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

Abstract

This work addresses the classic machine learning problem of online prediction with expert advice. A potential-based framework for the fixed horizon version of this problem was previously developed using verification arguments from optimal control theory (Kobzar, Kohn and Wang, New Potential-Based Bounds for Prediction with Expert Advice ( 2019)). This paper extends this framework to the random (geometric) stopping version.Taking advantage of these ideas, we construct potentials for the geometric version of prediction with expert advice from potentials used for the fixed horizon version. This construction leads to new explicit lower and upper bounds associated with specific adversary and player strategies for the geometric problem. We identify regimes where these bounds are state of the art.

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.…”