Follow The Approximate Sparse Leader for No-Regret Online Sparse Linear Approximation

TL;DR

This paper introduces an efficient online algorithm for sparse linear approximation that achieves sublinear regret bounds, with theoretical guarantees and supporting numerical experiments.

Contribution

The paper proposes Follow-The-Approximate-Sparse-Leader, a novel online meta-policy with proven regret bounds for sparse linear approximation tasks.

Findings

Achieves data-dependent sublinear regret bounds ranging from logarithmic to square-root.

Theoretical analysis confirms the effectiveness of the proposed policy.

Numerical simulations support the theoretical results and demonstrate practical efficacy.

Abstract

We consider the problem of \textit{online sparse linear approximation}, where one predicts the best sparse approximation of a sequence of measurements in terms of linear combination of columns of a given measurement matrix. Such online prediction problems are ubiquitous, ranging from medical trials to web caching to resource allocation. The inherent difficulty of offline recovery also makes the online problem challenging. In this letter, we propose Follow-The-Approximate-Sparse-Leader, an efficient online meta-policy to address this online problem. Through a detailed theoretical analysis, we prove that under certain assumptions on the measurement sequence, the proposed policy enjoys a data-dependent sublinear upper bound on the static regret, which can range from logarithmic to square-root. Numerical simulations are performed to corroborate the theoretical findings and demonstrate the efficacy of the proposed online policy.