Follow the Perturbed Leader: Optimism and Fast Parallel Algorithms for Smooth Minimax Games
This work addresses the challenge of efficient and parallelizable algorithms for online learning and minimax games, offering incremental improvements in regret bounds for predictable sequences.
The paper tackles the problem of improving regret guarantees in online learning when loss sequences are predictable, by modifying the Follow the Perturbed Leader algorithm with optimism, achieving better bounds while maintaining optimal worst-case performance. It applies this to minimax games, solving smooth convex-concave and nonconvex-nonconcave games up to O(T^{-1/2}) accuracy with T oracle calls, and features high parallelizability requiring O(T^{1/2}) iterations and O(T^{1/2}) parallel calls per iteration.
We consider the problem of online learning and its application to solving minimax games. For the online learning problem, Follow the Perturbed Leader (FTPL) is a widely studied algorithm which enjoys the optimal $O(T^{1/2})$ worst-case regret guarantee for both convex and nonconvex losses. In this work, we show that when the sequence of loss functions is predictable, a simple modification of FTPL which incorporates optimism can achieve better regret guarantees, while retaining the optimal worst-case regret guarantee for unpredictable sequences. A key challenge in obtaining these tighter regret bounds is the stochasticity and optimism in the algorithm, which requires different analysis techniques than those commonly used in the analysis of FTPL. The key ingredient we utilize in our analysis is the dual view of perturbation as regularization. While our algorithm has several applications, we consider the specific application of minimax games. For solving smooth convex-concave games, our algorithm only requires access to a linear optimization oracle. For Lipschitz and smooth nonconvex-nonconcave games, our algorithm requires access to an optimization oracle which computes the perturbed best response. In both these settings, our algorithm solves the game up to an accuracy of $O(T^{-1/2})$ using $T$ calls to the optimization oracle. An important feature of our algorithm is that it is highly parallelizable and requires only $O(T^{1/2})$ iterations, with each iteration making $O(T^{1/2})$ parallel calls to the optimization oracle.