We study a general adversarial online learning problem, in which we are given a decision set X' in a reflexive Banach space X and a sequence of reward vectors in the dual space of X. At each iteration, we choose an action from X', based on the observed sequence of previous rewards. Our goal is to minimize regret, defined as the gap between the realized reward and the reward of the best fixed action in hindsight. Using results from infinite dimensional convex analysis, we generalize the method of Dual Averaging (or Follow the Regularized Leader) to our setting and obtain upper bounds on the worst-case regret that generalize many previous results. Under the assumption of uniformly continuous rewards, we obtain explicit regret bounds in a setting where the decision set is the set of probability distributions on a compact metric space S. Importantly, we make no convexity assumptions on either the set S or the reward functions. We also prove a general lower bound on the worst-case regret for any online algorithm. We then apply these results to the problem of learning in repeated two-player zero-sum games on compact metric spaces. In doing so, we first prove that if both players play a Hannan-consistent strategy, then with probability 1 the empirical distributions of play weakly converge to the set of Nash equilibria of the game. We then show that, under mild assumptions, Dual Averaging on the (infinite-dimensional) space of probability distributions indeed achieves Hannan-consistency.

We study a general adversarial online learning problem, in which we are given a decision set X' in a reflexive Banach space X and a sequence of reward vectors in the dual space of X. At each iteration, we choose an action from X', based on the observed sequence of previous rewards. Our goal is to minimize regret, defined as the gap between the realized reward and the reward of the best fixed action in hindsight. Using results from infinite dimensional convex analysis, we generalize the method of Dual Averaging (or Follow the Regularized Leader) to our setting and obtain upper bounds on the worst-case regret that generalize many previous results. Under the assumption of uniformly continuous rewards, we obtain explicit regret bounds in a setting where the decision set is the set of probability distributions on a compact metric space S. Importantly, we make no convexity assumptions on either the set S or the reward functions.

Markov Decision Processes (MDPs) are an effective model to represent decision processes in the presence of transitional uncertainty and reward tradeoffs. However, due to the difficulty in exactly specifying the transition and reward functions in MDPs, researchers have proposed uncertain MDP models and robustness objectives in solving those models. Most approaches for computing robust policies have focused on the computation of maximin policies which maximize the value in the worst case amongst all realisations of uncertainty. Given the overly conservative nature of maximin policies, recent work has proposed minimax regret as an ideal alternative to the maximin objective for robust optimization. However, existing algorithms for handling minimax regret are restricted to models with uncertainty over rewards only and they are also limited in their scalability. Therefore, we provide a general model of uncertain MDPs that considers uncertainty over both transition and reward functions. Furthermore, we also consider dependence of the uncertainty across different states and decision epochs. We also provide a mixed integer linear program formulation for minimizing regret given a set of samples of the transition and reward functions in the uncertain MDP. In addition, we provide two myopic variants of regret, namely Cumulative Expected Myopic Regret (CEMR) and One Step Regret (OSR) that can be optimized in a scalable manner. Specifically, we provide dynamic programming and policy iteration based algorithms to optimize CEMR and OSR respectively. Finally, to demonstrate the effectiveness of our approaches, we provide comparisons on two benchmark problems from literature. We observe that optimizing the myopic variants of regret, OSR and CEMR are better than directly optimizing the regret.