Generalization Guarantees of Randomized Block Coordinate Descent for Nonconvex Learning

Abstract: Randomized block coordinate descent (RBCD) is a popular optimization algorithm for many large-scale machine learning problems. The existing theoretical analysis of RBCD mainly focuses on its convergence behavior from an optimization perspective, while its generalization behavior remains less understood for nonconvex problems. In this paper, we establish high-probability generalization guarantees for RBCD with nonconvex objectives. In the general nonconvex setting, we derive bounds on the gradients of both empirical and population risks, requiring only standard smoothness assumptions. With polynomially decaying step sizes, we tune the iteration number to balance optimization and generalization, thereby quantifying how early stopping leads to the optimal convergence rate for the population gradients. In the gradient-dominated setting, we provide bounds in terms of functional suboptimality, which nearly match the optimistic rates in convex and low-noise settings. Finally, we develop dimension-independent bounds for loss functions of a specific structure, making the generalization guarantees applicable to high-dimensional problems.