Mesh-Based Solutions for Nonparametric Penalized Regression

Brayan Ortiz; Noah Simon

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arXiv · 2112.03428

Mesh-Based Solutions for Nonparametric Penalized Regression

Brayan Ortiz·Noah Simon

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Abstract

It is often of interest to estimate regression functions non-parametrically. Penalized regression (PR) is one statistically-effective, well-studied solution to this problem. Unfortunately, in many cases, finding exact solutions to PR problems is computationally intractable. In this manuscript, we propose a mesh-based approximate solution (MBS) for those scenarios. MBS transforms the complicated functional minimization of NPR, to a finite parameter, discrete convex minimization; and allows us to leverage the tools of modern convex optimization. We show applications of MBS in a number of explicit examples (including both uni- and multi-variate regression), and explore how the number of parameters must increase with our sample-size in order for MBS to maintain the rate-optimality of NPR. We also give an efficient algorithm to minimize the MBS objective while effectively leveraging the sparsity inherent in MBS.

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Brayan Ortiz, Noah Simon. 2021-12-07. Mesh-Based Solutions for Nonparametric Penalized Regression. https://arxiv.org/abs/2112.03428

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