Fast convergence of sample-average approximation for saddle-point problems

Stochastic saddle point (SSP) problems are, in general, less studied compared to stochastic minimization problems. However, SSP problems emerge from machine learning (adversarial training, e.g., GAN, AUC maximization), statistics (robust estimation), and game theory (stochastic matrix games). Notwithstanding the existing results on the convergence of stochastic gradient algorithms, there is little analysis on the generalization, i.e., how learned on train data models behave with new data samples. In contrast to existing $\mathcal{O}(\frac{1}{n})$ in expectation result arXiv:2006.02067, arXiv:2010.12561 and $\mathcal{O}(\frac{\log (\frac{1}{\delta })}{\sqrt{n}})$ results with high probability arXiv:2105.03793 , we present $\mathcal{O}(\frac{\log (\frac{1}{\delta })}{n})$ result in high probability for strongly convex-strongly concave setting. The main idea is local norms analysis. We illustrate our results in a matrix-game problem.