Arbitrary Decisions are a Hidden Cost of Differentially Private Training

TL;DR

This paper investigates how the randomness introduced in differentially private training methods causes variability in model predictions, highlighting a hidden cost that impacts decision reliability and fairness.

Contribution

It provides a theoretical and empirical analysis of predictive multiplicity in DP training, quantifies its dependence on privacy levels, and emphasizes the need for auditing this effect.

Findings

Predictive multiplicity increases with higher privacy levels.

Randomness in DP training causes significant prediction variability.

Predictive multiplicity varies across individuals and groups.

Abstract

Mechanisms used in privacy-preserving machine learning often aim to guarantee differential privacy (DP) during model training. Practical DP-ensuring training methods use randomization when fitting model parameters to privacy-sensitive data (e.g., adding Gaussian noise to clipped gradients). We demonstrate that such randomization incurs predictive multiplicity: for a given input example, the output predicted by equally-private models depends on the randomness used in training. Thus, for a given input, the predicted output can vary drastically if a model is re-trained, even if the same training dataset is used. The predictive-multiplicity cost of DP training has not been studied, and is currently neither audited for nor communicated to model designers and stakeholders. We derive a bound on the number of re-trainings required to estimate predictive multiplicity reliably. We analyze--both theoretically and through extensive experiments--the predictive-multiplicity cost of three DP-ensuring algorithms: output perturbation, objective perturbation, and DP-SGD. We demonstrate that the degree of predictive multiplicity rises as the level of privacy increases, and is unevenly distributed across individuals and demographic groups in the data. Because randomness used to ensure DP during training explains predictions for some examples, our results highlight a fundamental challenge to the justifiability of decisions supported by differentially private models in high-stakes settings. We conclude that practitioners should audit the predictive multiplicity of their DP-ensuring algorithms before deploying them in applications of individual-level consequence.

Tables

Table 1. Table 1 . Summary of datasets used in our experimental evaluations.

Dataset	Size	Number of features	Train size	Test size
Synthetic	$\infty$	2	2000	20,000
Credit	653	46	489	164
Contraception	1,473	9	1,104	369
Mammography	830	5	622	208
Dermatology	358	34	268	90
CIFAR-10	60,000	$32\times 32\times 3$	50,000	10,000

Table 2. Table 2 . Summary statistics of the performance and predictive-multiplicity measures on real-world datasets. For tabular datasets, the performance metrics are the area under the ROC curve (AUC), and the harmonic mean of precision and recall ( F 1 subscript 𝐹 1 F_{1} score) on the test data. For CIFAR-10, the performance metric is the accuracy on the test data. For these, we report mean and standard deviation over the m 𝑚 m re-trained models. For disagreement, we report mean, standard deviation, minimum, median, maximum, the 90-th percentile, and the 95-th percentile over the examples in each respective test dataset. Observe that for every dataset there exist multiple examples with high level of predictive multiplicity even if the average level of predictive multiplicity for the given dataset is low. E.g., compare the 95-th percentile of disagreement on the CIFAR-10 dataset at ε = 2.22 𝜀 2.22 \varepsilon=2.22 (0.81) to its mean value (0.11).

		AUC		$F_1$ score		Disagreement
Dataset	$\epsilon$	Mean	Std.	Mean	Std.	Mean	Std.	Min	Median	Max	90 pctl.	95 pctl.
Contraception	0.50	57.51	6.72	48.72	7.86	0.90	0.10	0.48	0.93	1.00	0.99	1.00
	0.75	60.26	6.20	50.29	7.54	0.82	0.17	0.24	0.88	1.00	0.99	1.00
	1.00	62.50	5.47	51.56	7.09	0.73	0.23	0.11	0.79	1.00	0.98	1.00
	1.25	64.27	4.71	52.62	6.62	0.65	0.27	0.05	0.70	1.00	0.97	0.99
	1.50	65.62	4.00	53.53	6.14	0.57	0.30	0.02	0.60	1.00	0.96	0.99
	1.75	66.65	3.38	54.31	5.67	0.51	0.32	0.00	0.50	1.00	0.95	0.99
	2.00	67.43	2.86	54.98	5.21	0.45	0.33	0.00	0.42	1.00	0.94	0.98
	2.50	68.49	2.10	55.97	4.39	0.37	0.33	0.00	0.27	1.00	0.92	0.97
Credit	0.50	52.22	15.95	46.48	16.38	1.00	0.00	0.99	1.00	1.00	1.00	1.00
	0.75	53.72	15.70	47.84	15.70	0.99	0.01	0.98	0.99	1.00	1.00	1.00
	1.00	55.16	15.41	49.15	15.05	0.99	0.01	0.96	0.99	1.00	1.00	1.00
	1.25	56.56	15.06	50.39	14.46	0.98	0.02	0.94	0.98	1.00	1.00	1.00
	1.50	57.86	14.69	51.59	13.89	0.97	0.03	0.91	0.98	1.00	1.00	1.00
	1.75	59.10	14.31	52.72	13.35	0.96	0.03	0.89	0.97	1.00	1.00	1.00
	2.00	60.26	13.91	53.77	12.85	0.95	0.04	0.86	0.96	1.00	1.00	1.00
	2.50	62.41	13.12	55.70	12.05	0.93	0.06	0.80	0.95	1.00	1.00	1.00
Dermatology	0.50	62.19	19.76	48.81	17.88	0.96	0.03	0.89	0.96	1.00	1.00	1.00
	0.75	66.75	17.65	52.67	16.44	0.93	0.05	0.79	0.93	1.00	0.99	0.99
	1.00	70.44	15.83	55.88	15.21	0.89	0.08	0.69	0.90	1.00	0.98	0.99
	1.25	73.46	14.28	58.57	14.20	0.85	0.10	0.60	0.86	1.00	0.98	0.98
	1.50	75.94	12.97	60.93	13.30	0.82	0.12	0.52	0.83	1.00	0.97	0.98
	1.75	78.04	11.89	62.98	12.60	0.79	0.13	0.46	0.80	1.00	0.95	0.97
	2.00	79.80	10.96	64.78	12.00	0.75	0.15	0.39	0.77	0.99	0.94	0.96
	2.50	82.66	9.45	67.80	10.95	0.70	0.17	0.32	0.72	0.99	0.92	0.94
Mammography	0.50	75.64	8.95	69.22	9.88	0.62	0.28	0.20	0.61	1.00	0.98	1.00
	0.75	78.57	6.51	72.46	7.04	0.51	0.34	0.07	0.45	1.00	0.98	1.00
	1.00	80.36	5.26	74.39	5.48	0.44	0.36	0.02	0.33	1.00	0.97	0.99
	1.25	81.62	4.44	75.64	4.66	0.39	0.37	0.01	0.24	1.00	0.95	0.99
	1.50	82.54	3.82	76.56	4.14	0.35	0.37	0.00	0.17	1.00	0.93	0.99
	1.75	83.25	3.36	77.29	3.81	0.32	0.36	0.00	0.12	1.00	0.91	0.98
	2.00	83.81	2.98	77.85	3.56	0.29	0.35	0.00	0.08	1.00	0.89	0.98
	2.50	84.61	2.40	78.70	3.22	0.25	0.34	0.00	0.04	1.00	0.84	0.96

Equations

$$f_{\theta }(x)\triangleq 1[h_{\theta }(x)>q].$$

$$\mathrm{Pr}[T(\mathcal{D})\in A]\le \exp (\epsilon )\mathrm{Pr}[T({\mathcal{D}}^{\prime })\in A]+\delta .$$

$${\theta }_{\mathsf{priv}}={\theta }_{\mathsf{np}}+\xi ,\text{ where }\xi \sim \mathcal{N}(0,{\sigma }^2{I}_d)$$

$$\mu (x)\triangleq 2\underset{\theta ,{\theta }^{\prime }\sim P_{T(\mathcal{D})}}{\mathrm{Pr}}[f_{\theta }(x)\ne f_{{\theta }^{\prime }}(x)].$$

$$\mu (x)=4{\mathrm{Var}}_{\theta \sim P_{T(\mathcal{D})}}(f_{\theta }(x))=4p_x(1-p_x).$$

$$\mathsf{sigmoid}(t)\triangleq \frac{1}{1+\exp (-t)}.$$

$$\mu (x)=4p_x(1-p_x),\text{where px=Φ(∥x∥⋅σθnp⊺x).}$$

$$\hat{\mu }(x)\triangleq 4\frac{m}{m-1}{\hat{p}}_x(1-{\hat{p}}_x),$$

$$\alpha \le \frac{1}{(m-1)}+4\frac{m}{m-1}\sqrt{\frac{\log (2/\rho )}{2m}}\left(1+\sqrt{\frac{\log (2/\rho )}{2m}}\right).$$

$$\underset{j\in 1,\dots ,k}{\max }|\mu (x_j)-\hat{\mu }(x_j)|$$

$$+\frac{4m}{m-1}\sqrt{\frac{\log (2k/\rho )}{2m}}\left(1+\sqrt{\frac{\log (2k/\rho )}{2m}}\right).$$

$${\mu }_0=[1,1],$$

$${\mu }_1=[-1,-1],$$

$$\mu (x)$$

$$=2{\mathbb{E}}_{\theta ,{\theta }^{\prime }\sim P_{T(\mathcal{D})}}[1[f_{\theta }(x)\ne f_{{\theta }^{\prime }}(x)]]$$

$$=2{\mathbb{E}}_{\theta ,{\theta }^{\prime }\sim P_{T(\mathcal{D})}}[(f_{\theta }(x)-f_{{\theta }^{\prime }}(x){)}^2]$$

$$=2{\mathbb{E}}_{\theta \sim P_{T(\mathcal{D})}}[f_{\theta }^2(x)]-4{\mathbb{E}}_{\theta ,{\theta }^{\prime }\sim P_{T(\mathcal{D})}}[f_{\theta }(x)\cdot f_{{\theta }^{\prime }}(x)]$$

$$+2{\mathbb{E}}_{{\theta }^{\prime }\sim P_{T(\mathcal{D})}}[f_{{\theta }^{\prime }}^2(x)]$$

$$=4({\mathbb{E}}_{\theta \sim P_{T(\mathcal{D})}}[f_{\theta }(x){]}^2$$

$$-{\mathbb{E}}_{\theta \sim P_{T(\mathcal{D})}}[f_{\theta }(x)]\cdot {\mathbb{E}}_{{\theta }^{\prime }\sim P_{T(\mathcal{D})}}[f_{{\theta }^{\prime }}(x)])$$

$$=4{\mathrm{Var}}_{\theta \sim P_{T(\mathcal{D})}}(f_{\theta }(x))$$

$$=4p_x(1-p_x),$$

$$p_x={\mathbb{E}}_{{\theta }_{\mathsf{priv}}\sim P_{T(\mathcal{D})}}[f_{{\theta }_{\mathsf{priv}}}(x)]$$

$$\mathbb{E}[f_{{\theta }_{\mathsf{priv}}}(x)]$$

$$=\mathbb{E}[1[{\theta }_{\mathsf{priv}}^{⊺}x>0]]$$

$$=\mathrm{Pr}({\theta }_{\mathsf{priv}}^{⊺}x>0).$$

$${\theta }_{\mathsf{priv}}^{⊺}x$$

$$={\theta }_{\mathsf{np}}^{⊺}x+\sigma \sum _{i=1}^d{x}_i\xi$$

$$={\theta }_{\mathsf{np}}^{⊺}x+\sqrt{\sum _{i=1}^d{x}_i^2}\cdot \sigma \xi$$

$$={\theta }_{\mathsf{np}}^{⊺}x+\Vert x\Vert \sigma \xi .$$

$$p_x=\mathrm{Pr}({\theta }_{\mathsf{np}}^{⊺}x+\Vert x\Vert \sigma \xi >0)=\mathrm{Pr}\left(\xi >-\frac{{\theta }_{\mathsf{np}}^{⊺}x}{\Vert x\Vert \cdot \sigma }\right)=\Phi \left(\frac{{\theta }_{\mathsf{np}}^{⊺}x}{\Vert x\Vert \cdot \sigma }\right).$$

$$\mathbb{E}\left[\frac{m}{m-1}{\hat{p}}_x(1-{\hat{p}}_x)\right]$$

$$=\frac{m}{m-1}(\mathbb{E}[{\hat{p}}_x]-\mathrm{Var}({\hat{p}}_x)-\mathbb{E}[{\hat{p}}_x{]}^2)$$

$$=\frac{m}{m-1}\left(p_x-\frac{p_x(1-p_x)}{m}-p_x^2\right)$$

$$=p_x(1-p_x)$$

$$\left|\frac{m}{m-1}{\hat{p}}_x(1-{\hat{p}}_x)-p_x(1-p_x)\right|=$$

$$=\left|\frac{m}{m-1}(p_x+\nu )(1-p_x-\nu )-p_x(1-p_x)\right|$$

$$=\left|\left(\frac{m}{m-1}-1\right)p_x(1-p_x)+\frac{m}{m-1}\nu (1-2p_x-\nu )\right|$$

$$\le \frac{p_x(1-p_x)}{m-1}+\frac{m}{m-1}|\nu ||1-2p_x+\nu |$$

$$\le \frac{p_x(1-p_x)}{m-1}+\frac{m}{m-1}|\nu |(1+|\nu |)$$

$$\le \frac{p_x(1-p_x)}{m-1}+\frac{m}{m-1}\eta (1+\eta )$$

$$\le \frac{1}{4(m-1)}+\frac{m}{m-1}\eta (1+\eta ).$$

Full text

††ccs: Security and privacy Privacy protections††ccs: Security and privacy Social aspects of security and privacy††ccs: Computing methodologies Model verification and validation††ccs: Computing methodologies Machine learning††ccs: Computing methodologies Neural networks††journalyear: 2023††copyright: acmlicensed††conference: 2023 ACM Conference on Fairness, Accountability, and Transparency; June 12–15, 2023; Chicago, IL, USA††booktitle: 2023 ACM Conference on Fairness, Accountability, and Transparency (FAccT ’23), June 12–15, 2023, Chicago, IL, USA††price: 15.00††doi: 10.1145/3593013.3594103††isbn: 979-8-4007-0192-4/23/06

1. Introduction

In many high-stakes prediction tasks (e.g., lending, healthcare), training data used to fit parameters of machine-learning models are privacy-sensitive. A standard technical approach to ensure privacy is to use training procedures that satisfy differential privacy (DP) (Dwork et al., 2006, 2014). DP is a formal condition that, intuitively, guarantees a degree of plausible deniability on the inclusion of an individual sample in the training data. In order to satisfy this condition, non-trivial differentially-private training procedures use randomization (see, e.g., Chaudhuri et al. (2011); Abadi et al. (2016)). The noisy nature of DP mechanisms is key to guarantee plausible deniability of a record’s inclusion in the training data. Unfortunately, randomization comes at a cost: it often leads to decreased accuracy compared to non-private training (Jayaraman and Evans, 2019). Reduced accuracy, however, is not the only cost incurred by differentially-private training. DP mechanisms can also increase predictive multiplicity, discussed next.

In a prediction task, there can exist multiple models that achieve comparable levels of accuracy yet output drastically different predictions for the same input. This phenomenon is known as predictive multiplicity (Marx et al., 2020), and has been documented in multiple realistic machine-learning settings (Marx et al., 2020; Hsu and Calmon, 2022; Watson-Daniels et al., 2023). Predictive multiplicity can appear due to under-specification and randomness in the model’s training procedure (D’Amour et al., 2020; Black et al., 2021).

Predictive multiplicity formalizes the arbitrariness of decisions based on a model’s output. In practice, predictive multiplicity can lead to questions such as “Why has a model issued a negative decision on an individual’s loan application if other models with indistinguishable accuracy would have issued a positive decision?” or “Why has a model suggested a high dose of a medicine for an individual if other models with comparable average accuracy would have prescribed a lower dose?” These examples highlight that acting on predictions of a single model without regard for predictive multiplicity can result in arbitrary decisions. Models produced by training algorithms that exhibit high predictive multiplicity face fundamental challenges to their credibility and justifiability in high-stakes settings  (D’Amour et al., 2020; Black et al., 2022).

In this paper, we demonstrate a fundamental connection between privacy and predictive multiplicity: For a fixed training dataset and model class, DP training results in models that ensure the same degree of privacy and achieve comparable accuracy, yet assign conflicting outputs to individual inputs. DP training produces conflicting models even when non-private training results in a single optimal model. Thus, in addition to decreased accuracy, DP-ensuring training methods also incur an arbitrariness cost by exacerbating predictive multiplicity. We show that the degree of predictive multiplicity varies significantly across individuals and can disproportionately impact certain population groups. Fig. 1 illustrates the predictive-multiplicity cost of DP training in a simple synthetic scenario (see Section 5 for examples on real-world datasets).

Our main contributions are:

1. (1)

We provide the first analysis of the predictive-multiplicity cost of differentially-private training. 2. (2)

We analyze a method for estimating the predictive-multiplicity properties of randomized machine-learning algorithms using re-training. We derive the first bound on the sample complexity of estimating predictive multiplicity with this approach. Our bound enables practitioners to determine the number of re-trainings required to estimate the predictive-multiplicity cost of randomized training algorithms up to a desired level of accuracy. 3. (3)

We conduct a theoretical analysis of the predictive-multiplicity cost of the output perturbation mechanism (Chaudhuri et al., 2011) used to obtain a differentially-private logistic-regression model. We characterize the exact dependence of predictive multiplicity on the level of privacy for this method. 4. (4)

We conduct an empirical study of predictive multiplicity of two practical DP-ensuring learning algorithms: DP-SGD (Abadi et al., 2016) and objective perturbation (Chaudhuri et al., 2011). We use one synthetic dataset and five real-world datasets in the domains of finance, healthcare, and image classification. Our results confirm that, for these mechanisms, increasing the level of privacy invariably increases the level of predictive multiplicity. Moreover, we find that different examples exhibit different levels of predictive multiplicity. In particular, different demographic groups can have different average levels of predictive multiplicity.

In summary, the level of privacy in DP training significantly impacts the level of predictive multiplicity. This, in turn, means that decisions supported by differentially-private models can have an increased level of arbitrariness: a given decision would have been different had we used a different random seed in training, even when all other aspects of training are kept fixed and the optimal non-private model is unique. Before deploying DP-ensuring models in high-stakes situations, we suggest that practitioners quantify predictive multiplicity of these models over salient populations and—if possible to do so without violating privacy—measure predictive multiplicity of individual decisions during model operation. Such audits can help practitioners evaluate whether the increase in privacy threatens the justifiability of decisions, choose whether to enact a decision based on a model’s output, and determine whether to deploy a model in the first place.

2. Technical Background

2.1. Problem Setup and Notation

We consider a classification task on a training dataset, denoted as $\mathcal{D}~{}\triangleq~{}\{(x_{i},y_{i})\}_{i=1}^{n}$, and consisting of examples $x_{i}\in\mathcal{X}$ along with their respective labels $y_{i}\in\mathcal{Y}$. In this work, we focus on the setting of binary classification, $\mathcal{Y}=\{0,1\}$. The goal of a classification task is to use the dataset to train a classifier $f_{\theta}:\mathcal{X}\to\mathcal{Y}$, which accurately predicts labels for input examples in a given test dataset $\mathcal{D}_{\mathrm{test}}\in 2^{\mathcal{X}\times\mathcal{Y}}$, where $2^{\mathcal{X}\times\mathcal{Y}}$ denotes the power set over $\mathcal{X}\times\mathcal{Y}$. Each classifier $f_{\theta}(x)$ is parameterized by a vector $\theta\in\Theta\subseteq\mathbb{R}^{d}$.

A classifier associates a confidence score to each predicted input $x$, denoted as $h_{\theta}(x)\in[0,1]$. If the confidence score is higher than some threshold $q\in[0,1]$, then the decision is positive. Otherwise, it is negative. The classifier’s prediction is thus obtained by applying a threshold to the confidence score:

[TABLE]

In the rest of the paper, we use the standard threshold of $q=0.5$.

We study randomized training algorithms $T:(\mathcal{X}\times\mathcal{Y})^{n}\to\Theta$, which produce a parameter vector of a classifier in a randomized way. Thus, given a training dataset, $T(\mathcal{D})$ is a random variable. We denote by $P_{T(\mathcal{D})}$ the model distribution, the probability distribution over $\Theta$ generated by the random variable $T(\mathcal{D})$.

In general, the source of randomness in the training procedure could include, e.g., random initializations of $\theta$ prior to training. However, we consider only those sources which are introduced by the privacy-preserving techniques, as we explain in the next section. Throughout this paper, the datasets, as well as any input example $x\in\mathcal{X}$, are not random variables but fixed values. The only randomness we consider in our notation is due to the internal randomization of the training procedure $T(\cdot)$. Finally, $I_{d}$ denotes the $d$-by-$d$ identity matrix, and $\mathbb{1}(\cdot)$ denotes the indicator function.

2.2. Differentially Private Learning

Learning with differential privacy (DP) is one of the standard approaches to train models on privacy-sensitive data (Dwork et al., 2006, 2014). A randomized learning algorithm $T(\cdot)$ is $(\varepsilon,\delta)$-differentially private (DP) if for any two neighbouring datasets (i.e., datasets differing by at most one example) $\mathcal{D},\mathcal{D}^{\prime}\in(\mathcal{X}\times\mathcal{Y})^{n}$, for any subset of parameter vectors $A\subseteq\Theta$, it holds that

[TABLE]

In other words, the respective probability distributions of models produced on any two neighbouring datasets should be similar to a degree defined by parameters $(\varepsilon,\delta)$. The parameters represent the level of privacy: low $\varepsilon$ and low $\delta$ mean better privacy. DP mathematically encodes a notion of plausible deniability of the inclusion of an example in the dataset.

There are multiple ways to ensure DP in machine learning. We describe next the output perturbation mechanism, which we theoretically analyze in Section 3.

Output perturbation mechanism (Rubinstein et al., 2012; Chaudhuri et al., 2011; Wu et al., 2017).

Output perturbation is a simple mechanism for achieving DP that takes an output parameter vector of a non-private training procedure, and privatizes it by adding random noise, e.g., sampled from the isotropic Gaussian distribution. Concretely, suppose that $T_{\mathsf{np}}:(\mathcal{X}\times\mathcal{Y})^{n}\rightarrow\Theta$ is a non-private learning algorithm. Denoting its output parameters as $\theta_{\mathsf{np}}=T_{\mathsf{np}}(\mathcal{D})$, we obtain the privatized parameters $\theta_{\mathsf{priv}}\in\Theta$ as:

[TABLE]

The exact level of DP provided by this procedure depends on the choice the non-private training algorithm $T_{\mathsf{np}}(\mathcal{D})$. In particular, to achieve $(\varepsilon,\delta)$-DP, it is sufficient to set the noise scale $\sigma=C\cdot\nicefrac{{\sqrt{2\log(1.25/\delta)}}}{{\varepsilon}}$, where $C~{}\triangleq~{}\max_{\mathcal{D}\sim\mathcal{D}^{\prime}}\|T_{\mathrm{np}}(\mathcal{D})-T_{\mathrm{np}}(\mathcal{D}^{\prime})\|_{2}$ is the sensitivity of the non-private training algorithm, the maximum discrepancy in terms of the $\ell_{2}$ distance between parameter vectors obtained by training on any two neighbouring datasets $\mathcal{D},\mathcal{D}^{\prime}$.

Denoting by $T(\mathcal{D})=T_{\mathrm{np}}(\mathcal{D})+\xi$ the output-perturbation procedure in Eq. 3, we treat $T(\mathcal{D})$ as a random variable over the randomness of the injected noise $\xi$. Other methods to achieve DP such as objective perturbation (Chaudhuri et al., 2011) also inject noise as part of training. In those cases, we similarly consider $T(\mathcal{D})$ as a random variable over such injected noise, and treat all other aspects of training such as pre-training initialization as fixed.

2.3. Predictive Multiplicity

Predictive multiplicity occurs when multiple classification models achieve comparable average accuracy yet produce conflicting predictions on a given example (Marx et al., 2020). To quantify predictive multiplicity in randomized training, we need to measure dissimilarity of predictions among the models sampled from the probability distribution $P_{T(\mathcal{D})}$ induced by differentially-private training. For this, we use a definition of disagreement which has appeared in different forms in (Black et al., 2022; D’Amour et al., 2020; Marx et al., 2020). For a given fixed input example $x\in\mathcal{X}$, we define the disagreement $\mu(x)$ as:

[TABLE]

In the above definition, $\theta,\theta^{\prime}\sim P_{T(\mathcal{D})}$ denotes two models sampled independently from $P_{T(\mathcal{D})}$. We use a scaling factor of two in order to ensure that $\mu(x)$ is in the $[0,1]$ range for the ease of interpretation. A disagreement value $\mu(x)\approx 1$ indicates that the prediction for $x$ is approximately equal to an unbiased coin flip. Moreover, a disagreement $\mu(x)\approx 0$ implies that, with high probability, the prediction for $x$ does not significantly change if two models are independently sampled from $P_{T(\mathcal{D})}$ (i.e., by re-training a model twice with different random seeds).

In the literature, a commonly studied source of variance of outcomes of training algorithms is from re-sampling of the dataset $\mathcal{D}$, usually under the assumption that it is an i.i.d. sample from some data distribution. We do not study variance arising from dataset re-sampling, and are only interested in the predictive-multiplicity properties of the randomized training procedure $T(\cdot)$ itself. Thus, we fix both the dataset $\mathcal{D}$ used in training and the input example $x$ for which we compute the level of predictive multiplicity, and make sure that the randomness is only due to internal randomization of the training procedure $T(\cdot)$.

When evaluating dissimilarity across models, many prior works that study predictive multiplicity (e.g., (Marx et al., 2020; Semenova et al., 2022; Hsu and Calmon, 2022; Watson-Daniels et al., 2023)) only consider models that surpass a certain accuracy threshold. Although conditioning on model accuracy is theoretically valid, it can bring about confusion in the context of private learning, as in practice such conditioning would demand special mechanisms in order to satisfy DP (see, e.g., (Papernot and Steinke, 2022)). In particular, first applying a DP training method that guarantees an $(\varepsilon,\delta)$-level of privacy, and then selecting or discarding the resulting model based on accuracy, would result in models that violate the initial $(\varepsilon,\delta)$-DP guarantees. We note, however, that our results and experiments involving estimation of predictive multiplicity in Sections 4 and 5 extend to the case in which we add additional conditioning on top of model distribution $P_{T(\mathcal{D})}$ to control for accuracy.

Before proceeding with our analyses of disagreement, we first state a simple yet useful relation between disagreement and statistical variance. Observe that for a given input $x$, the output prediction $f_{\theta}(x)$ is a random variable over the randomness of the training procedure $\theta\sim P_{T(\mathcal{D})}$. As we assume that the decisions are binary, and training runs are independent, we have that $f_{\theta}(x)\sim\mathrm{Bernoulli}(p_{x})$ for some input-specific parameter $p_{x}$. Having noted this fact, we show that disagreement, defined in Eq. 4, can be expressed as a continuous transformation of $p_{x}$:

Proposition 0.

multvar For binary classifiers, disagreement for a given example $x\in\mathcal{X}$ is proportional to variance of decisions over the distribution of models generated by the training algorithm:

[TABLE]

We provide the proof of this and all the following formal statements in Appendix A. Additionally, in Appendix B, we provide an analysis using an alternative measure of predictive multiplicity.

3. Predictive Multiplicity of Output Perturbation

To demonstrate how DP training can lead to an increase in predictive multiplicity, we theoretically analyze the multiplicity properties of the output-perturbation mechanism described in Section 2.2.

Following Chaudhuri et al. (2011) and Wu et al. (2017), we study the case of logistic regression. In a logistic-regression model parameterized by vector $\theta\in\mathbb{R}^{d}$, we compute the confidence score for an input $x\in\mathcal{X}\subseteq\mathbb{R}^{d}$ as $h_{\theta}(x)=\mathsf{sigmoid}(\theta^{\intercal}x)$, where

[TABLE]

Recall that the classifier’s prediction is obtained by applying a threshold to the confidence score by Eq. 1, in this case as $f_{\theta}(x)=\mathbb{1}[\mathsf{sigmoid}(\theta^{\intercal}x)>0.5]$. Note that the quantity $\theta^{\intercal}x$ is interchangeable with confidence, as one can be obtained from the other using an invertible transformation. We show the exact relationship between disagreement and the scale of noise $\sigma$ in this setting:

Proposition 0.

Let $\theta_{\mathsf{np}}=T_{\mathsf{np}}(\mathcal{D})$ be a non-private parameter vector of a logistic-regression model. Suppose that the privatized $\theta_{\mathsf{priv}}$ is obtained using Gaussian noise of scale $\sigma$ as in Eq. 3. Then, the disagreement of a private logistic-regression model parameterized by $\theta_{\mathsf{priv}}$ is:

[TABLE]

We visualize the relationship in Fig. 2, assuming the input space is normalized so that $\|x\|=1$. There are two main takeaways from this result. First, disagreement is high when the level of privacy is high. Second, the level of multiplicity is unevenly distributed across input examples. This is because the exact relationship between multiplicity and privacy also depends on the confidence of the non-private model, $\theta_{\mathsf{np}}^{\intercal}\,x$, with lower-confidence examples generally having higher multiplicity in this setting. We note that, in this illustration, the simple relationship between confidence and predictive multiplicity is an artifact of normalized features, i.e., $\|x\|=1$. In general, examples with high-confidence predictions can display high predictive multiplicity after DP-ensuring training, as illustrated in Section 5.2.

Other methods for DP training, such as gradient perturbation (Abadi et al., 2016), are not as straightforward to analyze theoretically. In the next sections, we study predictive multiplicity of these algorithms using a Monte-Carlo method.

4. Measuring Predictive Multiplicity of Randomized Algorithms

Theoretically characterizing predictive multiplicity of DP algorithms beyond the output-perturbation mechanism and for more complex model classes is a challenging problem (see, e.g. (Hsu and Calmon, 2022, Section 4)). For instance, the accuracy and generalization behavior of the DP-SGD algorithm (Abadi et al., 2016) used for DP training of neural networks is an active area of research (e.g., (Wang et al., 2023)). Even in simpler model classes, where training amounts to solving a convex optimization problem (e.g., support vector machines), DP mechanisms such as objective perturbation (Chaudhuri et al., 2011) display a complex interplay between privacy, accuracy, and distortion of model parameters.

For these theoretically intractable cases, we adopt a simple Monte-Carlo strategy (D’Amour et al., 2020; Black et al., 2021): Train multiple models on the same dataset with different randomization seeds, and compute statistics of the outputs of these models. Note that this procedure does not preserve differential privacy, which we discuss in more detail in Section 7.2.

In this section, we formalize this simple and intuitive approach, and provide the first sample complexity bound for estimating predictive multiplicity. Our bound has a closed-form expression, so a practitioner can use it to determine how many re-trainings are required to estimate predictive multiplicity up to a given approximation error.

At first, re-training might appear as a blunt approach for analyzing predictive multiplicity in DP. Our results indicate that this is not the case. Surprisingly, we prove that, if one wants to estimate disagreement in Eq. 4 for $k$ input examples, the number of required re-trainings increases logarithmically in $k$. This result demonstrates that re-training can be an effective strategy to estimate predictive multiplicity regardless of the intricacies of a specific DP mechanism, and that a moderate number of re-trainings is sufficient to estimate disagreement for a large number of examples.

Recall that, according to Proposition 1, disagreement of an example $x$ is proportional to the variance of outputs within the model distribution $P_{T(\mathcal{D})}$. We use this connection to provide an unbiased estimator for disagreement.

Proposition 0.

Suppose we have $m$ models sampled from the model distribution: $\theta_{1},\theta_{2},\ldots,\theta_{m}\sim P_{T(\mathcal{D})}$. Then, the following expression is an unbiased estimator for disagreement $\mu(x)$ for a single example $x\in\mathcal{X}$:

[TABLE]

where $\hat{p}_{x}=\frac{1}{m}\sum_{i=1}^{m}f_{\theta_{i}}(x)$ is the sample mean of $f_{\theta}(x)$.

How many models $\theta_{1},\theta_{2},\ldots,\theta_{m}$ do we need to sample in order to estimate disagreement? To answer this, we provide an upper bound on estimation accuracy given the number of samples from the model distribution, as well as a bound on the number of samples required for a given level of estimation accuracy.

Proposition 0.

For $m$ models sampled from the model distribution, $\theta_{1},\theta_{2},\ldots,\theta_{m}\sim P_{T(\mathcal{D})}$, with probability at least $1-\rho$, for $\rho\in(0,1]$ the additive estimation error $\alpha~{}\triangleq~{}|\hat{\mu}(x)-\mu(x)|$ satisfies:

[TABLE]

For example, this bound yields that 5,000 re-trainings result in the estimation error of at most $0.08$ with probability $95\%$. In Section A.3, we provide a closed-form expression for computing the number of samples $m$ required to achieve a given error level $\alpha$. We also provide a visualization of the bound in Fig. 8(a) (Appendix).

In practice, one might need to estimate disagreement for multiple examples, e.g., to compute average disagreement over a test dataset. When doing so naïvely, the re-training costs could mount to infeasible levels if we assume that each estimation requires the same number of models, $m$, for each input example. In contrast, we show that in such cases sample complexity grows only logarithmically.

Proposition 0.

Let $x_{1},x_{2},\ldots,x_{k}\in\mathcal{X}$. If $\theta_{1},\theta_{2},\ldots,\theta_{m}\sim P_{T(\mathcal{D})}$ are i.i.d. samples from the model distribution, then with probability at least $1-\rho$, for $\rho\in(0,1]$ the maximum additive error satisfies:

[TABLE]

This positive result shows that auditing models for predictive multiplicity for large populations and datasets is practical, as the sample complexity grows slowly in the number of examples.

5. Empirical Studies

In this section, we empirically explore the predictive multiplicity of DP algorithms. We use a low-dimensional synthetic dataset in order to visualize the level of multiplicity across the input space. To study predictive-multiplicity effects in realistic settings, we use real-world tabular datasets representative of high-stakes domains, namely lending and healthcare, and one image dataset. The code to reproduce our experiments is available at:

github.com/spring-epfl/dp_multiplicity

5.1. Experimental Setup

Datasets and Tasks

We use the following datasets:

• •

A Synthetic dataset containing data belonging to two classes with class-conditional distributions $X_{0}\sim\mathcal{N}(\mu_{0},\Sigma_{0})$ and $X_{1}\sim\mathcal{N}(\mu_{1},\Sigma_{1})$, respectively. We set the distribution parameters to be:

[TABLE]

The classes in this synthetic dataset are well-separable by a linear model (see Fig. 1)

• •

Credit Approval tabular dataset (Credit). The task is to predict whether a credit card application should be approved or rejected based on several attributes which describe the application and the applicant.

• •

Contraceptive Method Choice tabular dataset (Contraception) based on 1987 National Indonesia Contraceptive Prevalence Survey. The task is to predict the choice of a contraception method based on demographic and socio-economic characteristics of a married couple.

• •

Mammographic Mass tabular dataset (Mammography) collected at the Institute of Radiology of the University Erlangen-Nuremberg in 2003 – 2006. The task is to predict whether a screened tumor is malignant or benign based on several clinical attributes.

• •

Dermatology tabular dataset. The task is to predict a dermatological disease based on a set of clinical and histopathological attributes.

• •

CIFAR-10 (Krizhevsky et al., 2009), an image dataset of pictures labeled as one of ten classes. The task is to predict the class.

We take the realistic tabular datasets (Credit, Contraception, Mammography, and Dermatology) from the University of California Irvine Machine Learning (UCIML) dataset repository (Dua and Graff, 2017). In Appendix B, we provide additional details about processing of the datasets, and a summary of their characteristics (Table 1).

For the synthetic dataset, we obtain the training dataset by sampling 1,000 examples from each of the distributions. In order to have precise estimates of population accuracy, we sample a larger test dataset of 20,000 examples. For tabular datasets, we use a random $75\%$ subset for training, and use the rest as a held-out test dataset for model evaluations. For CIFAR-10, we use the default 50K/10K train-test split.

Models and Training Algorithms

For the synthetic and tabular datasets, we use logistic regression with objective perturbation (Chaudhuri et al., 2011). For the image dataset, we train a convolutional neural network on ScatterNet features (Oyallon and Mallat, 2015) using DP-SGD (Abadi et al., 2016), following the approach by Tramer and Boneh (2021). We provide more details in Appendix B.

Metrics

The goal of our experiments is to quantify predictive multiplicity and explain the factors which impact it. For all settings, we measure disagreement to capture the dissimilarity of predictions, and predictive performance of the models to quantify the effect of performance on multiplicity. Concretely, we measure:

• •

Disagreement for examples on a test dataset, computed using the unbiased estimator in Section 4. As this disagreement metric is tailored to binary classification, we use a special procedure for the ten-class task on CIFAR-10: we treat each multi-class classifier as ten binary classifiers, and we report average disagreement across those ten per-class classifiers. Additionally, in Appendix B, we also report predictive multiplicity in terms of confidence scores instead of predictions following the recent approach by Watson-Daniels et al. (2023).

• •

Performance on a test dataset. For tabular datasets, we report the standard area under the ROC Curve (AUC for short). For CIFAR-10, we report accuracy.

Experiment Outline

For a given dataset and a value of the privacy parameter $\varepsilon$, we train multiple models on exactly the same data with different randomization seeds.

For the synthetic and tabular datasets, we use several values of $\varepsilon$ between 0.5 (which provides a good guaranteed level of privacy (see, e.g. Wood et al., 2018, Section 4)) and 2.5, with $\delta=0$. For each value of $\varepsilon$ we train $m=5{\small,}000$ models. For CIFAR-10, we train $m=50$ neural-network models because of computational constraints. We use DP-SGD parameters that provide privacy guarantees from $\varepsilon\approx 2$ to $\varepsilon\approx 7$ at the standard choice of $\delta=10^{-5}$.

5.2. Predictive Multiplicity and Privacy

First, we empirically study how multiplicity evolves with increasing privacy. In Fig. 1, we visualize the two-dimensional synthetic examples and their disagreement for different privacy levels. As privacy increases, so do the areas for which model decisions exhibit high disagreement (darker areas). Although the regions with higher disagreement correlate with model confidence and accuracy, the level of privacy contributes significantly. For instance, some points which are relatively far from the decision boundary, which means they are confidently classified as either class, can nevertheless have high predictive multiplicity.

Fig. 3 shows the experimental results for our tabular datasets and CIFAR-10. On the left side, we show the relationship between the privacy level and performance. On the right, between the privacy level and disagreement. As with the theoretical analysis and the results on synthetic data, we can clearly see that models with higher level of privacy (low $\varepsilon$) invariably exhibit higher predictive multiplicity. Notably, even for datasets such as Mammography and CIFAR-10 for which average disagreement is relatively low, there exist examples whose disagreement is 100%. See Table 2 in the Appendix for detailed information on the distribution of the disagreement values across the test data.

Implications. The increase in the privacy level results in making more decisions which are partially or fully explained by randomness in training. Let us give an example with a concrete data record from the Mammography dataset representing a 56-year-old patient labeled as having a malignant tumor. Classifiers with low level of privacy $\varepsilon=2.5$ predict the correct malignant class for this individual most of the time (approx. 55% disagreement). If we set the level of privacy to the high $\varepsilon=0.5$, this record is classified close to 42% of the time as benign, and 58% of the time as malignant (approx. 97% disagreement). Thus, if one were to use a model with the high level of privacy to inform treatment of this patient, the model’s decision would have been close in its utility to a coin flip.

5.3. What Causes the Increase in Predictive Multiplicity?

In the previous section, we showed that the increase in privacy causes an increase in predictive multiplicity. It is not clear, however, what is the exact mechanism through which DP impacts predictive multiplicity. Hypothetically, the contribution to multiplicity could be through two pathways:

1. (1)

Direct: The increase in predictive multiplicity is the result of the variability in the learning process stemming from randomization, regardless of the performance decrease. 2. (2)

Indirect: The increase in predictive multiplicity is the result of the decrease in performance.

These two options are not mutually exclusive, and it is possible that both play a role. In both cases, the desire for a given level of privacy—which determines the degree of randomization added during training—is ultimately the cause of the increase in multiplicity. Nevertheless, how randomization contributes to the increase has practical implications: If our results are explained by pathway (2), we should be able to reduce the impact of privacy on predictive multiplicity by designing algorithms which achieve better accuracy at the same privacy level.

For output perturbation, our analysis in Section 3 shows that multiplicity is directly caused by randomization—pathway (1)—as only the privacy level, confidence, and the norm of a predicted example impact disagreement. Therefore, performance does not have a direct impact on predictive multiplicity in output perturbation.

In Fig. 4, to quantify the impact of performance on predictive multiplicity for the case of objective perturbation, we show the top 5% disagreement values for varying levels of accuracy on the synthetic dataset. We use the synthetic dataset to ensure that test accuracy estimates are reliable, as we have a large test dataset in this case. We see that, for a given level of accuracy, different privacy parameters can result in different disagreement. This suggests that randomization caused by DP training can have a direct effect on predictive multiplicity, so we observe pathway (1).

Implications. This observation indicates that there exist cases for which improving accuracy of a DP-ensuring algorithm at a given privacy level will not necessarily lower predictive multiplicity.

5.4. Disparities in Predictive Multiplicity

The visualizations in Fig. 1 show that different examples can exhibit highly varying levels of predictive multiplicity. This observation holds for real-world datasets too. Fig. 5(a) shows the distributions of the disagreement values across the population of examples in the test data for tabular datasets. For example, for lower privacy levels (high $\varepsilon$) on the Contraception dataset, there are groups of individuals with different values of predictive multiplicity. As the level of privacy increases (low $\varepsilon$), the disagreement tends to concentrate around 1, with decisions for a majority of examples largely explained by randomness in training.

Next, we verify if the differences in the level of disagreement also exist across demographic groups. In Fig. 5(b), we show average disagreement across points from three different age groups in the Contraception dataset. As before, for low levels of privacy (high $\varepsilon$) we see more disparity in disagreement. The disparities even out as we increase the privacy level (low $\varepsilon$), with groups having average disagreement closer to 1. Thus, disagreement is not only unevenly distributed across individuals, but across salient demographic groups.

Implications. As some groups and individuals can have higher predictive multiplicity than others, evaluations of training algorithms in terms of their predictive multiplicity must account for such disparities. For instance, our experiments on the Contraception dataset (in Fig. 5(b)) show that, for different privacy levels, decisions for individuals in the 16–30 age bracket exhibit higher predictive multiplicity than of patients between 30 and 40 years old. Predictions for individuals under 30, therefore, systematically exhibit more dependence on randomness in training than on the relevant features for prediction. This highlights the need to conduct disaggregated evaluations as opposed to only evaluating average disagreement on whole datasets.

6. Related work

Rashomon Effect and Predictive Multiplicity.

The Rashomon effect, observed and termed by Breiman (2001), describes the phenomenon where a multitude of distinct models achieve similar average loss. The Rashomon effect occurs even for simple models such as linear regression, decision trees, and shallow neural networks (Auer et al., 1995). When no privacy constraints are present, predictive multiplicity can be viewed as a facet of the Rashomon effect in classification tasks, where similarly-accurate models produce conflicting outputs. One of the main challenges in studying predictive multiplicity is measuring it. Semenova et al. (2022) proposed the Rashomon ratio to measure the Rashomon effect and used a Monte Carlo technique to sample decision tree models for estimation. Marx et al. (2020) quantified predictive multiplicity using optimization formulations to find the worst-case disagreement among all candidate models while controlling for accuracy. Recently, Hsu and Calmon (2022); Watson-Daniels et al. (2023) proposed other metrics for quantifying predictive multiplicity: Rashomon capacity and viable prediction range. Black et al. (2022) proposed measures of predictive multiplicity which are applicable to randomized learning. Our Proposition 3 complements the prior work by providing a closed-form expression for sample complexity of estimating predictive multiplicity which arises due to randomness in training.

Side Effects of Differential Privacy.

To the best of our knowledge, our work is the first one to study the properties of DP training in terms of predictive multiplicity. Multiple works, however, have studied other unintended consequences of private learning. In particular, a number of works (Bagdasaryan et al., 2019; Sanyal et al., 2022; Ganev et al., 2022) show that DP training comes at a cost of decreased performance for groups which are under-represented in the data. Relatedly, Cummings et al. (2019) show that DP training is incompatible with some notions of algorithmic fairness.

7. Discussion

Our theoretical and empirical results show that training with DP and, more broadly, applying randomization in training increases predictive multiplicity. We demonstrated that higher privacy levels result in higher multiplicity. If a training algorithm exhibits high predictive multiplicity for a given input example, the decisions supported by a model’s output for this example lose their justifiability: these decisions depend on the randomness used in training rather than on relevant properties or features of this example. The connection between privacy in learning and decision arbitrariness might not be obvious to practitioners. This lack of awareness is potentially damaging in high-stakes settings (e.g., medical diagnostics, lending, education), where decisions of significant—and potentially life-changing—consequence could be significantly influenced by randomness used to ensure privacy.

In this concluding section, we discuss whether predictive multiplicity is indeed a valid concern for DP-ensuring algorithms, and outline a path forward.

7.1. Can the Increase in Predictive Multiplicity be Beneficial?

Despite the harms of arbitrariness, one might argue that multiplicity can, in some cases, be beneficial.

Opportunities for Satisfying Desirable Properties Beyond Accuracy?

Black et al. (2022) and Semenova et al. (2022) argue that multiplicity presents a valuable opportunity. In non-private training, the existence of many models that achieve comparable accuracy creates an opportunity for selecting a model which satisfies both an acceptable accuracy level and other useful properties beyond performance, such as fairness (Coston et al., 2021), interpretability (Fisher et al., 2019), or generalizability (Semenova et al., 2022). In order to leverage this opportunity, one needs to deliberately steer training towards the model which satisfies desirable properties beyond accuracy, or search the “Rashomon set” of good models (Fisher et al., 2019). However, with randomization alone (e.g., adding Gaussian noise to gradients in training), model designers cannot steer training without compromising DP guarantees, and can only arrive at a model which satisfies additional desirable properties by chance. Thus, this positive side of the multiplicity phenomenon is not necessarily present in DP-ensuring training.

It is an open problem to find whether specially-crafted noise distributions or post-processing techniques could be designed to provide the same level of privacy as the standard approaches, and at the same time attain additional useful properties such as fairness.

Predictive Multiplicity is Individually Fair?

Individual fairness (Dwork et al., 2012) is a formalization of the “treat like alike” principle: an individually fair classifier makes similar decisions for individuals who are thought to be similar. A way to formally satisfy individual fairness is, in fact, through randomization of decisions. This could lead to an argument that predictive multiplicity is individually fair. For instance, suppose that a predictive model used to assist with hiring decisions is applied to several individuals who are all equally qualified to get the job. Consider two possible decision rules for selecting the candidate to hire with different multiplicity levels. The first rule has high multiplicity: produce a random decision. The second rule has low multiplicity: select a candidate based on lexicographic order. As the second decision rule results in a breach of individual fairness and, possibly, a systemic exclusion of some candidates, the first rule with high multiplicity seems preferable.

This argument, however, only holds if there is randomness at the prediction stage. This is not the case for standard DP-ensuring algorithms such as the ones we study. DP training produces one deterministic classifier that is used for all predictions. Thus, once training is done, there is no randomization of decisions as in the example above. Thus, the decisions due to such DP-ensuring models are no different than arbitrary rules such as selection based on lexicographic ordering.

Overcoming the Algorithmic Leviathan?

Creel and Hellman (2022) consider a setting where different decision-making systems which have high impact on an individual’s livelihood, e.g., credit scoring systems from competing bureaus in the USA (Citron and Pasquale, 2014), are trained in ways that lead to all of them outputting the same decisions. This algorithmic monoculture would completely remove the possibility of accessing resources for some individuals, as turning to a competing decision-maker would not change the outcome. In this case, Creel and Hellman argue that high predictive multiplicity could be a desirable property as it enables to access resources across the decision-makers.

In some high-stakes settings, such as healthcare, an algorithmic monoculture might not pose a concern. Indeed, one would wish that predictive models used as a part of a diagnostic procedure for a disease output a consistent decision so that patients can be treated (or not treated) as needed. In this scenario, in fact, predictive multiplicity could potentially harm patients by either delaying a patient’s treatment, or recommending a treatment when the patient is healthy. In such settings, the positive impact of predictive multiplicity in avoiding an algorithmic Leviathan loses meaning.

Regardless of whether algorithmic monoculture is a legitimate concern or not for a given application, it is helpful for model designers and decision subjects to be informed of the level of predictive multiplicity, whether to gauge the likelihood of recourse, or brace for the arbitrariness of decisions.

7.2. Open Problems

Reporting Mutiplicity

Potential mitigations of the harms of predictive multiplicity could be to abstain from outputting a prediction with high multiplicity, or to communicate the magnitude of multiplicity to the stakeholders. Doing so is challenging: any sort of communication of disagreement values could partially reveal information about the privacy-sensitive training data and break DP guarantees. Consider, as before, the setting of using a predictive model to assist in a medical diagnosis. Whether a model abstains from predictions or outputs them along with disagreement estimates, there is a certain amount of information leakage about the training data to doctors. If the disagreement estimates are computed on privacy-sensitive data and are used without appropriate privatization—whether published or used to decide on abstention—they can reveal information about the data. To address this issue, one could use privacy-preserving technologies such as DP to abstain from making a prediction based on a high disagreement value or report the disagreement estimate in a privacy-preserving way. Studying whether effective privatization of disagreement computations is possible is an open problem for future work.

General Characterization of the Predictive-Multiplicity Costs of DP

We have theoretically characterized the predictive-multiplicity behavior of the output-perturbation mechanism as applied to logistic regression. Doing so for other mechanisms and model families is a non-trivial undertaking. In this work, we resort to empirical measurement with re-training. An open problem is finding whether we can characterize these behaviors for a wider range of model families, mechanisms, or even for any general mechanism which satisfies DP.

7.3. Recommendations Moving Forward

As discussed in the previous sections, existing techniques do not enable model designers to eliminate, or even mitigate, the implications of predictive multiplicity when using DP-ensuring models. We have pointed out which open problems would need to be solved in order to reduce the impact of predictive multiplicity in high-stakes privacy-sensitive scenarios. Until DP mechanisms that mitigate multiplicity become available, the negative effects of multiplicity can only be countered by auditing for multiplicity prior to deployment. Therefore, in order to understand the impact of privacy on the justifiability of model decisions, model designers should directly measure predictive multiplicity when using DP training, e.g., using the methods we introduce in Section 4. If at the desired level of privacy the training algorithm exhibits high predictive multiplicity (either in general or for certain populations), model designers should carefully consider whether the use of such models is justified in the first place.

Acknowledgements.

This work is partially funded by the Swiss National Science Foundation under grant 200021-188824, and the US National Science Foundation under grants CAREER 1845852, FAI 2040880, and CIF 1900750. Hsiang Hsu acknowledges support from Meta Ph.D. Fellowship. The authors would like to thank Salil Vadhan, Borja Balle, Jakab Tardos, and the anonymous reviewers at FAccT 2023 for their helpful feedback.

Appendix A Omitted Proofs and Derivations

A.1. Section 2

First, we provide an explanation on the range of disagreement without normalization:

Proposition 0 (Range of non-normalized disagreement).

The expression $\Pr[f_{\theta}(x)\neq f_{\theta^{\prime}}(x)]$ has range of $[0,0.5]$.

Proof.

As $f_{\theta}(x)\in\{0,1\}$, we can assume $\Pr[f_{\theta}(x)=1]=p$, and thus $\Pr[f_{\theta}(x)\neq f_{\theta^{\prime}}(x)]=\Pr[f_{\theta}(x)=0\;\text{and}\;f_{\theta^{\prime}}(x)=1]+\Pr[f_{\theta}(x)=1\;\text{and}\;f_{\theta^{\prime}}(x)=0]=2p(1-p)\in[0,0.5]$. ∎

Next, we provide a proof that disagreement is proportional to variance in our setup:

Proof of Proposition 1.

As $f_{\theta}(x)\in\{0,1\}$, we have that

[TABLE]

where $p_{x}(1-p_{x})$ is the population variance of the r.v. $f_{\theta}(x)\sim\mathrm{Bernoulli}(p_{x})$. ∎

A.2. Section 3

Proof of Proposition 1.

First, observe that the expression

[TABLE]

can be expressed as:

[TABLE]

Denoting by $\xi~{}\triangleq~{}\mathcal{N}(0,1)$ and $\xi_{d}~{}\triangleq~{}\mathcal{N}(0,I_{d})$, we can see that the score $\theta_{\mathsf{priv}}^{\intercal}\,x$ is equal to:

[TABLE]

Plugging in the closed form in Eq. 14 into Eq. 13, we get:

[TABLE]

∎

A.3. Section 4

Proof of Proposition 1.

The $\nicefrac{{1}}{{m-1}}$ term comes from Bessel’s correction. Observe that

[TABLE]

Therefore, $\mathbb{E}[\hat{\mu}(x)]=4p_{x}(1-p_{x})=\mu(x)$. ∎

Proof of Proposition 2.

As $\hat{\mu}(x)$ is a continuous transformation of $\hat{p}_{x}$, we could bound the deviation $|\hat{\mu}(x)-\mu(x)|$ by $|\hat{p}_{x}-p_{x}|$. Suppose $\hat{p}_{x}=p_{x}+\nu$ and $\nu\in[-\eta,\eta]$, we have

[TABLE]

By Chernoff-Hoeffding inequality, we have the following concentration bounds on the sample mean $\hat{p}_{x}$,

[TABLE]

Thus with probability at least $1-\rho$, we have:

[TABLE]

Combining LABEL:eq:variance-estimation-1 and Eq. 18, we have

[TABLE]

Plugging $\eta=\sqrt{\log(2/\rho)/2m}$ into Eq. (19) yields the desired result. Note that by solving $\frac{1}{(m-1)}+\frac{4m}{m-1}\eta(1+\eta)\leq\alpha$ with $\eta=\sqrt{\log(2/\rho)/2m}$ with conditions $\alpha>0$ and $0<\rho<1$, we have:

[TABLE]

where $t=\log(2/\rho).$ ∎

Proof of Proposition 3.

Since the samples are i.i.d., we have the following union bound for the concentration of sample mean,

[TABLE]

Therefore, with probability $1-\rho$, $|\hat{p}_{x_{i}}-p_{x_{i}}|\leq\sqrt{\log(2k/\rho)/2m}$ for $i=1,\ldots,k$, and the desired result follows the derivation in Proposition 2.

∎

Appendix B Additional Experimental Details

B.1. Details on the Experimental Setup

Datasets

For illustrative purposes, we use the following classes as our target labels. For the Credit dataset, we use “Approved” as the target label. For the Contraception dataset, we use “long-term method”. For the dermatology dataset, we use “seboreic dermatitis” diagnosis. For the Mammography dataset, we use “malignant”.

CIFAR-10

We use the convolutional neural network trained over the ScatterNet features (Oyallon and Mallat, 2015) following Tramer and Boneh (2021, Table 9, Appendix). We use DP-SGD with batch size of 2048, learning rate of 4, Nesterov momentum of 0.9, and gradient clipping norm of 0.1. We vary the gradient noise multiplier $\sigma$ to achieve the privacy levels of $\varepsilon\approx 2.22,2.73,3.62.4.39,5.59$ as computed by the Moments accountant (Abadi et al., 2016).

Software

We use the following software:

• •

diffprivlib (Holohan et al., 2019) for the implementation of objective-perturbation for logistic regression.

• •

PyTorch (Paszke et al., 2019) for implementing neural networks.

• •

opacus (Yousefpour et al., 2021) for training PyTorch neural networks with DP-SGD.

• •

numpy (Harris et al., 2020), scipy (Virtanen et al., 2020), and pandas (pandas development team, 2020) for numeric analyses.

• •

seaborn (Waskom, 2021) for visualizations.

B.2. Multiplicity of Predictions vs. Scores

Recall that the models we consider are not only capable of outputting a binary prediction but also a confidence score. The disagreement metric in Eq. 4, however, only uses the predictions after applying a threshold. To verify if the trends we observe persist also at the level of confidence scores, we additionally evaluate viable prediction range, a metric for measuring multiplicity of the confidence scores proposed by Watson-Daniels et al. (2023):

[TABLE]

Fig. 6 shows the viable prediction range for different values in the input space for logistic regression trained with objective perturbation on our synthetic dataset. The regions with high viable prediction range overlap with the regions with high disagreement (see Fig. 1). This is also consistent with the results on the tabular datasets, for which Fig. 7 shows both disagreement and viable prediction range increasing on average as the level of privacy increases.

Implications. Models trained with a high level of privacy exhibit high multiplicity both of their confidence scores (in terms of viable prediction range) and of “hard” predictions after applying a threshold (in terms of disagreement).

B.3. Additional Figures and Tables

The rest of the document contains additional figures and tables.