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Stochastic Majorization-Minimization Algorithms for Large-Scale Optimization

Julien Mairal 1, * 
* Corresponding author
1 LEAR - Learning and recognition in vision
Inria Grenoble - Rhône-Alpes, LJK - Laboratoire Jean Kuntzmann, Grenoble INP - Institut polytechnique de Grenoble - Grenoble Institute of Technology
Abstract : Majorization-minimization algorithms consist of iteratively minimizing a majorizing surrogate of an objective function. Because of its simplicity and its wide applicability, this principle has been very popular in statistics and in signal processing. In this paper, we intend to make this principle scalable. We introduce a stochastic majorization-minimization scheme which is able to deal with large-scale or possibly infinite data sets. When applied to convex optimization problems under suitable assumptions, we show that it achieves an expected convergence rate of $O(1/\sqrt{n})$ after $n$ iterations, and of $O(1/n)$ for strongly convex functions. Equally important, our scheme almost surely converges to stationary points for a large class of non-convex problems. We develop several efficient algorithms based on our framework. First, we propose a new stochastic proximal gradient method, which experimentally matches state-of-the-art solvers for large-scale $\ell_1$-logistic regression. Second, we develop an online DC programming algorithm for non-convex sparse estimation. Finally, we demonstrate the effectiveness of our approach for solving large-scale structured matrix factorization problems.
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Submitted on : Tuesday, September 10, 2013 - 2:03:46 PM
Last modification on : Friday, January 28, 2022 - 9:02:02 PM
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  • HAL Id : hal-00835840, version 2
  • ARXIV : 1306.4650



Julien Mairal. Stochastic Majorization-Minimization Algorithms for Large-Scale Optimization. NIPS 2013 - Advances in Neural Information Processing Systems, Dec 2013, South Lake Tahoe, United States. pp.2283-2291. ⟨hal-00835840v2⟩



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