A Saddle Point Algorithm for Robust Data-Driven Factor Model Problems

TL;DR

This paper introduces a saddle-point algorithm for robust factor model problems, leveraging linear minimization oracles and providing semi-closed form solutions for specific divergences, with demonstrated high-dimensional effectiveness.

Contribution

It presents a novel first-order saddle-point algorithm with explicit solutions for key divergence-based LMOs, improving robustness and efficiency in high-dimensional factor modeling.

Findings

The algorithm effectively solves high-dimensional factor models.

Semi-closed form solutions are derived for Frobenius, KL, and Wasserstein divergences.

Numerical experiments show superior performance over standard solvers.

Abstract

We study the factor model problem, which aims to uncover low-dimensional structures in high-dimensional datasets. Adopting a robust data-driven approach, we formulate the problem as a saddle-point optimization. Our primary contribution is a first-order algorithm that solves this reformulation by leveraging a linear minimization oracle (LMO). We further develop semi-closed form solutions (up to a scalar) for three specific LMOs, corresponding to the Frobenius norm, Kullback-Leibler divergence, and Gelbrich (aka Wasserstein) distance. The analysis includes explicit quantification of these LMOs' regularity conditions, notably the Lipschitz constants of the dual function, which govern the algorithm's convergence performance. Numerical experiments confirm our method's effectiveness in high-dimensional settings, outperforming standard off-the-shelf optimization solvers.