General Procedure to Provide High-Probability Guarantees for Stochastic Saddle Point Problems

Abstract

This paper considers smooth strongly convex and strongly concave (SC-SC) stochastic saddle point (SSP) problems. Suppose there is an arbitrary oracle that in expectation returns an $\epsilon$-solution in the sense of certain gaps, which can be the duality gap or its weaker variants. We propose a general PB-SSP framework to guarantee an $\epsilon$ small duality gap solution with high probability via only $\mathcal{O}\big(\log \frac{1}{p}\cdot\text{poly}(\log \kappa)\big)$ calls of this oracle, where $p\in(0,1)$ is the confidence level and $\kappa$ is the condition number. When applied to the sample average approximation (SAA) oracle, in addition to equipping the solution with high probability, our approach even improves the sample complexity by a factor of $\text{poly}(\kappa)$, since the high-probability argument enables us to circumvent some key difficulties of the uniform stability analysis of SAA.