Continuous Prediction with Experts' Advice

TL;DR

This paper introduces a continuous-time stochastic calculus approach to the experts' advice problem, resulting in a parameter-free algorithm with improved quantile regret guarantees and an anytime algorithm matching optimal regret rates.

Contribution

It develops a novel continuous-time analysis framework for online learning, leading to new algorithms with enhanced regret bounds and insights into adversarial gain scenarios.

Findings

Designed a continuous-time, parameter-free algorithm with improved quantile regret.

Developed a discrete-time algorithm with similar guarantees and analysis.

Created an anytime algorithm matching optimal fixed-time regret in Brownian motion gain scenarios.

Abstract

Prediction with experts' advice is one of the most fundamental problems in online learning and captures many of its technical challenges. A recent line of work has looked at online learning through the lens of differential equations and continuous-time analysis. This viewpoint has yielded optimal results for several problems in online learning. In this paper, we employ continuous-time stochastic calculus in order to study the discrete-time experts' problem. We use these tools to design a continuous-time, parameter-free algorithm with improved guarantees for the quantile regret. We then develop an analogous discrete-time algorithm with a very similar analysis and identical quantile regret bounds. Finally, we design an anytime continuous-time algorithm with regret matching the optimal fixed-time rate when the gains are independent Brownian Motions; in many settings, this is the most difficult case. This gives some evidence that, even with adversarial gains, the optimal anytime and fixed-time regrets may coincide.