Generalization Bounds for Stochastic Saddle Point Problems

TL;DR

This paper establishes new generalization bounds for empirical saddle point solutions in stochastic saddle point problems, covering various assumptions and demonstrating near-optimal sample complexity in applications like policy learning and game theory.

Contribution

It introduces the first generalization bounds for ESP solutions in SSP problems, including cases without strong convexity or bounded domains.

Findings

Achieves an $ ext{O}(1/n)$ generalization bound for Lipschitz continuous, strongly convex-strongly concave functions.

Provides bounds under weaker assumptions, including non-strong convexity and unbounded domains.

Shows regularized ESP solutions have near-optimal sample complexity in practical examples.

Abstract

This paper studies the generalization bounds for the empirical saddle point (ESP) solution to stochastic saddle point (SSP) problems. For SSP with Lipschitz continuous and strongly convex-strongly concave objective functions, we establish an $\mathcal{O}(1/n)$ generalization bound by using a uniform stability argument. We also provide generalization bounds under a variety of assumptions, including the cases without strong convexity and without bounded domains. We illustrate our results in two examples: batch policy learning in Markov decision process, and mixed strategy Nash equilibrium estimation for stochastic games. In each of these examples, we show that a regularized ESP solution enjoys a near-optimal sample complexity. To the best of our knowledge, this is the first set of results on the generalization theory of ESP.