Stochastic Online Convex Optimization. Application to probabilistic time series forecasting

TL;DR

This paper develops a stochastic online convex optimization framework that achieves fast regret bounds and applies it to improve probabilistic time series forecasting, especially for non-stationary data.

Contribution

It introduces a general stochastic online convex optimization framework with fast regret bounds and applies it to calibrate probabilistic forecasters for non-stationary time series.

Findings

Algorithms achieve best-known stochastic regret rates.

Applicable to non-stationary sub-Gaussian time series.

Fast, any-time valid regret bounds.

Abstract

We introduce a general framework of stochastic online convex optimization to obtain fast-rate stochastic regret bounds. We prove that algorithms such as online newton steps and a scale-free 10 version of Bernstein online aggregation achieve best-known rates in unbounded stochastic settings. We apply our approach to calibrate parametric probabilistic forecasters of non-stationary sub-gaussian time series. Our fast-rate stochastic regret bounds are any-time valid. Our proofs combine self-bounded and Poissonnian inequalities for martingales and sub-gaussian random variables, respectively, under a stochastic exp-concavity assumption.