Pde Approach to the Problem of Online Prediction With Expert Advice: A Construction of Potential-Based Strategies

Abstract

We consider a sequence of repeated prediction games and formally pass to the limit. The supersolutions of the resulting non-linear parabolic partial differential equation

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].…”

New Potential-Based Bounds for Prediction with Expert Advice

“…When N > 2, for any subset S ⊂ {−1, 1} N and any q ∈ {±1} N we have Rokhlin (2017) proposed a fixed-horizon potential corresponding to p e providing a PDE perspective on the suboptimal bound in Corollary 2.2 of Cesa-Bianchi and Lugosi (2006). Our approach gives the best known upper bound and therefore provides a PDE perspective on Theorem 2.2 of Cesa-Bianchi and Lugosi (2006).…”

“…Rakhlin et al (2012) proposed a principled way of deriving potential-based player strategies by bounding above the value function, conditional on the realized losses, in a manner that is consistent with its recursive minmax form. Rokhlin (2017) suggested using supersolutions of the asymptotic PDE as potentials for player strategies in the scaling limit. The present paper extends these ideas by applying related arguments to the original problem (not a scaling limit), and by providing numerous examples (including lower as well as upper bounds).…”

New Potential-Based Bounds for the Geometric-Stopping Version of Prediction with Expert Advice

“…3. Rokhlin (2017) proposed a fixed-horizon potential corresponding to the exponentially weighted average (Exp) player p e providing a PDE perspective on the bound in Corollary 2.2 of Cesa-Bianchi and Lugosi (2006). Our PDE-based approach gives the best known upper bound for the geometric stopping problem and it is straightforward to extend this approach to the fixed horizon problem and provide a PDE perspective on the best known bound in Theorem 2.2 of Cesa-Bianchi and Lugosi (2006).…”

“…Extending the ideas of Rakhlin et al (2012) and Rokhlin (2017), Kobzar et al (2019) derived potential-based player and adversary strategies using sub-and supersolutions of the asymptotic PDE as potentials, and provided numerous examples (including lower as well as upper bounds) in the fixed horizon setting. This paper further extends this framework and leverages the Laplace's transform relationship between the finite horizon and geometric problems, specifically:…”