No-Regret Learning with Unbounded Losses: The Case of Logarithmic Pooling

TL;DR

This paper develops an online learning algorithm for aggregating expert probability forecasts using logarithmic pooling, achieving sublinear regret in a semi-adversarial setting with unbounded losses.

Contribution

It introduces a novel semi-adversarial framework for online expert aggregation with unbounded log loss and proposes an online mirror descent algorithm with provable regret bounds.

Findings

Achieves $O( oot{T} ext{log} T)$ regret in the proposed setting.

Extends online learning theory to unbounded loss functions.

Demonstrates effectiveness of logarithmic pooling in adversarial scenarios.

Abstract

For each of $T$ time steps, $m$ experts report probability distributions over $n$ outcomes; we wish to learn to aggregate these forecasts in a way that attains a no-regret guarantee. We focus on the fundamental and practical aggregation method known as logarithmic pooling -- a weighted average of log odds -- which is in a certain sense the optimal choice of pooling method if one is interested in minimizing log loss (as we take to be our loss function). We consider the problem of learning the best set of parameters (i.e. expert weights) in an online adversarial setting. We assume (by necessity) that the adversarial choices of outcomes and forecasts are consistent, in the sense that experts report calibrated forecasts. Imposing this constraint creates a (to our knowledge) novel semi-adversarial setting in which the adversary retains a large amount of flexibility. In this setting, we present an algorithm based on online mirror descent that learns expert weights in a way that attains $O(\sqrt{T} \log T)$ expected regret as compared with the best weights in hindsight.