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We describe an approximate dynamic programming (ADP) approach to computeapproximations of the optimal strategies and of the minimal losses that can beguaranteed in discounted repeated games with vector-valued losses. Such gamesprominently arise in the analysis of regret in repeated decision-making inadversarial environments, also known as adversarial online learning. At thecore of our approach is a characterization of the lower Pareto frontier of theset of expected losses that a player can guarantee in these games as the uniquefixed point of a set-valued dynamic programming operator. When applied to theproblem of regret minimization with discounted losses, our approach yieldsalgorithms that achieve markedly improved performance bounds compared tooff-the-shelf online learning algorithms like Hedge. These results thus suggestthe significant potential of ADP-based approaches in adversarial onlinelearning.

An Approximate Dynamic Programming Approach to Adversarial Online Learning