Complexity of Block Coordinate Descent with Proximal Regularization and Applications to Wasserstein CP-dictionary Learning
Complexity of Block Coordinate Descent with Proximal Regularization and Applications to Wasserstein CP-dictionary Learning
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DOI：
10.48550/arxiv.2306.02420
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发表时间：
2023-06
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影响因子：
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通讯作者：
Dohyun Kwon;Hanbaek Lyu
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文献类型：
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作者：
Dohyun Kwon;Hanbaek Lyu
本文研究了具有近端正则化的Gauss-Seidel型块坐标下降方法（BCD-PR），它是一种经典的在约束条件下最小化一般非凸目标的方法，具有广泛的实际应用。从理论上建立了该算法的最坏情况复杂度界。也就是说，我们证明了对于具有块约束的一般非凸光滑目标，经典BCD-PR算法在O（1/epsilon）迭代内收敛到一个epsilon平稳点。在温和的条件下，即使算法在每一步中执行得不准确，这个结果仍然成立。作为一个应用，我们提出了一个可证明的高效的“Wasserstein CP-dictionary学习”算法，该算法寻求一组可以很好地近似给定的d维联合概率分布的初等概率分布。我们的算法是BCD-PR的一个版本，它在对偶空间中运行，其中原始问题在熵上和近端上都是正则化的。
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.