Generalization Bounds for Stochastic Saddle Point Problems

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 $O\left(1/n\right)$ generalization bound by using a probabilistic 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 three examples: batch policy learning in Markov decision process, stochastic composite optimization problem, 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.