A Differential Equations Approach to Optimizing Regret Trade-offs
This work addresses a foundational problem in online learning and regret optimization, offering theoretical insights for researchers, but it is incremental as it builds on prior results like Cover's theorem and focuses on a specific discounted setting.
The paper tackles the problem of finding provably optimal algorithms for predicting binary sequences with time-discounted payoffs, focusing on regret trade-offs, and shows that the optimal payoff functions satisfy the Hermite differential equation, leading to a unique solution in this setting.
We consider the classical question of predicting binary sequences and study the {\em optimal} algorithms for obtaining the best possible regret and payoff functions for this problem. The question turns out to be also equivalent to the problem of optimal trade-offs between the regrets of two experts in an "experts problem", studied before by \cite{kearns-regret}. While, say, a regret of $Θ(\sqrt{T})$ is known, we argue that it important to ask what is the provably optimal algorithm for this problem --- both because it leads to natural algorithms, as well as because regret is in fact often comparable in magnitude to the final payoffs and hence is a non-negligible term. In the basic setting, the result essentially follows from a classical result of Cover from '65. Here instead, we focus on another standard setting, of time-discounted payoffs, where the final "stopping time" is not specified. We exhibit an explicit characterization of the optimal regret for this setting. To obtain our main result, we show that the optimal payoff functions have to satisfy the Hermite differential equation, and hence are given by the solutions to this equation. It turns out that characterization of the payoff function is qualitatively different from the classical (non-discounted) setting, and, namely, there's essentially a unique optimal solution.