Mixability of Integral Losses: a Key to Efficient Online Aggregation of Functional and Probabilistic Forecasts

TL;DR

This paper extends online prediction with expert advice to function-valued forecasts, demonstrating that integral loss functions used in probabilistic forecasting are mixable, enabling efficient aggregation of functional predictions.

Contribution

It proves that integral loss functions are mixable or exp-concave, facilitating effective online aggregation of functional and probabilistic forecasts.

Findings

Integral loss functions are shown to be mixable or exp-concave.

Various probabilistic loss functions are proven to be mixable.

The results enable efficient online aggregation of functional forecasts.

Abstract

In this paper we extend the setting of the online prediction with expert advice to function-valued forecasts. At each step of the online game several experts predict a function, and the learner has to efficiently aggregate these functional forecasts into a single forecast. We adapt basic mixable (and exponentially concave) loss functions to compare functional predictions and prove that these adaptations are also mixable (exp-concave). We call this phenomenon mixability (exp-concavity) of integral loss functions. As an application of our main result, we prove that various loss functions used for probabilistic forecasting are mixable (exp-concave). The considered losses include Sliced Continuous Ranked Probability Score, Energy-Based Distance, Optimal Transport Costs and Sliced Wasserstein-2 distance, Beta-2 and Kullback-Leibler divergences, Characteristic function and Maximum Mean Discrepancies.