Approximate Leave-one-out Cross Validation for Regression with $\ell_1$ Regularizers (extended version)

• URL: http://arxiv.org/abs/2310.17629v1
• Date: Thu, 26 Oct 2023 17:48:10 GMT
• Title: Approximate Leave-one-out Cross Validation for Regression with $\ell_1$ Regularizers (extended version)
• Authors: Arnab Auddy, Haolin Zou, Kamiar Rahnama Rad, Arian Maleki
• Abstract summary: We present a novel theory for a wide class of problems in the generalized linear model family with non-differentiable regularizers. We show that |ALO - LO| goes to zero as p goes to infinity while n/p and SNR are fixed and bounded.
• Score: 12.029919627622954
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: The out-of-sample error (OO) is the main quantity of interest in risk estimation and model selection. Leave-one-out cross validation (LO) offers a (nearly) distribution-free yet computationally demanding approach to estimate OO. Recent theoretical work showed that approximate leave-one-out cross validation (ALO) is a computationally efficient and statistically reliable estimate of LO (and OO) for generalized linear models with differentiable regularizers. For problems involving non-differentiable regularizers, despite significant empirical evidence, the theoretical understanding of ALO's error remains unknown. In this paper, we present a novel theory for a wide class of problems in the generalized linear model family with non-differentiable regularizers. We bound the error |ALO - LO| in terms of intuitive metrics such as the size of leave-i-out perturbations in active sets, sample size n, number of features p and regularization parameters. As a consequence, for the $\ell_1$-regularized problems, we show that |ALO - LO| goes to zero as p goes to infinity while n/p and SNR are fixed and bounded.

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