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Approximate Leave-one-out Cross Validation for Regression with ℓ1 Regularizers
Proceedings of The 27th International Conference on Artificial Intelligence and Statistics, PMLR 238:2377-2385, 2024.
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 method 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 twice 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 the non-differentiable ℓ1 regularizer. 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 signal-to-noise ratio (SNR). As a consequence, for the ℓ1 regularized problems, we show that |ALO−LO|p→∞⟶0 while n/p and SNR remain bounded.