New Potential-Based Bounds for Prediction with Expert Advice

Abstract

This work addresses the classic machine learning problem of online prediction with expert advice. We consider the finite-horizon version of this zero-sum, two-person game.Using verification arguments from optimal control theory, we view the task of finding better lower and upper bounds on the value of the game (regret) as the problem of finding better sub-and supersolutions of certain partial differential equations (PDEs). These sub-and supersolutions serve as the potentials for player and adversary strategies, which lead to the corresponding bounds. Our techniques extend in a nonasymptotic setting the recent work of Drenska and Kohn (J. Nonlinear Sci. 2020), which showed that the asymptotically optimal value function is the unique solution of an associated nonlinear PDE.To get explicit bounds, we use closed-form solutions of specific PDEs. Our bounds hold for any fixed number of experts and any time-horizon; in certain regimes (which we identify) they improve upon the previous state-of-the-art.For up to three experts, our bounds provide the asymptotically optimal leading order term. Therefore, in this setting, we provide a continuum perspective on recent work on optimal strategies.

Finite-time 4-expert prediction problem

“…Following this setting, for the case of N = 4 experts in the geometric horizon setting, Bayraktar, Ekren and Zhang [4] showed that the comb strategy is asymptotically optimal by explicitly solving the corresponding nonlinear PDE. And very recently in [14], Kobzar, Kohn and Wang found lower and upper bounds for the optimal regret for finite stopping problem by constructing certain sub-and supersolutions of (1.1) following the method of [16]. Their results are only tight for N = 3 and improved those of [1].…”

A PDE Approach to the Prediction of a Binary Sequence with Advice from Two History-Dependent Experts

“…A different class of examples focuses directly on the experts' outcomes, letting those be determined directly (rather than through a time series) by the market. (A PDE-based discussion of one such problem can be found in [9], and a PDE perspective on potential-based strategies can be found in [14,15,21]; PDE methods have also been applied to another class of problems known as "drifting games" [11].) In the present setting the experts' outcomes are highly constrained, since (i) their progress must be consistent with a time series, and (ii) their predictions depend, at a given time, on the past d items in that time series.…”