Opportunistic strategies for generalized no-regret problems

This paper considers a generalized no-regret problem with vector-valued rewards, defined in terms of a desired reward set of the agent. For each mixed action q of the opponent, the agent has a set R^*(q) where the average reward should reside. In addition, the agent has a response mixed action p which brings the expected reward under these two actions, r(p,q), to R ^*(q). If a strategy of the agent ensures that the average reward converges to R^*(q̄_n), where q̄_n is the empirical distribution of the opponent's actions, for any strategy of the opponent, we say that it is a no-regret strategy with respect to R^*(q). When the multifunction q R ^*(q) is convex, as is the case in the standard no-regret problem, no-regret strategies can be devised. Our main interest in this paper is in cases where this convexity property does not hold. The best that can be guaranteed in general then is the convergence of the average reward to R ^c(q̄_n), the convex hull of R ^*(q̄_n). However, as the game unfolds, it may turn out that the opponent's choices of actions are limited in some way. If these restrictions were known in advance, the agent could possibly ensure convergence of the average reward to some desired subset of R ^c(q̄_n), or even approach R^*(q̄ n) itself. We formulate appropriate goals for opportunistic no-regret strategies, in the sense that they may exploit such limitations on the opponent's action sequence in an on-line manner, without knowing them beforehand. As the main technical tool, we propose a class of approachability algorithms that rely on a calibrated forecast of the opponent's actions, which are opportunistic in the above mentioned sense. As an application, we consider the online no-regret problem with average cost constraints, introduced in Mannor, Tsitsiklis, and Yu (2009). We show, in particular, that our algorithm does attain the best-response-in-hindsight for this problem if the opponent's play happens to be stationary, or close to stationary in a certain sense.