Approximate Leave-One-Out Cross Validation for Regression With ℓ₁ Regularizers

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 $|{\mathrm { ALO}}-{\mathrm { LO}}|$ 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 $|{\mathrm { ALO}}-{\mathrm { LO}}| \xrightarrow {p\rightarrow \infty } 0$ while $n/p$ and signal-to-noise ratio (SNR) are bounded.