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

TL;DR

This paper introduces a novel, elementary potential-based method for analyzing drifting games, providing asymptotically optimal bounds with simple proofs, advancing understanding in online learning and boosting.

Contribution

It proposes a new approach using PDE-based potential guesses and elementary proofs to establish matching upper and lower bounds in drifting games.

Findings

Derived asymptotically optimal potentials for drifting games.

Provided elementary proofs for upper bounds using Taylor expansion.

Established matching lower bounds with probabilistic and combinatorial arguments.

Abstract

We develop a new approach to drifting games, a class of two-person games with many applications to boosting and online learning settings. 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 elementary, combining Taylor expansion with probabilistic or combinatorial arguments. 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.