[Paper Review] Near-Optimal Algorithms for Minimax Optimization

Tianyi Lin, Chi Jin+1 more|arXiv (Cornell University)|Feb 5, 2020

Sparse and Compressive Sensing TechniquesEngineering69 references45 citations

TL;DR

The paper introduces near-optimal first-order algorithms for smooth and strongly-convex-strongly-concave minimax problems, achieving the lower-bound gradient complexity up to log factors. It also extends acceleration to other settings including nonconvex cases.

ABSTRACT

This paper resolves a longstanding open question pertaining to the design of near-optimal first-order algorithms for smooth and strongly-convex-strongly-concave minimax problems. Current state-of-the-art first-order algorithms find an approximate Nash equilibrium using $ ilde{O}(κ_{\mathbf x}+κ_{\mathbf y})$ or $ ilde{O}(\min\{κ_{\mathbf x}\sqrt{κ_{\mathbf y}}, \sqrt{κ_{\mathbf x}}κ_{\mathbf y}\})$ gradient evaluations, where $κ_{\mathbf x}$ and $κ_{\mathbf y}$ are the condition numbers for the strong-convexity and strong-concavity assumptions. A gap still remains between these results and the best existing lower bound $ ildeΩ(\sqrt{κ_{\mathbf x}κ_{\mathbf y}})$. This paper presents the first algorithm with $ ilde{O}(\sqrt{κ_{\mathbf x}κ_{\mathbf y}})$ gradient complexity, matching the lower bound up to logarithmic factors. Our algorithm is designed based on an accelerated proximal point method and an accelerated solver for minimax proximal steps. It can be easily extended to the settings of strongly-convex-concave, convex-concave, nonconvex-strongly-concave, and nonconvex-concave functions. This paper also presents algorithms that match or outperform all existing methods in these settings in terms of gradient complexity, up to logarithmic factors.

Motivation & Objective

• Motivate and address the gap between known upper and lower gradient complexity bounds for minimax optimization.
• Design near-optimal first-order algorithms for strongly-convex-strongly-concave and strongly-convex-concave minimax problems.
• Extend the accelerated framework to convex-concave, nonconvex-strongly-concave, and nonconvex-concave settings.
• Provide an accelerated solver for minimax proximal steps and a practical, provably efficient implementation.
• Offer a comparative analysis against existing results, highlighting improvements in gradient complexity.

Proposed method

• Develop an accelerated proximal point framework (APPA) with inexact proximal subproblem solutions.
• Use an accelerated solver for minimax proximal steps to solve subproblems of the form min_x max_y f(x,y) + ell||x - x̃||^2.
• Introduce Maximin-AG2, an algorithm that alternates accelerated minimization over x with accelerated ascent over y via AGD/AGA subroutines.
• Employ Nesterov’s accelerated gradient techniques and an accelerated solver to achieve tilde-O(√(κ_x κ_y)) gradient complexity in the strongly-convex-strongly-concave setting.
• Provide algorithmic components (AGD, Inexact-APPA, and accelerated minimax solvers) with formal convergence guarantees.
• Extend the framework to convex-concave, nonconvex-strongly-concave, and nonconvex-concave settings with near-optimal rates.

Experimental results

Research questions

• RQ1 Can first-order methods achieve the lower bound tilde-O(√(κ_x κ_y)) gradient complexity for strongly-convex-strongly-concave minimax problems?
• RQ2 What algorithmic structure yields near-optimal rates for strongly-convex-concave minimax problems and their extensions?
• RQ3 How can acceleration be integrated into proximal-point and minimax steps to handle a broad class of minimax problems (convex-concave and nonconvex-concave)?
• RQ4 Can an accelerated proximal-step solver generalize to nonconvex settings while preserving favorable gradient complexities?
• RQ5 How do the proposed methods compare to existing upper and lower bounds across settings (convex-concave, strongly convex-concave, nonconvex-concave)?

Key findings

• Achieves tilde-O(√(κ_x κ_y)) gradient complexity for strongly-convex-strongly-concave minimax, matching known lower bounds up to log factors.
• Obtains tilde-O(√(κ_x/ε)) gradient complexity for strongly-convex-concave minimax, matching the lower bound up to log factors.
• Attains tilde-O(ε^{-1}) gradient complexity in convex-concave settings, aligning with lower bounds and existing upper bounds up to log factors.
• Develops accelerated solvers for minimax proximal steps that work under less restrictive smoothness assumptions (via APPA).
• Delivers accelerated algorithms for nonconvex-strongly-concave and nonconvex-concave minimax problems with improved stationarity-based rates (e.g., ε^{-2.5} to ε^{-3} ranges).
• Provides a unified framework (APPA + Maximin-AG2) that covers multiple minimax regimes with provable guarantees.