Nonparametric Finite Mixture Models with Possible Shape Constraints: A Cubic Newton Approach

TL;DR

This paper introduces a cubic Newton method for nonparametric finite mixture models with shape constraints, providing new computational guarantees and demonstrating improved scalability over existing methods.

Contribution

It develops a novel cubic regularized Newton algorithm with theoretical guarantees for structured mixture models, extending previous work to self-concordant objectives with polyhedral constraints.

Findings

The proposed method offers improved runtime and scalability.

The algorithm provides worst-case and local computational guarantees.

Bounds are derived for Gaussian mixtures approximating infinite-dimensional estimators.

Abstract

We explore computational aspects of maximum likelihood estimation of the mixture proportions of a nonparametric finite mixture model -- a convex optimization problem with old roots in statistics and a key member of the modern data analysis toolkit. Motivated by problems in shape constrained inference, we consider structured variants of this problem with additional convex polyhedral constraints. We propose a new cubic regularized Newton method for this problem and present novel worst-case and local computational guarantees for our algorithm. We extend earlier work by Nesterov and Polyak to the case of a self-concordant objective with polyhedral constraints, such as the ones considered herein. We propose a Frank-Wolfe method to solve the cubic regularized Newton subproblem; and derive efficient solutions for the linear optimization oracles that may be of independent interest. In the particular case of Gaussian mixtures without shape constraints, we derive bounds on how well the finite mixture problem approximates the infinite-dimensional Kiefer-Wolfowitz maximum likelihood estimator. Experiments on synthetic and real datasets suggest that our proposed algorithms exhibit improved runtimes and scalability features over existing benchmarks.