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An Online Convex Optimization Approach to Blackwell's Approachability

The notion of approachability in repeated games with vector payoffs was introduced by Blackwell in the 1950s, along with geometric conditions for approachability and corresponding strategies that rely on computing {\em steering directions} as projections from the current average payoff vector to the (convex) target set. Recently, Abernethy, Batlett and Hazan (2011) proposed a class of approachability algorithms that rely on the no-regret properties of Online Linear Programming for computing a suitable sequence of steering directions. This is first carried out for target sets that are convex cones, and then generalized to any convex set by embedding it in a higher-dimensional convex cone. In this paper we present a more direct formulation that relies on the support function of the set, along with suitable Online Convex Optimization algorithms, which leads to a general class of approachability algorithms. We further show that Blackwell's original algorithm and its convergence follow as a special case.