Pseudonorm Approachability and Applications to Regret Minimization
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Blackwell's celebrated approachability theory provides a general framework for a variety of learning problems, including regret minimization. However, Blackwell's proof and implicit algorithm measure approachability using the ℓ_2 (Euclidean) distance. We argue that in many applications such as regret minimization, it is more useful to study approachability under other distance metrics, most commonly the ℓ_∞-metric. But, the time and space complexity of the algorithms designed for ℓ_∞-approachability depend on the dimension of the space of the vectorial payoffs, which is often prohibitively large. Thus, we present a framework for converting high-dimensional ℓ_∞-approachability problems to low-dimensional pseudonorm approachability problems, thereby resolving such issues. We first show that the ℓ_∞-distance between the average payoff and the approachability set can be equivalently defined as a pseudodistance between a lower-dimensional average vector payoff and a new convex set we define. Next, we develop an algorithmic theory of pseudonorm approachability, analogous to previous work on approachability for ℓ_2 and other norms, showing that it can be achieved via online linear optimization (OLO) over a convex set given by the Fenchel dual of the unit pseudonorm ball. We then use that to show, modulo mild normalization assumptions, that there exists an ℓ_∞-approachability algorithm whose convergence is independent of the dimension of the original vectorial payoff. We further show that that algorithm admits a polynomial-time complexity, assuming that the original ℓ_∞-distance can be computed efficiently. We also give an ℓ_∞-approachability algorithm whose convergence is logarithmic in that dimension using an FTRL algorithm with a maximum-entropy regularizer.
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