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Register a new userImproved Rates for Differentially Private Stochastic Convex Optimization with Heavy-Tailed Data
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Huanyu Zhang
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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We study stochastic convex optimization with heavy-tailed data under the constraint of differential privacy. Most prior work on this problem is restricted to the case where the loss function is Lipschitz. Instead, as introduced by Wang, Xiao, Devadas, and Xu, we study general convex loss functions with the assumption that the distribution of gradients has bounded $k$-th moments. We provide improved upper bounds on the excess population risk under approximate differential privacy of $tilde{O}left(sqrt{frac{d}{n}}+left(frac{d}{epsilon n}right)^{frac{k-1}{k}}right)$ and $tilde{O}left(frac{d}{n}+left(frac{d}{epsilon n}right)^{frac{2k-2}{k}}right)$ for convex and strongly convex loss functions, respectively. We also prove nearly-matching lower bounds under the constraint of pure differential privacy, giving strong evidence that our bounds are tight.
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We study differentially private (DP) algorithms for stochastic convex optimization (SCO). In this problem the goal is to approximately minimize the population loss given i.i.d. samples from a distribution over convex and Lipschitz loss functions. A long line of existing work on private convex optimization focuses on the empirical loss and derives asymptotically tight bounds on the excess empirical loss. However a significant gap exists in the known bounds for the population loss. We show that, up to logarithmic factors, the optimal excess population loss for DP algorithms is equal to the larger of the optimal non-private excess population loss, and the optimal excess empirical loss of DP algorithms. This implies that, contrary to intuition based on private ERM, private SCO has asymptotically the same rate of $1/sqrt{n}$ as non-private SCO in the parameter regime most common in practice. The best previous result in this setting gives rate of $1/n^{1/4}$. Our approach builds on existing differentially private algorithms and relies on the analysis of algorithmic stability to ensure generalization.
In shuffle privacy, each user sends a collection of randomized messages to a trusted shuffler, the shuffler randomly permutes these messages, and the resulting shuffled collection of messages must satisfy differential privacy. Prior work in this model has largely focused on protocols that use a single round of communication to compute algorithmic primitives like means, histograms, and counts. In this work, we present interactive shuffle protocols for stochastic convex optimization. Our optimization protocols rely on a new noninteractive protocol for summing vectors of bounded $ell_2$ norm. By combining this sum subroutine with techniques including mini-batch stochastic gradient descent, accelerated gradient descent, and Nesterovs smoothing method, we obtain loss guarantees for a variety of convex loss functions that significantly improve on those of the local model and sometimes match those of the central model.
Finding efficient, easily implementable differentially private (DP) algorithms that offer strong excess risk bounds is an important problem in modern machine learning. To date, most work has focused on private empirical risk minimization (ERM) or private population loss minimization. However, there are often other objectives--such as fairness, adversarial robustness, or sensitivity to outliers--besides average performance that are not captured in the classical ERM setup. To this end, we study a completely general family of convex, Lipschitz loss functions and establish the first known DP excess risk and runtime bounds for optimizing this broad class. We provide similar bounds under additional assumptions of smoothness and/or strong convexity. We also address private stochastic convex optimization (SCO). While $(epsilon, delta)$-DP ($delta > 0$) has been the focus of much recent work in private SCO, proving tight population loss bounds and runtime bounds for $(epsilon, 0)$-DP remains a challenging open problem. We provide the tightest known $(epsilon, 0)$-DP population loss bounds and fastest runtimes under the presence of (or lack of) smoothness and strong convexity. Our methods extend to the $delta > 0$ setting, where we offer the unique benefit of ensuring differential privacy for arbitrary $epsilon > 0$ by incorporating a new form of Gaussian noise. Finally, we apply our theory to two learning frameworks: tilted ERM and adversarial learning. In particular, our theory quantifies tradeoffs between adversarial robustness, privacy, and runtime. Our results are achieved using perhaps the simplest DP algorithm: output perturbation. Although this method is not novel conceptually, our novel implementation scheme and analysis show that the power of this method to achieve strong privacy, utility, and runtime guarantees has not been fully appreciated in prior works.
In this paper we study the problem of stochastic multi-armed bandits (MAB) in the (local) differential privacy (DP/LDP) model. Unlike the previous results which need to assume bounded reward distributions, here we mainly focus on the case the reward distribution of each arm only has $(1+v)$-th moment with some $vin (0, 1]$. In the first part, we study the problem in the central $epsilon$-DP model. We first provide a near-optimal result by developing a private and robust Upper Confidence Bound (UCB) algorithm. Then, we improve the result via a private and robust version of the Successive Elimination (SE) algorithm. Finally, we show that the instance-dependent regret bound of our improved algorithm is optimal by showing its lower bound. In the second part of the paper, we study the problem in the $epsilon$-LDP model. We propose an algorithm which could be seen as locally private and robust version of the SE algorithm, and show it could achieve (near) optimal rates for both instance-dependent and instance-independent regrets. All of the above results can also reveal the differences between the problem of private MAB with bounded rewards and heavy-tailed rewards. To achieve these (near) optimal rates, we develop several new hard instances and private robust estimators as byproducts, which might could be used to other related problems. Finally, experimental results also support our theoretical analysis and show the effectiveness of our algorithms.
We study the differentially private Empirical Risk Minimization (ERM) and Stochastic Convex Optimization (SCO) problems for non-smooth convex functions. We get a (nearly) optimal bound on the excess empirical risk and excess population loss with subquadratic gradient complexity. More precisely, our differentially private algorithm requires $O(frac{N^{3/2}}{d^{1/8}}+ frac{N^2}{d})$ gradient queries for optimal excess empirical risk, which is achieved with the help of subsampling and smoothing the function via convolution. This is the first subquadratic algorithm for the non-smooth case when $d$ is super constant. As a direct application, using the iterative localization approach of Feldman et al. cite{fkt20}, we achieve the optimal excess population loss for stochastic convex optimization problem, with $O(min{N^{5/4}d^{1/8},frac{ N^{3/2}}{d^{1/8}}})$ gradient queries. Our work makes progress towards resolving a question raised by Bassily et al. cite{bfgt20}, giving first algorithms for private ERM and SCO with subquadratic steps. We note that independently Asi et al. cite{afkt21} gave other algorithms for private ERM and SCO with subquadratic steps.
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