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

TL;DR

Through a detailed theoretical analysis, it is proved 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.

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.

Figures (2)

• Figure 1: Time-averaged regret vs time (a) for synthetic data with $\bm{u}_t = \bm{u}$, (b) for synthetic data with variable $\bm{u}_t$ such that $\bm{u}_{t,i}\sim \mathcal{U}[0,1]$ i.i.d. $\forall i\in \Lambda$, with $M=256,N=512,K=10$, (c) for synthetic data with variable $\bm{u}_t$ such that $u_{t,i}\sim \mathcal{U}[0,1]$ is updated if $t$ is a power of $2$, (d) for ensemble average over different DIGITS figures with $N=784,\ M=392,\ K=10$.
• Figure 2: Execution time vs $T$.

Theorems & Definitions (6)

• Lemma 1
• Theorem 1
• proof
• Theorem 2: Theorem 13.3,zhang2023mathematical
• Lemma 2
• proof