We give new upper and lower bounds on the minimax sample complexity of differentially private mean estimation of distributions with bounded $k$-th moments. Roughly speaking, in the univariate case, we show that $n = Θ\left(\frac{1}{α^2} + \frac{1}{α^{\frac{k}{k-1}}\varepsilon}\right)$ samples are necessary and sufficient to estimate the mean to $α$-accuracy under $\varepsilon$-differential privacy, or any of its common relaxations. This result demonstrates a qualitatively different behavior compared to estimation absent privacy constraints, for which the sample complexity is identical for all $k \geq 2$. We also give algorithms for the multivariate setting whose sample complexity is a factor of $O(d)$ larger than the univariate case.
University of Waterloo（滑铁卢大学）；Meta；Cornell University（康奈尔大学）
We study stochastic convex optimization with heavy-tailed data under the constraint of differential privacy (DP). 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 \cite{WangXDX20}, 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 concentrated DP for convex and strongly convex loss functions. Along the way, we derive new algorithms for private mean estimat…
King Abdullah University of Science and Technology（阿卜杜拉国王科技大学）；State University of New York at Buffalo（纽约州立大学布法罗分校）
We study the problem of Differentially Private Stochastic Convex Optimization (DP-SCO) with heavy-tailed data. Specifically, we focus on the $\ell_1$-norm linear regression in the $ε$-DP model. While most of the previous work focuses on the case where the loss function is Lipschitz, here we only need to assume the variates has bounded moments. Firstly, we study the case where the $\ell_2$ norm of data has bounded second order moment. We propose an algorithm which is based on the exponential mechanism and show that it is possible to achieve an upper bound of $\tilde{O}(\sqrt{\frac{d}{nε}})$ (wi…
State University of New York at Buffalo（纽约州立大学布法罗分校）；King Abdullah University of Science and Technology（阿卜杜拉国王科技大学）；MIT（麻省理工学院）
In this paper, we consider the problem of designing Differentially Private (DP) algorithms for Stochastic Convex Optimization (SCO) on heavy-tailed data. The irregularity of such data violates some key assumptions used in almost all existing DP-SCO and DP-ERM methods, resulting in failure to provide the DP guarantees. To better understand this type of challenges, we provide in this paper a comprehensive study of DP-SCO under various settings. First, we consider the case where the loss function is strongly convex and smooth. For this case, we propose a method based on the sample-and-aggregate f…
As one of the most fundamental problems in machine learning, statistics and differential privacy, Differentially Private Stochastic Convex Optimization (DP-SCO) has been extensively studied in recent years. However, most of the previous work can only handle either regular data distribution or irregular data in the low dimensional space case. To better understand the challenges arising from irregular data distribution, in this paper we provide the first study on the problem of DP-SCO with heavy-tailed data in the high dimensional space. In the first part we focus on the problem over some polyto…
Modern policy gradient algorithms such as Proximal Policy Optimization (PPO) rely on an arsenal of heuristics, including loss clipping and gradient clipping, to ensure successful learning. These heuristics are reminiscent of techniques from robust statistics, commonly used for estimation in outlier-rich (``heavy-tailed'') regimes. In this paper, we present a detailed empirical study to characterize the heavy-tailed nature of the gradients of the PPO surrogate reward function. We demonstrate that the gradients, especially for the actor network, exhibit pronounced heavy-tailedness and that it in…
Privacy of SGD under Gaussian or Heavy-Tailed Noise: Guarantees without Gradient Clipping
Inria, CNRS, Ecole Normale Supérieure PSL Research University（法国国家科学研究中心、巴黎高等师范学院PSL研究大学）；Department of Management Science and Information Systems（管理科学与信息系统系）；Faculty of Engineering and Natural Sciences（工程与自然科学学院）；Department of Mathematics（数学系）
University of Wisconsin-Madison（威斯康星大学麦迪逊分校）；University of California, San Diego（加州大学圣地亚哥分校）；Simons Institute for the Theory of Computing（西蒙斯计算理论研究所）
We study the fundamental task of outlier-robust mean estimation for heavy-tailed distributions in the presence of sparsity. Specifically, given a small number of corrupted samples from a high-dimensional heavy-tailed distribution whose mean $μ$ is guaranteed to be sparse, the goal is to efficiently compute a hypothesis that accurately approximates $μ$ with high probability. Prior work had obtained efficient algorithms for robust sparse mean estimation of light-tailed distributions. In this work, we give the first sample-efficient and polynomial-time robust sparse mean estimator for heavy-taile…
We consider the task of heavy-tailed statistical estimation given streaming $p$-dimensional samples. This could also be viewed as stochastic optimization under heavy-tailed distributions, with an additional $O(p)$ space complexity constraint. We design a clipped stochastic gradient descent algorithm and provide an improved analysis, under a more nuanced condition on the noise of the stochastic gradients, which we show is critical when analyzing stochastic optimization problems arising from general statistical estimation problems. Our results guarantee convergence not just in expectation but wi…
Northwestern University（西北大学）；University of Washington（华盛顿大学）
The empirical success of deep learning is often attributed to SGD's mysterious ability to avoid sharp local minima in the loss landscape, as sharp minima are known to lead to poor generalization. Recently, empirical evidence of heavy-tailed gradient noise was reported in many deep learning tasks, and it was shown in Şimşekli (2019a,b) that SGD can escape sharp local minima under the presence of such heavy-tailed gradient noise, providing a partial solution to the mystery. In this work, we analyze a popular variant of SGD where gradients are truncated above a fixed threshold. We show that it ac…
Accelerated stochastic first-order method for convex optimization under heavy-tailed noise
Department of Mathematics, Linköping University, Sweden（林雪平大学数学系）；Department of Industrial and Systems Engineering, University of Minnesota, USA（明尼苏达大学工业与系统工程系）
Second-order Optimization under Heavy-Tailed Noise: Hessian Clipping and Sample Complexity Limits
KAUST Center of Excellence for Generative AI（知识增强人工智能卓越中心）；ETH Zurich（苏黎世联邦理工学院）；ETH AI Center（苏黎世联邦理工学院人工智能中心）；Georgia Institute of Technology（佐治亚理工学院）
In this paper, we propose a new accelerated stochastic first-order method called clipped-SSTM for smooth convex stochastic optimization with heavy-tailed distributed noise in stochastic gradients and derive the first high-probability complexity bounds for this method closing the gap in the theory of stochastic optimization with heavy-tailed noise. Our method is based on a special variant of accelerated Stochastic Gradient Descent (SGD) and clipping of stochastic gradients. We extend our method to the strongly convex case and prove new complexity bounds that outperform state-of-the-art results …
