Tactics for Improving Least Squares Estimation

Qiang Heng; Hua Zhou; Kenneth Lange

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arXiv · 2501.02475

Tactics for Improving Least Squares Estimation

Qiang Heng·Hua Zhou·Kenneth Lange

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Abstract

This paper deals with tactics for fast computation in least squares regression in high dimensions. These tactics include: (a) the majorization-minimization (MM) principle, (b) smoothing by Moreau envelopes, and (c) the proximal distance principle for constrained estimation. In iteratively reweighted least squares, the MM principle can create a surrogate function that trades case weights for adjusted responses. Reduction to ordinary least squares then permits the reuse of the Gram matrix and its Cholesky decomposition across iterations. This tactic is pertinent to estimation in L2E regression and generalized linear models. For problems such as quantile regression, non-smooth terms of an objective function can be replaced by their Moreau envelope approximations and majorized by spherical quadratics. Finally, penalized regression with distance-to-set penalties also benefits from this perspective. Our numerical experiments validate the speed and utility of deweighting and Moreau envelope approximations. Julia software implementing these experiments is available on our web page.

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Qiang Heng, Hua Zhou, Kenneth Lange. 2025-01-05. Tactics for Improving Least Squares Estimation. https://arxiv.org/abs/2501.02475

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