Asymptotic Dynamics of Alternating Minimization for Bilinear Regression

This study investigates the asymptotic dynamics of alternating minimization applied to optimize a bilinear non-convex function with normally distributed covariates. This is achieved by employing the replica method to a multi-temperature glassy system which unfolds the algorithm's time evolution. Our results show that the dynamics can be described effectively by a two-dimensional discrete stochastic process, where each step depends on all previous time steps, revealing the structure of the memory dependence in the evolution of alternating minimization. The theoretical framework developed in this work can be applied to the analysis of various iterative algorithms, extending beyond the scope of alternating minimization.
Comment: 28 pages, 4 figures, Submission to SciPost