\delta>0$ that

[TABLE]

The Borel-Cantelli lemma thus implies that for any $m\in\mathbb{N}$

[TABLE]

For the assertion of the proposition, for each $m\in\mathbb{N}$ define the random variables

[TABLE]

We again trace the argument for the proof of Proposition 2 of the paper [2]; we have by inequality (18) that

[TABLE]

But the convergence (19) guarantees that $D_{k}<\alpha_{k}$ when $\max_{m\leq i<k}\{D_{i}\}\leq\frac{\lambda_{0}}{(1+\epsilon)\overline{\rho}}$, at least for all large enough $k$, so that $\mathbb{E}[V_{k,m}\mid\mathcal{F}_{k-1}]\leq V_{k-1,m}+E_{k-1,m}$, where $E_{k,m}=0$ eventually (with probability $1$). The Robbins-Siegmund lemma A.4 completes the proof.