A New Approach to Drifting Games, Based on Asymptotically Optimal Potentials

We develop a new approach to drifting games, a class of two-person games with many applications to boosting and online learning settings, including Prediction with Expert Advice and the Hedge game. Our approach involves (a) guessing an asymptotically optimal potential by solving an associated partial differential equation (PDE); then (b) justifying the guess, by proving upper and lower bounds on the final-time loss whose difference scales like a negative power of the number of time steps. The proofs of our potential-based upper bounds are elementary, using little more than Taylor expansion. The proofs of our potential-based lower bounds are also rather elementary, combining Taylor expansion with probabilistic or combinatorial arguments. Most previous work on asymptotically optimal strategies has used potentials obtained by solving a discrete dynamic programming principle; the arguments are complicated by their discrete nature. Our approach is facilitated by the fact that the potentials we use are explicit solutions of PDEs; the arguments are based on basic calculus. Not only is our approach more elementary, but we give new potentials and derive corresponding upper and lower bounds that match each other in the asymptotic regime.