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We consider the problem of online learning and its application to solvingminimax 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 simplemodification of FTPL which incorporates optimism can achieve better regretguarantees, while retaining the optimal worst-case regret guarantee forunpredictable sequences. A key challenge in obtaining these tighter regretbounds is the stochasticity and optimism in the algorithm, which requiresdifferent analysis techniques than those commonly used in the analysis of FTPL.The key ingredient we utilize in our analysis is the dual view of perturbationas regularization. While our algorithm has several applications, we considerthe specific application of minimax games. For solving smooth convex-concavegames, our algorithm only requires access to a linear optimization oracle. ForLipschitz and smooth nonconvex-nonconcave games, our algorithm requires accessto an optimization oracle which computes the perturbed best response. In boththese settings, our algorithm solves the game up to an accuracy of$O(T^{-1/2})$ using $T$ calls to the optimization oracle. An important featureof 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 callsto the optimization oracle.

Follow the Perturbed Leader: Optimism and Fast Parallel Algorithms for Smooth Minimax Games