A Unified Analysis of First-Order Methods for Smooth Games via Integral Quadratic Constraints

Abstract

The theory of integral quadratic constraints (IQCs) allows the certification of exponential convergence of interconnected systems containing nonlinear or uncertain elements. In this work, we adapt the IQC theory to study first-order methods for smooth and strongly-monotone games and show how to design tailored quadratic constraints to get tight upper bounds of convergence rates. Using this framework, we recover the existing bound for the gradient method~(GD), derive sharper bounds for the proximal point method~(PPM) and optimistic gradient method~(OG), and provide \emph{for the first time} a global convergence rate for the negative momentum method~(NM) with an iteration complexity $\mathcal{O}({\kappa }^{1.5})$, which matches its known lower bound. In addition, for time-varying systems, we prove that the gradient method with optimal step size achieves the fastest provable worst-case convergence rate with quadratic Lyapunov functions. Finally, we further extend our analysis to stochastic games and study the impact of multiplicative noise on different algorithms. We show that it is impossible for an algorithm with one step of memory to achieve acceleration if it only queries the gradient once per batch (in contrast with the stochastic strongly-convex optimization setting, where such acceleration has been demonstrated). However, we exhibit an algorithm which achieves acceleration with two gradient queries per batch.

Journal of Machine Learning Research

References in corpus (6)

• On Finding Local Nash Equilibria (and Only Local Nash Equilibria) in Zero-Sum Games
• The proximal point method revisited
• Differentiable Game Mechanics
• Dualities in Convex Algebraic Geometry
• On the Suboptimality of Negative Momentum for Minimax Optimization
• A Unified Analysis of Stochastic Optimization Methods Using Jump System Theory and Quadratic Constraints

Cited by in corpus (6)

• The Analysis of Optimization Algorithms, A Dissipativity Approach
• Internal Model-Based Online Optimization
• On the Suboptimality of Negative Momentum for Minimax Optimization
• Accelerated Algorithms for Smooth Convex-Concave Minimax Problems with $\mathcal{O}(1/k^2)$ Rate on Squared Gradient Norm
• Near-optimal Local Convergence of Alternating Gradient Descent-Ascent for Minimax Optimization
• Complex Momentum for Optimization in Games