Relax and penalize: a new bilevel approach to mixed-binary hyperparameter optimization

TL;DR

This paper introduces a novel bilevel optimization method for mixed-binary hyperparameters in machine learning, using a continuous reformulation with penalties to ensure consistent solutions, and demonstrates competitive results in two applications.

Contribution

The paper presents a new bilevel approach that directly handles mixed-binary hyperparameters via a continuous reformulation with penalties, avoiding relaxation and rounding issues.

Findings

Method is guaranteed to produce mixed-binary solutions under certain conditions.

Approach is compatible with existing continuous bilevel solvers.

Achieves competitive performance on hyperparameter estimation tasks.

Abstract

In recent years, bilevel approaches have become very popular to efficiently estimate high-dimensional hyperparameters of machine learning models. However, to date, binary parameters are handled by continuous relaxation and rounding strategies, which could lead to inconsistent solutions. In this context, we tackle the challenging optimization of mixed-binary hyperparameters by resorting to an equivalent continuous bilevel reformulation based on an appropriate penalty term. We propose an algorithmic framework that, under suitable assumptions, is guaranteed to provide mixed-binary solutions. Moreover, the generality of the method allows to safely use existing continuous bilevel solvers within the proposed framework. We evaluate the performance of our approach for two specific machine learning problems, i.e., the estimation of the group-sparsity structure in regression problems and the data distillation problem. The reported results show that our method is competitive with state-of-the-art approaches based on relaxation and rounding