Proximal Point Method for Online Saddle Point Problem
This work addresses nonstationary two-player game optimization for researchers in online learning and game theory, though it appears incremental as it builds on existing proximal point methods.
The paper tackles the online saddle point problem for time-varying convex-concave games by proposing three proximal point method variants (OPPM, OptOPPM, and OptOPPM with multiple predictors), which achieve near-optimal duality gap bounds and maintain nearly constant bounds in stationary environments.
This paper focuses on the online saddle point problem, which involves a sequence of two-player time-varying convex-concave games. Considering the nonstationarity of the environment, we adopt the duality gap and the dynamic Nash equilibrium regret as performance metrics for algorithm design. We present three variants of the proximal point method: the Online Proximal Point Method (OPPM), the Optimistic OPPM (OptOPPM), and the OptOPPM with multiple predictors. Each algorithm guarantees upper bounds for both the duality gap and dynamic Nash equilibrium regret, achieving near-optimality when measured against the duality gap. Specifically, in certain benign environments, such as sequences of stationary payoff functions, these algorithms maintain a nearly constant metric bound. Experimental results further validate the effectiveness of these algorithms. Lastly, this paper discusses potential reliability concerns associated with using dynamic Nash equilibrium regret as a performance metric. The technical appendix and code can be found at https://github.com/qingxin6174/PPM-for-OSP.