Complexity of Block Coordinate Descent with Proximal Regularization and
  Applications to Wasserstein CP-dictionary Learning
Dohyun Kwon Hanbaek Lyu
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.