A Stochastic View of Optimal Regret through Minimax Duality

We study the regret of optimal strategies for online convex optimization games. Using von Neumann’s minimax theorem, we show that the optimal regret in this adversarial setting is closely related to the behavior of the empirical minimization algorithm in a stochastic process setting: it is equal to the maximum, over joint distributions of the adversary’s action sequence, of the difference between a sum of minimal expected losses and the minimal empirical loss. We show that the optimal regret has a natural geometric interpretation, since it can be viewed as the gap in Jensen’s inequality for a concave functional–the minimizer over the player’s actions of expected loss–defined on a set of probability distributions. We use this expression to obtain upper and lower bounds on the regret of an optimal strategy for a variety of online learning problems. Our method provides upper bounds without the need to construct a learning algorithm; the lower bounds provide explicit optimal strategies for the adversary.

💡 Research Summary

The paper investigates the regret of optimal strategies in online convex optimization games by establishing a deep connection between the adversarial setting and a stochastic process framework. Using von Neumann’s minimax theorem, the authors transform the original zero‑sum game between a learner and an adversary into a dual formulation where the adversary’s action sequence is described by a joint probability distribution. In this representation the optimal regret (R^{*}) can be written explicitly as

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