Supermartingales in Prediction with Expert Advice

TL;DR

This paper explores the use of game-theoretic supermartingales in defensive forecasting for prediction with expert advice, showing equivalence with the Aggregating Algorithm and extending to new settings with conditional advice and multiple loss functions.

Contribution

It introduces a unified approach using supermartingales for prediction with expert advice, demonstrating its equivalence to existing algorithms and extending applicability to new complex scenarios.

Findings

Defensive forecasting matches the performance of the Aggregating Algorithm.

Algorithms can be adapted for experts giving advice conditional on future decisions.

Application to settings with multiple loss functions is feasible.

Abstract

We apply the method of defensive forecasting, based on the use of game-theoretic supermartingales, to prediction with expert advice. In the traditional setting of a countable number of experts and a finite number of outcomes, the Defensive Forecasting Algorithm is very close to the well-known Aggregating Algorithm. Not only the performance guarantees but also the predictions are the same for these two methods of fundamentally different nature. We discuss also a new setting where the experts can give advice conditional on the learner's future decision. Both the algorithms can be adapted to the new setting and give the same performance guarantees as in the traditional setting. Finally, we outline an application of defensive forecasting to a setting with several loss functions.