Self-concordant smoothing in proximal quasi-Newton algorithms for large-scale convex composite optimization

TL;DR

This paper introduces a self-concordant smoothing technique for large-scale convex composite optimization, enabling efficient proximal quasi-Newton algorithms with convergence guarantees and practical performance improvements.

Contribution

It proposes a novel self-concordant smoothing framework that enhances proximal quasi-Newton methods for convex problems with nonsmooth components, including l1 and group lasso penalties.

Findings

Algorithms outperform state-of-the-art methods on synthetic and real data.

Prox-GGN-SCORE reduces computational cost via low-rank Hessian approximations.

Convergence guarantees are established for the proposed methods.

Abstract

We introduce a notion of self-concordant smoothing for minimizing the sum of two convex functions, one of which is smooth and the other nonsmooth. The key highlight is a natural property of the resulting problem's structure that yields a variable metric selection method and a step length rule especially suited to proximal quasi-Newton algorithms. Also, we efficiently handle specific structures promoted by the nonsmooth term, such as l1-regularization and group lasso penalties. A convergence analysis for the class of proximal quasi-Newton methods covered by our framework is presented. In particular, we obtain guarantees, under standard assumptions, for two algorithms: Prox-N-SCORE (a proximal Newton method) and Prox-GGN-SCORE (a proximal generalized Gauss-Newton method). The latter uses a low-rank approximation of the Hessian inverse, reducing most of the cost of matrix inversion and making it effective for overparameterized machine learning models. Numerical experiments on synthetic and real data demonstrate the efficiency of both algorithms against state-of-the-art approaches. A Julia implementation is publicly available at https://github.com/adeyemiadeoye/SelfConcordantSmoothOptimization.jl.