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

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 general 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, whthich govern the algorithm's convergence performance. Numerical experiments confirm our meod's effectiveness in high-dimensional settings, outperforming standard off-the-shelf optimization solvers.