Randomized Bregman Coordinate Descent Methods for Non-Lipschitz Optimization

• URL: http://arxiv.org/abs/2001.05202v1
• Date: Wed, 15 Jan 2020 09:57:38 GMT
• Title: Randomized Bregman Coordinate Descent Methods for Non-Lipschitz Optimization
• Authors: Tianxiang Gao, Songtao Lu, Jia Liu, Chris Chu
• Abstract summary: A new textitrandomized Bregman (block) coordinate descent (CD) method is proposed. We show that the proposed method is $O(epsilon-2n)$ to achieve $epsilon-stationary point, where $n$ is the number of blocks of coordinates.
• Score: 31.474280642125734
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We propose a new \textit{randomized Bregman (block) coordinate descent} (RBCD) method for minimizing a composite problem, where the objective function could be either convex or nonconvex, and the smooth part are freed from the global Lipschitz-continuous (partial) gradient assumption. Under the notion of relative smoothness based on the Bregman distance, we prove that every limit point of the generated sequence is a stationary point. Further, we show that the iteration complexity of the proposed method is $O(n\varepsilon^{-2})$ to achieve $\epsilon$-stationary point, where $n$ is the number of blocks of coordinates. If the objective is assumed to be convex, the iteration complexity is improved to $O(n\epsilon^{-1} )$. If, in addition, the objective is strongly convex (relative to the reference function), the global linear convergence rate is recovered. We also present the accelerated version of the RBCD method, which attains an $O(n\varepsilon^{-1/\gamma} )$ iteration complexity for the convex case, where the scalar $\gamma\in [1,2]$ is determined by the \textit{generalized translation variant} of the Bregman distance. Convergence analysis without assuming the global Lipschitz-continuous (partial) gradient sets our results apart from the existing works in the composite problems.

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