Distributionally Robust Linear Regression With Block Lewis Weights

• URL: http://arxiv.org/abs/2607.00252v1
• Date: Tue, 30 Jun 2026 23:01:08 GMT
• Title: Distributionally Robust Linear Regression With Block Lewis Weights
• Abstract summary: We present an algorithm for the group distributionally robust (GDR) least squares problem.<n>Our algorithm improves over known interior point methods for moderate accuracy regimes.<n>We also give algorithms that smoothly interpolate between minimizing the average least squares loss and the distributionally robust loss.
• Score: 4.787652251527859
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We present an algorithm for the group distributionally robust (GDR) least squares problem. Given $m$ groups, a parameter vector in $\mathbb{R}^d$, and stacked design matrices and responses $\mathbf{A}$ and $\mathbf{b}$, our algorithm obtains a $(1+\varepsilon)$-multiplicative optimal solution using $\widetilde{O}(\min\{\mathsf{rank}(\mathbf{A}),m\}^{1/3}\varepsilon^{-2/3})$ linear-system-solves of matrices of the form $\mathbf{A}^{\top}\mathbf{B}\mathbf{A}$ for block-diagonal $\mathbf{B}$. Our technical methods follow from a recent geometric construction, block Lewis weights, that relates the empirical GDR problem to a carefully chosen least squares problem and an application of accelerated proximal methods. Our algorithm improves over known interior point methods for moderate accuracy regimes and matches the state-of-the-art guarantees for the special case of $\ell_{\infty}$ regression. We also give algorithms that smoothly interpolate between minimizing the average least squares loss and the distributionally robust loss.

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