An Efficient Smoothing Damped Newton Method for Large-Scale Mathematical Programs with Equilibrium Constraints

TL;DR

This paper introduces a highly efficient, package-free smoothing damped Newton method for large-scale mathematical programs with equilibrium constraints, specifically applied to hyperparameter optimization in machine learning.

Contribution

The paper presents a novel SDNM algorithm that leverages MPEC structure for improved efficiency and convergence in large-scale hyperparameter optimization problems.

Findings

SDNM outperforms existing algorithms on LIBSVM datasets.

SDNM converges quadratically under certain conditions.

The method is package-free and fully exploits MPEC structure.

Abstract

Bilevel hyperparameter optimization has received growing attention thanks to the fast development of machine learning. Due to the tremendous size of data sets, the scale of bilevel hyperparameter optimization problem could be extremely large, posing great challenges in designing efficient numerical algorithms. In this paper, we focus on solving the large-scale mathematical programs with equilibrium constraints (MPEC) derived from hyperparameter selection of L1 support vector classification (L1-SVC). We propose a highly efficient smoothing damped Newton method (SDNM) for solving such MPEC. Compared with most existing algorithms where subproblems are solved by packages, our approach fully takes advantage of the structure of MPEC and therefore is package-free. Moreover, the proposed SDNM converges to C-stationary point under MPEC-LICQ with subproblem enjoys a quadratic convergence rate under proper assumptions. Extensive numerical results over LIBSVM dataset show the superior performance of SDNM over other state-of-art algorithms.