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

arXiv: https://arxiv.org/abs/2306.02420 · HTML: https://mlsanity.com/paper/2306.02420/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 45 references that are in the map.
- First posted on arXiv 4 Jun 2023.

## Builds on (most cited first, 10 of 10)

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Coordinate Descent Algorithms](https://mlsanity.com/paper/1502.04759/) (2015)
- [Global Convergence of Block Coordinate Descent in Deep Learning](https://mlsanity.com/paper/1803.00225/) (2018)
- [A Smoothed Dual Approach for Variational Wasserstein Problems](https://mlsanity.com/paper/1503.02533/) (2015)
- [When Are Nonconvex Problems Not Scary?](https://mlsanity.com/paper/1510.06096/) (2015)
- [Convergent Block Coordinate Descent for Training Tikhonov Regularized Deep Neural Networks](https://mlsanity.com/paper/1711.07354/) (2017)
- [Iteration Complexity Analysis of Block Coordinate Descent Methods](https://mlsanity.com/paper/1310.6957/) (2013)
- [Minimax Lower Bounds for Kronecker-Structured Dictionary Learning](https://mlsanity.com/paper/1605.05284/) (2016)
- [STARK: Structured Dictionary Learning Through Rank-one Tensor Recovery](https://mlsanity.com/paper/1711.04887/) (2017)

## Built on by (most cited first, 2 of 2)

- [Stochastic optimization with arbitrary recurrent data sampling](https://mlsanity.com/paper/2401.07694/) (2024)
- [Convergence and Complexity Guarantee for Inexact First-order Riemannian Optimization Algorithms](https://mlsanity.com/paper/2405.03073/) (2024)

## Abstract (arXiv)

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.