Sparsity-Based Interpolation of External, Internal and Swap Regret

TL;DR

This paper introduces a unified algorithm for $oldsymbol{ ext{phi-regret}}$ minimization in online learning, achieving adaptive bounds that interpolate between external, internal, and swap regret, with efficient computation and improved performance in various regimes.

Contribution

It presents a novel, instance-adaptive algorithm for $oldsymbol{ ext{phi-regret}}$ minimization that unifies and improves upon existing regret bounds across different regimes.

Findings

Achieves optimal bounds for external, internal, and swap regret.

Provides a computationally efficient algorithm matching standard swap-regret complexity.

Utilizes Haar-wavelet-inspired features and sparsity-aware online regression techniques.

Abstract

Focusing on the expert problem in online learning, this paper studies the interpolation of several performance metrics via $\phi$-regret minimization, which measures the total loss of an algorithm by its regret with respect to an arbitrary action modification rule $\phi$. With $d$ experts and $T\gg d$ rounds in total, we present a single algorithm achieving the instance-adaptive $\phi$-regret bound \begin{equation*} \tilde O\left(\min\left\{\sqrt{d-d^{\mathrm{unif}}_\phi+1},\sqrt{d-d^{\mathrm{self}}_\phi}\right\}\cdot\sqrt{T}\right), \end{equation*} where $d^{\mathrm{unif}}_\phi$ is the maximum amount of experts modified identically by $\phi$, and $d^{\mathrm{self}}_\phi$ is the amount of experts that $\phi$ trivially modifies to themselves. By recovering the optimal $O(\sqrt{T\log d})$ external regret bound when $d^{\mathrm{unif}}_\phi=d$, the standard $\tilde O(\sqrt{T})$ internal regret bound when $d^{\mathrm{self}}_\phi=d-1$ and the optimal $\tilde O(\sqrt{dT})$ swap regret bound in the worst case, we improve upon existing algorithms in the intermediate regimes. In addition, the computational complexity of our algorithm matches that of the standard swap-regret minimization algorithm due to (Blum and Mansour, 2007). Technically, building on the well-known reduction from $\phi$-regret minimization to external regret minimization on stochastic matrices, our main idea is to further convert the latter to online linear regression using Haar-wavelet-inspired matrix features. Then, by associating the complexity of each $\phi$ instance with its sparsity under the feature representation, we apply techniques from comparator-adaptive online learning to exploit the sparsity in this regression subroutine.