Integral Mixability: a Tool for Efficient Online Aggregation of Functional and Probabilistic 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 one a single forecast. We adapt basic mixable loss functions to compare functional predictions and prove that these "integral" expansions are also mixable. We call this phenomena integral mixability. As an application, we consider various loss functions for prediction of probability distributions and show that they are mixable by using our main result. The considered loss functions include Continuous ranking probability score (CRPS), Optimal transport costs (OT), Beta-2 and Kullback-Leibler (KL) divergences.1 Merge, mix, average, etc.

CRPS Learning

“…Now we discuss the learning theory with respect to the CRPS ( 22), which can be regarded as an integral based loss in terms of learning Korotin et al (2019).…”

“…Unfortunately, the quantile loss is not exp-concave, thus an application of the EWA and EWAG does not guarantee optimal convergence properties by standard arguments from the previous section. In contrast, the CRPS is exp-concave for random variables with bounded support, see Korotin et al (2019). Thus, when considering learning algorithms with the structure (24) we receive that the corresponding EWA algorithms satisfy optimalities (8) and for gradient based algorithms additionally (9).…”

“…A polynomial CRPS learning algorithm is applied in Thorey et al (2018). Korotin et al (2019Korotin et al ( , 2020 analyze properties of exponential weighting learning algorithms for forecasting distributions, including a CRPS based algorithm. Also, Zhang et al (2020) show an application for probabilistic electricity demand forecasting where the combination weights are optimized with respect to the CRPS.…”

Online Aggregation of Probability Forecasts with Confidence

“…The definition ( 17) is a special case of this definition (up to a factor), where µ(u) = 1 b−a for u ∈ [a, b] and µ(u) = 0 otherwise. It can be proved that the function ( 18) is η-mixable for 0 < η ≤ 2 and η-exponentially concave for 0 < η ≤ 1 2 (see Korotin et al (2019)). The CRPS score measures the difference between the forecast F and a perfect forecast H(u − y) which puts all mass on the verification y.…”

Online Learning with Continuous Ranked Probability Score

“…The definition ( 19) is a special case of this definition (up to a factor), where µ(u) = 1 b−a for u ∈ [a, b] and µ(u) = 0 otherwise. In can be proved that the function ( 20) is η-mixable for 0 < η ≤ 2 and η-exponentially concave for 0 < η ≤ 1 2 (see Korotin et al 2019). The CRPS score measures the difference between the forecast F and a perfect forecast H(u − y) which puts all mass on the verification y.…”