Exponential Weight Approachability, Applications to Calibration and Regret Minimization

Abstract

Basic ideas behind the “exponential weight algorithm” (designed for aggregation or minimization of regret) can be transposed into the theory of Blackwell approachability. Using them, we develop an algorithm—that we call “exponential weight approachability”—bounding the distance of average vector payoffs to some product set, with a logarithmic dependency in the dimension of the ambient space. The classic strategy of Blackwell would get instead a polynomial dependency. This result has important consequences, in several frameworks that emerged both in game theory and machine learning. The most striking application is the construction of algorithms that are calibrated with respect to the family of all balls (we treat in details the case of the uniform norm), with dimension independent and optimal, up to logarithmic factors, rates of convergence. Calibration can also be achieved with respect to all Borel sets, covering and improving the previously known results. Exponential weight approachability can also be used to design an optimal and natural algorithm that minimizes refined notions of regret.