No-Regret Learning with Unbounded Losses: The Case of Logarithmic Pooling
This work addresses the challenge of no-regret aggregation for forecasters in a semi-adversarial environment, which is incremental as it adapts existing methods to a new constrained setting.
The paper tackles the problem of learning optimal expert weights for logarithmic pooling in an online adversarial setting with calibrated forecasts, achieving an expected regret bound of O(√T log T) compared to the best weights in hindsight.
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.