A Saddle Point Algorithm for Robust Data-Driven Factor Model Problems
This work addresses the factor model problem for data analysis, offering an incremental improvement in optimization methods for high-dimensional datasets.
The paper tackles the factor model problem by developing a robust data-driven saddle-point optimization algorithm that leverages linear minimization oracles, and it demonstrates effectiveness in high-dimensional settings with outperformance over standard solvers.
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.