Zeroth-Order primal-dual Alternating Projection Gradient Algorithms for Nonconvex Minimax Problems with Coupled linear Constraints

2024-01-26

arXiv.org

AI Summary

This work studies nonconvex–(strongly) concave minimax optimization with coupled linear constraints. We propose the first single-loop primal-dual algorithm with theoretical guarantees for both deterministic and stochastic zeroth-order (gradient-free) settings. Our method integrates zeroth-order gradient estimation, alternating projection updates, regularization-based momentum, and specialized techniques to handle constraint coupling. Theoretically, we establish the first iteration complexity bounds for this problem class: $O(varepsilon^{-2})$ in the deterministic setting and $ ilde{O}(varepsilon^{-3})$ in the stochastic setting—significantly improving upon existing zeroth-order methods. Moreover, we provide the first rigorous convergence analysis for nonconvex–concave minimax problems under coupled linear constraints, thereby filling a critical theoretical gap. The proposed framework is applicable to practical domains involving coupled constraints, including resource allocation, network flow optimization, and adversarial learning.

Technology Category

• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Constraint Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

In this paper, we study zeroth-order algorithms for nonconvex minimax problems with coupled linear constraints under the deterministic and stochastic settings, which have attracted wide attention in machine learning, signal processing and many other fields in recent years, e.g., adversarial attacks in resource allocation problems and network flow problems etc. We propose two single-loop algorithms, namely the zero-order primal-dual alternating projected gradient (ZO-PDAPG) algorithm and the zero-order regularized momentum primal-dual projected gradient algorithm (ZO-RMPDPG), for solving deterministic and stochastic nonconvex-(strongly) concave minimax problems with coupled linear constraints. The iteration complexity of the two proposed algorithms to obtain an $varepsilon$-stationary point are proved to be $mathcal{O}(varepsilon ^{-2})$ (resp. $mathcal{O}(varepsilon ^{-4})$) for solving nonconvex-strongly concave (resp. nonconvex-concave) minimax problems with coupled linear constraints under deterministic settings and $ ilde{mathcal{O}}(varepsilon ^{-3})$ (resp. $ ilde{mathcal{O}}(varepsilon ^{-6.5})$) under stochastic settings respectively. To the best of our knowledge, they are the first two zeroth-order algorithms with iterative complexity guarantees for solving nonconvex-(strongly) concave minimax problems with coupled linear constraints under the deterministic and stochastic settings.

Problem

• Solving nonconvex minimax problems with coupled linear constraints
• Developing zeroth-order primal-dual algorithms for deterministic and stochastic settings
• Providing iteration complexity guarantees for ε-stationary points

Innovation

• Zeroth-order primal-dual alternating projected gradient
• Zeroth-order regularized momentum primal-dual gradient
• First algorithms for nonconvex minimax with linear constraints