Stochastic gradient descent ascent (SGDA) and its variants have been the workhorse for solving minimax problems. However, in contrast to the well-studied stochastic gradient descent (SGD) with differential privacy (DP) constraints, there is little work on understanding the generalization (utility) of SGDA with DP constraints. In this paper, we use the algorithmic stability approach to establish the generalization (utility) of DP-SGDA in different settings. In particular, for the convex-concave setting, we prove that the DP-SGDA can achieve an optimal utility rate in terms of the weak primal-dual population risk in both smooth and non-smooth cases. To our best knowledge, this is the first-ever-known result for DP-SGDA in the non-smooth case. We further provide its utility analysis in the nonconvex-strongly-concave setting which is the first-ever-known result in terms of the primal population risk. The convergence and generalization results for this nonconvex setting are new even in the non-private setting. Finally, numerical experiments are conducted to demonstrate the effectiveness of DP-SGDA for both convex and nonconvex cases.
In the field of machine learning, many problems can be formulated as the minimax problem, including reinforcement learning, generative adversarial networks, to just name a few. So the minimax problem has attracted a huge amount of attentions from researchers in recent decades. However, there is relatively little work on studying the privacy of the general minimax paradigm. In this paper, we focus on the privacy of the general minimax setting, combining differential privacy together with minimax optimization paradigm. Besides, via algorithmic stability theory, we theoretically analyze the high …
In this work, we study local minimax convergence estimation rates subject to $ε$-differential privacy. Unlike worst-case rates, which may be conservative, algorithms that are locally minimax optimal must adapt to easy instances of the problem. We construct locally minimax differentially private estimators for one-parameter exponential families and estimating the tail rate of a distribution. In these cases, we show that optimal algorithms for simple hypothesis testing, namely the recent optimal private testers of Canonne et al. (2019), directly inform the design of locally minimax estimation al…
In this paper, we consider nonconvex minimax optimization, which is gaining prominence in many modern machine learning applications such as GANs. Large-scale edge-based collection of training data in these applications calls for communication-efficient distributed optimization algorithms, such as those used in federated learning, to process the data. In this paper, we analyze Local stochastic gradient descent ascent (SGDA), the local-update version of the SGDA algorithm. SGDA is the core algorithm used in minimax optimization, but it is not well-understood in a distributed setting. We prove th…
School of Computer Science（计算机学院）；Wuhan University（武汉大学）；The University of Sydney（悉尼大学）；School of Cyber Science and Technology（网络空间科学与技术学院）；Shenzhen Campus of Sun Yat-sen University（中山大学深圳校区）；Nanyang Technological University（南洋理工大学）
Local SGD is a promising approach to overcome the communication overhead in distributed learning by reducing the synchronization frequency among worker nodes. Despite the recent theoretical advances of local SGD in empirical risk minimization, the efficiency of its counterpart in minimax optimization remains unexplored. Motivated by large scale minimax learning problems, such as adversarial robust learning and training generative adversarial networks (GANs), we propose local Stochastic Gradient Descent Ascent (local SGDA), where the primal and dual variables can be trained locally and averaged…
In this paper, we develop a general framework to design differentially private expectation-maximization (EM) algorithms in high-dimensional latent variable models, based on the noisy iterative hard-thresholding. We derive the statistical guarantees of the proposed framework and apply it to three specific models: Gaussian mixture, mixture of regression, and regression with missing covariates. In each model, we establish the near-optimal rate of convergence with differential privacy constraints, and show the proposed algorithm is minimax rate optimal up to logarithm factors. The technical tools …
Provable Responsible AI and Data Analytics (PRADA) Lab（可证明责任AI和数据分析实验室）；King Abdullah University of Science and Technology（卡瓦尔大学科学与技术大学）；State University of New York at Buffalo（纽约州立大学布法罗分校）；Technische Universität Berlin（柏林技术大学）
Center for Applied Statistics and School of Statistics, Renmin University of China（应用统计中心和中国人民大学统计学院）；Department of Statistics and Data Science, National University of Singapore（统计与数据科学系，新加坡国立大学）
We study the problem of multi-armed bandits with $ε$-global Differential Privacy (DP). First, we prove the minimax and problem-dependent regret lower bounds for stochastic and linear bandits that quantify the hardness of bandits with $ε$-global DP. These bounds suggest the existence of two hardness regimes depending on the privacy budget $ε$. In the high-privacy regime (small $ε$), the hardness depends on a coupled effect of privacy and partial information about the reward distributions. In the low-privacy regime (large $ε$), bandits with $ε$-global DP are not harder than the bandits without p…
University of Science and Technology of China（中国科学技术大学）；State University of New York at Buffalo（纽约州立大学布法罗分校）；Guangzhou University（广州大学）；Zhejiang University（浙江大学）；King Abdullah University of Science and Technology（阿卜杜拉国王科技大学）
This paper takes an initial step to systematically investigate the generalization bounds of algorithms for solving nonconvex-(strongly)-concave (NC-SC/NC-C) stochastic minimax optimization measured by the stationarity of primal functions. We first establish algorithm-agnostic generalization bounds via uniform convergence between the empirical minimax problem and the population minimax problem. The sample complexities for achieving $ε$-generalization are $\tilde{\mathcal{O}}(dκ^2ε^{-2})$ and $\tilde{\mathcal{O}}(dε^{-4})$ for NC-SC and NC-C settings, respectively, where $d$ is the dimension and…
ETH Zurich（苏黎世联邦理工学院）；Max Planck ETH Center for Learning Systems（马克斯·普朗克-苏黎世联邦理工学习系统中心）；Amazon（亚马逊）；University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；University of Washington（华盛顿大学）
We study differentially private (DP) algorithms for smooth stochastic minimax optimization, with stochastic minimization as a byproduct. The holy grail of these settings is to guarantee the optimal trade-off between the privacy and the excess population loss, using an algorithm with a linear time-complexity in the number of training samples. We provide a general framework for solving differentially private stochastic minimax optimization (DP-SMO) problems, which enables the practitioners to bring their own base optimization algorithm and use it as a black-box to obtain the near-optimal privacy…
Finding Differentially Private Second Order Stationary Points in Stochastic Minimax Optimization
King Abdullah University of Science and Technology（国王阿卜杜勒-阿齐兹大学）；Provable Responsible AI and Data Analytics (PRADA) Lab（可证责任AI和数据分析师实验室）；State University of New York at Buffalo（纽约州立大学布法罗分校）；Seoul National University（首尔国立大学）
Stochastic first-order methods such as Stochastic Extragradient (SEG) or Stochastic Gradient Descent-Ascent (SGDA) for solving smooth minimax problems and, more generally, variational inequality problems (VIP) have been gaining a lot of attention in recent years due to the growing popularity of adversarial formulations in machine learning. However, while high-probability convergence bounds are known to reflect the actual behavior of stochastic methods more accurately, most convergence results are provided in expectation. Moreover, the only known high-probability complexity results have been de…
Gaoling School of Artificial Intelligence, Renmin University of China（中国人民大学高瓴人工智能学院）；Beijing Key Laboratory of Big Data Management and Analysis Methods（北京市大数据管理与分析方法重点实验室）
本文研究极小极大问题的泛化界，通过均匀局部收敛获得了非凸-强凹设定下更尖锐的高概率泛化误差界，并基于此推导出比现有结果快 n 倍的超额原风险界。
Robust Kernel Hypothesis Testing under Data Corruption
Centre for Artificial Intelligence（人工智能中心）；Gatsby Computational Neuroscience Unit（加布特计算神经科学单位）；University College London（伦敦大学学院）；Inria London（伦敦研究所）；Department of Statistics and Data Science（统计与数据科学系）；Department of Applied Statistics（应用统计学系）；Yonsei University（延世大学）
In large-scale statistical learning, data collection and model fitting are moving increasingly toward peripheral devices---phones, watches, fitness trackers---away from centralized data collection. Concomitant with this rise in decentralized data are increasing challenges of maintaining privacy while allowing enough information to fit accurate, useful statistical models. This motivates local notions of privacy---most significantly, local differential privacy, which provides strong protections against sensitive data disclosures---where data is obfuscated before a statistician or learner can eve…
University of Pennsylvania（宾夕法尼亚大学）；Rutgers University（罗格斯大学）
We propose differentially private algorithms for parameter estimation in both low-dimensional and high-dimensional sparse generalized linear models (GLMs) by constructing private versions of projected gradient descent. We show that the proposed algorithms are nearly rate-optimal by characterizing their statistical performance and establishing privacy-constrained minimax lower bounds for GLMs. The lower bounds are obtained via a novel technique, which is based on Stein's Lemma and generalizes the tracing attack technique for privacy-constrained lower bounds. This lower bound argument can be of …
University of Waterloo（滑铁卢大学）；Northeastern University（东北大学）
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 …
