Complexity of Block Coordinate Descent with Proximal Regularization and Applications to Wasserstein CP-dictionary Learning

TL;DR

This paper analyzes the complexity of block coordinate descent with proximal regularization for nonconvex problems, providing convergence bounds and applying it to Wasserstein CP-dictionary learning.

Contribution

It establishes the worst-case complexity bounds for BCD-PR on nonconvex objectives and introduces an efficient dual-space algorithm for Wasserstein CP-dictionary learning.

Findings

BCD-PR converges to an epsilon-stationary point within O(1/epsilon) iterations.

The results hold even with inexact updates under mild conditions.

The proposed algorithm effectively approximates joint probability distributions.

Abstract

We consider the block coordinate descent methods of Gauss-Seidel type with proximal regularization (BCD-PR), which is a classical method of minimizing general nonconvex objectives under constraints that has a wide range of practical applications. We theoretically establish the worst-case complexity bound for this algorithm. Namely, we show that for general nonconvex smooth objectives with block-wise constraints, the classical BCD-PR algorithm converges to an epsilon-stationary point within O(1/epsilon) iterations. Under a mild condition, this result still holds even if the algorithm is executed inexactly in each step. As an application, we propose a provable and efficient algorithm for `Wasserstein CP-dictionary learning', which seeks a set of elementary probability distributions that can well-approximate a given set of d-dimensional joint probability distributions. Our algorithm is a version of BCD-PR that operates in the dual space, where the primal problem is regularized both entropically and proximally.