Implicit differentiation of Lasso-type models for hyperparameter optimization

TL;DR

This paper introduces an efficient implicit differentiation method for Lasso-type models that scales well with high-dimensional data and improves hyperparameter optimization accuracy compared to standard approaches.

Contribution

The work develops a novel implicit differentiation algorithm for Lasso models that avoids matrix inversion and exploits sparsity, enhancing hyperparameter tuning efficiency.

Findings

Outperforms standard hyperparameter optimization methods in accuracy.

Scales effectively to high-dimensional datasets.

Demonstrates improved error minimization on held-out data.

Abstract

Setting regularization parameters for Lasso-type estimators is notoriously difficult, though crucial in practice. The most popular hyperparameter optimization approach is grid-search using held-out validation data. Grid-search however requires to choose a predefined grid for each parameter, which scales exponentially in the number of parameters. Another approach is to cast hyperparameter optimization as a bi-level optimization problem, one can solve by gradient descent. The key challenge for these methods is the estimation of the gradient with respect to the hyperparameters. Computing this gradient via forward or backward automatic differentiation is possible yet usually suffers from high memory consumption. Alternatively implicit differentiation typically involves solving a linear system which can be prohibitive and numerically unstable in high dimension. In addition, implicit differentiation usually assumes smooth loss functions, which is not the case for Lasso-type problems. This work introduces an efficient implicit differentiation algorithm, without matrix inversion, tailored for Lasso-type problems. Our approach scales to high-dimensional data by leveraging the sparsity of the solutions. Experiments demonstrate that the proposed method outperforms a large number of standard methods to optimize the error on held-out data, or the Stein Unbiased Risk Estimator (SURE).