Sparsest Univariate Learning Models Under Lipschitz Constraint

TL;DR

This paper introduces novel continuous-domain methods for univariate regression that incorporate Lipschitz constraints to enhance stability and interpretability, and develops algorithms to find the sparsest solutions with theoretical guarantees.

Contribution

It proposes two Lipschitz-constrained regression formulations with proven representer theorems and develops algorithms to identify the sparsest, interpretable solutions.

Findings

Both formulations admit global minimizers that are continuous and piecewise-linear.

Algorithms effectively find the sparsest CPWL solutions.

Numerical experiments demonstrate the practical effectiveness of the methods.

Abstract

Beside the minimization of the prediction error, two of the most desirable properties of a regression scheme are stability and interpretability. Driven by these principles, we propose continuous-domain formulations for one-dimensional regression problems. In our first approach, we use the Lipschitz constant as a regularizer, which results in an implicit tuning of the overall robustness of the learned mapping. In our second approach, we control the Lipschitz constant explicitly using a user-defined upper-bound and make use of a sparsity-promoting regularizer to favor simpler (and, hence, more interpretable) solutions. The theoretical study of the latter formulation is motivated in part by its equivalence, which we prove, with the training of a Lipschitz-constrained two-layer univariate neural network with rectified linear unit (ReLU) activations and weight decay. By proving representer theorems, we show that both problems admit global minimizers that are continuous and piecewise-linear (CPWL) functions. Moreover, we propose efficient algorithms that find the sparsest solution of each problem: the CPWL mapping with the least number of linear regions. Finally, we illustrate numerically the outcome of our formulations.