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We consider the block coordinate descent methods of Gauss-Seidel type withproximal regularization (BCD-PR), which is a classical method of minimizinggeneral nonconvex objectives under constraints that has a wide range ofpractical applications. We theoretically establish the worst-case complexitybound for this algorithm. Namely, we show that for general nonconvex smoothobjectives with block-wise constraints, the classical BCD-PR algorithmconverges to an epsilon-stationary point within O(1/epsilon) iterations. Undera mild condition, this result still holds even if the algorithm is executedinexactly in each step. As an application, we propose a provable and efficientalgorithm for `Wasserstein CP-dictionary learning', which seeks a set ofelementary probability distributions that can well-approximate a given set ofd-dimensional joint probability distributions. Our algorithm is a version ofBCD-PR that operates in the dual space, where the primal problem is regularizedboth entropically and proximally.

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