A Polynomial-time Algorithm for Online Sparse Linear Regression with Improved Regret Bound under Weaker Conditions

TL;DR

This paper presents a new polynomial-time algorithm for online sparse linear regression that achieves improved regret bounds under weaker assumptions, utilizing novel techniques like adaptive sampling and batching online Newton steps.

Contribution

Introduces a polynomial-time algorithm for OSLR with better regret bounds under the weaker compatibility condition, using innovative estimation and tuning methods.

Findings

Achieves tighter regret bounds compared to previous algorithms.

Extends to OSLR with additional observations, improving prior results.

Employs novel analysis techniques for covariance and regret decomposition.

Abstract

In this paper, we study the problem of online sparse linear regression (OSLR) where the algorithms are restricted to accessing only $k$ out of $d$ attributes per instance for prediction, which was proved to be NP-hard. Previous work gave polynomial-time algorithms assuming the data matrix satisfies the linear independence of features, the compatibility condition, or the restricted isometry property. We introduce a new polynomial-time algorithm, which significantly improves previous regret bounds (Ito et al., 2017) under the compatibility condition that is weaker than the other two assumptions. The improvements benefit from a tighter convergence rate of the $\ell_1$-norm error of our estimators. Our algorithm leverages the well-studied Dantzig Selector, but importantly with several novel techniques, including an algorithm-dependent sampling scheme for estimating the covariance matrix, an adaptive parameter tuning scheme, and a batching online Newton step with careful initializations. We also give novel and non-trivial analyses, including an induction method for analyzing the $\ell_1$-norm error, careful analyses on the covariance of non-independent random variables, and a decomposition on the regret. We further extend our algorithm to OSLR with additional observations where the algorithms can observe additional $k_0$ attributes after each prediction, and improve previous regret bounds (Kale et al., 2017; Ito et al., 2017).