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Previous algorithms can solve convex-concave minimax problems $\min_{x \in \mathcal{X}} \max_{y \in \mathcal{Y}} f(x,y)$ with $\mathcal{O}(\epsilon^{-2/3})$ second-order oracle calls using Newton-type methods. This result has been…

Optimization and Control · Mathematics 2025-06-11 Lesi Chen , Chengchang Liu , Luo Luo , Jingzhao Zhang

Minimax optimization problems have attracted significant attention in recent years due to their widespread application in numerous machine learning models. To solve the minimax problem, a wide variety of stochastic optimization methods have…

Machine Learning · Computer Science 2024-06-12 Hongchang Gao

The subgradient method is one of the most fundamental algorithmic schemes for nonsmooth optimization. The existing complexity and convergence results for this method are mainly derived for Lipschitz continuous objective functions. In this…

Optimization and Control · Mathematics 2024-11-01 Xiao Li , Lei Zhao , Daoli Zhu , Anthony Man-Cho So

In this paper, we focus on solving a class of constrained non-convex non-concave saddle point problems in a decentralized manner by a group of nodes in a network. Specifically, we assume that each node has access to a summand of a global…

Optimization and Control · Mathematics 2019-11-01 Weijie Liu , Aryan Mokhtari , Asuman Ozdaglar , Sarath Pattathil , Zebang Shen , Nenggan Zheng

This paper shows that the implicit bias of gradient descent on linearly separable data is exactly characterized by the optimal solution of a dual optimization problem given by a smoothed margin, even for general losses. This is in contrast…

Machine Learning · Computer Science 2020-11-13 Ziwei Ji , Matus Telgarsky

User-level differentially private stochastic convex optimization (DP-SCO) has garnered significant attention due to the paramount importance of safeguarding user privacy in modern large-scale machine learning applications. Current methods,…

Machine Learning · Computer Science 2025-02-14 Badih Ghazi , Ravi Kumar , Daogao Liu , Pasin Manurangsi

Differentially private (DP) stochastic convex optimization (SCO) is ubiquitous in trustworthy machine learning algorithm design. This paper studies the DP-SCO problem with streaming data sampled from a distribution and arrives sequentially.…

Machine Learning · Computer Science 2022-06-17 Yuxuan Han , Zhicong Liang , Zhipeng Liang , Yang Wang , Yuan Yao , Jiheng Zhang

In this paper, we propose and analyze zeroth-order stochastic approximation algorithms for nonconvex and convex optimization, with a focus on addressing constrained optimization, high-dimensional setting and saddle-point avoiding. To handle…

Optimization and Control · Mathematics 2019-01-16 Krishnakumar Balasubramanian , Saeed Ghadimi

We study the limits and capability of public-data assisted differentially private (PA-DP) algorithms. Specifically, we focus on the problem of stochastic convex optimization (SCO) with either labeled or unlabeled public data. For…

Machine Learning · Computer Science 2024-03-07 Enayat Ullah , Michael Menart , Raef Bassily , Cristóbal Guzmán , Raman Arora

We consider stochastic convex optimization for heavy-tailed data with the guarantee of being differentially private (DP). Most prior works on differentially private stochastic convex optimization for heavy-tailed data are either restricted…

Machine Learning · Computer Science 2024-09-11 Chenhan Jin , Kaiwen Zhou , Bo Han , James Cheng , Tieyong Zeng

We study the task of $(\epsilon, \delta)$-differentially private online convex optimization (OCO). In the online setting, the release of each distinct decision or iterate carries with it the potential for privacy loss. This problem has a…

Cryptography and Security · Computer Science 2023-12-21 Naman Agarwal , Satyen Kale , Karan Singh , Abhradeep Guha Thakurta

We present a novel accelerated primal-dual (APD) method for solving a class of deterministic and stochastic saddle point problems (SPP). The basic idea of this algorithm is to incorporate a multi-step acceleration scheme into the…

Optimization and Control · Mathematics 2013-09-24 Yunmei Chen , Guanghui Lan , Yuyuan Ouyang

We consider a generic convex optimization problem associated with regularized empirical risk minimization of linear predictors. The problem structure allows us to reformulate it as a convex-concave saddle point problem. We propose a…

Optimization and Control · Mathematics 2015-09-10 Yuchen Zhang , Lin Xiao

This paper improves the state-of-the-art rate of a first-order algorithm for solving entropy regularized optimal transport. The resulting rate for approximating the optimal transport (OT) has been improved from…

Optimization and Control · Mathematics 2023-01-25 Yiling Luo , Yiling Xie , Xiaoming Huo

In this paper we propose a randomized primal-dual proximal block coordinate updating framework for a general multi-block convex optimization model with coupled objective function and linear constraints. Assuming mere convexity, we establish…

Optimization and Control · Mathematics 2017-01-25 Xiang Gao , Yangyang Xu , Shuzhong Zhang

We consider convex-concave saddle point problems with a separable structure and non-strongly convex functions. We propose an efficient stochastic block coordinate descent method using adaptive primal-dual updates, which enables flexible…

Machine Learning · Statistics 2015-11-24 Zhanxing Zhu , Amos J. Storkey

Convex optimization finds many real-life applications, where--optimized on real data--optimization results may expose private data attributes (e.g., individual health records, commercial information), thus leading to privacy breaches. To…

Optimization and Control · Mathematics 2024-06-25 Vladimir Dvorkin , Ferdinando Fioretto , Pascal Van Hentenryck , Pierre Pinson , Jalal Kazempour

In this paper, we introduce faster accelerated primal-dual algorithms for minimizing a convex function subject to strongly convex function constraints. Prior to our work, the best complexity bound was $\mathcal{O}(1/{\varepsilon})$,…

Optimization and Control · Mathematics 2024-11-28 Zhenwei Lin , Qi Deng

We propose a family of recursive cutting-plane algorithms to solve feasibility problems with constrained memory, which can also be used for first-order convex optimization. Precisely, in order to find a point within a ball of radius…

Optimization and Control · Mathematics 2023-06-21 Moïse Blanchard , Junhui Zhang , Patrick Jaillet

We consider the stochastic approximation problem where a convex function has to be minimized, given only the knowledge of unbiased estimates of its gradients at certain points, a framework which includes machine learning methods based on…

Machine Learning · Computer Science 2013-06-11 Francis Bach , Eric Moulines