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

Junfan Li, Shizhong Liao, Zenglin Xu, Liqiang Nie
Proceedings of Thirty Eighth Conference on Learning Theory, PMLR 291:3623-3670, 2025.

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).

Cite this Paper

BibTeX
@InProceedings{pmlr-v291-li25c, title = {A Polynomial-time Algorithm for Online Sparse Linear Regression with Improved Regret Bound under Weaker Conditions}, author = {Li, Junfan and Liao, Shizhong and Xu, Zenglin and Nie, Liqiang}, booktitle = {Proceedings of Thirty Eighth Conference on Learning Theory}, pages = {3623--3670}, year = {2025}, editor = {Haghtalab, Nika and Moitra, Ankur}, volume = {291}, series = {Proceedings of Machine Learning Research}, month = {30 Jun--04 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v291/main/assets/li25c/li25c.pdf}, url = {https://proceedings.mlr.press/v291/li25c.html}, 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).} }
Endnote
%0 Conference Paper %T A Polynomial-time Algorithm for Online Sparse Linear Regression with Improved Regret Bound under Weaker Conditions %A Junfan Li %A Shizhong Liao %A Zenglin Xu %A Liqiang Nie %B Proceedings of Thirty Eighth Conference on Learning Theory %C Proceedings of Machine Learning Research %D 2025 %E Nika Haghtalab %E Ankur Moitra %F pmlr-v291-li25c %I PMLR %P 3623--3670 %U https://proceedings.mlr.press/v291/li25c.html %V 291 %X 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).
APA
Li, J., Liao, S., Xu, Z. & Nie, L.. (2025). A Polynomial-time Algorithm for Online Sparse Linear Regression with Improved Regret Bound under Weaker Conditions. Proceedings of Thirty Eighth Conference on Learning Theory, in Proceedings of Machine Learning Research 291:3623-3670 Available from https://proceedings.mlr.press/v291/li25c.html.

Related Material