Min-Max Optimization Made Simple: Approximating the Proximal Point Method via Contraction Maps
This provides a simpler and more intuitive analysis for min-max optimization, which is incremental but improves upon existing methods for researchers and practitioners in optimization.
The paper tackles convex/concave min-max optimization by developing a first-order method that approximates the Proximal Point method using contraction maps, achieving near-optimal convergence rates with O(log 1/ε) gradient calls per iteration.
In this paper we present a first-order method that admits near-optimal convergence rates for convex/concave min-max problems while requiring a simple and intuitive analysis. Similarly to the seminal work of Nemirovski and the recent approach of Piliouras et al. in normal form games, our work is based on the fact that the update rule of the Proximal Point method (PP) can be approximated up to accuracy $ε$ with only $O(\log 1/ε)$ additional gradient-calls through the iterations of a contraction map. Then combining the analysis of (PP) method with an error-propagation analysis we establish that the resulting first order method, called Clairvoyant Extra Gradient, admits near-optimal time-average convergence for general domains and last-iterate convergence in the unconstrained case.