Online Min-Max Optimization: From Individual Regrets to Cumulative Saddle Points

• URL: http://arxiv.org/abs/2602.10565v1
• Date: Wed, 11 Feb 2026 06:29:37 GMT
• Title: Online Min-Max Optimization: From Individual Regrets to Cumulative Saddle Points
• Authors: Abhijeet Vyas, Brian Bullins,
• Abstract summary: We study an online version of min-max optimization based on cumulative saddle points under a variety of performance measures beyond convex-concave settings.<n>For a dynamic notion of regret compatible with individual regrets, we derive bounds under a two-sided Polyak-ojasiewicz (PL) condition.
• Score: 9.267347813727824
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We propose and study an online version of min-max optimization based on cumulative saddle points under a variety of performance measures beyond convex-concave settings. After first observing the incompatibility of (static) Nash equilibrium (SNE-Reg$_T$) with individual regrets even for strongly convex-strongly concave functions, we propose an alternate \emph{static} duality gap (SDual-Gap$_T$) inspired by the online convex optimization (OCO) framework. We provide algorithms that, using a reduction to classic OCO problems, achieve bounds for SDual-Gap$_T$~and a novel \emph{dynamic} saddle point regret (DSP-Reg$_T$), which we suggest naturally represents a min-max version of the dynamic regret in OCO. We derive our bounds for SDual-Gap$_T$~and DSP-Reg$_T$~under strong convexity-strong concavity and a min-max notion of exponential concavity (min-max EC), and in addition we establish a class of functions satisfying min-max EC~that captures a two-player variant of the classic portfolio selection problem. Finally, for a dynamic notion of regret compatible with individual regrets, we derive bounds under a two-sided Polyak-Łojasiewicz (PL) condition.

Related papers

• Non-stationary Bandit Convex Optimization: A Comprehensive Study [28.086802754034828]
  Bandit Convex Optimization is a class of sequential decision-making problems.<n>We study this problem in non-stationary environments.<n>We aim to minimize the regret under three standard measures of non-stationarity.
  arXiv  Detail & Related papers  (2025-06-03T15:18:41Z)
• Stochastic Momentum Methods for Non-smooth Non-Convex Finite-Sum Coupled Compositional Optimization [68.22688819802622]
  We propose a new state-of-the-art complexity of $O(/epsilon)$ for finding an (nearly) $'level KKT solution.<n>By applying our hinge-of-the-art complexity of $O(/epsilon)$ for finding an (nearly) $'level KKT solution, we achieve a new state-of-the-art complexity of $O(/epsilon)$ for finding an (nearly) $'level KKT solution.
  arXiv  Detail & Related papers  (2025-06-03T06:31:59Z)
• Stochastic Zeroth-Order Optimization under Strongly Convexity and Lipschitz Hessian: Minimax Sample Complexity [59.75300530380427]
  We consider the problem of optimizing second-order smooth and strongly convex functions where the algorithm is only accessible to noisy evaluations of the objective function it queries. We provide the first tight characterization for the rate of the minimax simple regret by developing matching upper and lower bounds.
  arXiv  Detail & Related papers  (2024-06-28T02:56:22Z)
• Bridging the Gap Between General and Down-Closed Convex Sets in Submodular Maximization [8.225819874406238]
  We show that Mualem citemualem23re shows that this approach cannot smooth between down- and non-down-closed constraints. In this work, we suggest novel offline and online algorithms based on a natural decomposition of the body into two distinct convex bodies. We also empirically demonstrate the superiority of our proposed algorithms across three offline and two online applications.
  arXiv  Detail & Related papers  (2024-01-17T14:56:42Z)
• Nesterov Meets Optimism: Rate-Optimal Separable Minimax Optimization [108.35402316802765]
  We propose a new first-order optimization algorithm -- AcceleratedGradient-OptimisticGradient (AG-OG) Ascent. We show that AG-OG achieves the optimal convergence rate (up to a constant) for a variety of settings. We extend our algorithm to extend the setting and achieve the optimal convergence rate in both bi-SC-SC and bi-C-SC settings.
  arXiv  Detail & Related papers  (2022-10-31T17:59:29Z)
• Adaptivity and Non-stationarity: Problem-dependent Dynamic Regret for Online Convex Optimization [70.4342220499858]
  We introduce novel online algorithms that can exploit smoothness and replace the dependence on $T$ in dynamic regret with problem-dependent quantities. Our results are adaptive to the intrinsic difficulty of the problem, since the bounds are tighter than existing results for easy problems and safeguard the same rate in the worst case.
  arXiv  Detail & Related papers  (2021-12-29T02:42:59Z)
• Dynamic Regret of Convex and Smooth Functions [93.71361250701075]
  We investigate online convex optimization in non-stationary environments. We choose the dynamic regret as the performance measure. We show that it is possible to further enhance the dynamic regret by exploiting the smoothness condition.
  arXiv  Detail & Related papers  (2020-07-07T14:10:57Z)
• Optimal Epoch Stochastic Gradient Descent Ascent Methods for Min-Max Optimization [61.66927778694731]
  Epoch gradient descent method (a.k.a. Epoch-GD) proposed by Hazan and Kale (2011) was deemed a breakthrough for strongly convex minimization. Our result is the first one that shows Epoch-GDA can achieve the optimal rate of $O (1/T)$ for the duality gap of SCSC min-max problems.
  arXiv  Detail & Related papers  (2020-02-13T02:16:57Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.