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Register a new userA Block Decomposition Algorithm for Sparse Optimization
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Added by
Ganzhao Yuan
Publication date
2019
fields
Informatics Engineering
and research's language is
English
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Sparse optimization is a central problem in machine learning and computer vision. However, this problem is inherently NP-hard and thus difficult to solve in general. Combinatorial search methods find the global optimal solution but are confined to small-sized problems, while coordinate descent methods are efficient but often suffer from poor local minima. This paper considers a new block decomposition algorithm that combines the effectiveness of combinatorial search methods and the efficiency of coordinate descent methods. Specifically, we consider a random strategy or/and a greedy strategy to select a subset of coordinates as the working set, and then perform a global combinatorial search over the working set based on the original objective function. We show that our method finds stronger stationary points than Amir Beck et al.s coordinate-wise optimization method. In addition, we establish the convergence rate of our algorithm. Our experiments on solving sparse regularized and sparsity constrained least squares optimization problems demonstrate that our method achieves state-of-the-art performance in terms of accuracy. For example, our method generally outperforms the well-known greedy pursuit method.
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This paper proposes TriPD, a new primal-dual algorithm for minimizing the sum of a Lipschitz-differentiable convex function and two possibly nonsmooth convex functions, one of which is composed with a linear mapping. We devise a randomized block-coordinate version of the algorithm which converges under the same stepsize conditions as the full algorithm. It is shown that both the original as well as the block-coordinate scheme feature linear convergence rate when the functions involved are either piecewise linear-quadratic, or when they satisfy a certain quadratic growth condition (which is weaker than strong convexity). Moreover, we apply the developed algorithms to the problem of multi-agent optimization on a graph, thus obtaining novel synchronous and asynchronous distributed methods. The proposed algorithms are fully distributed in the sense that the updates and the stepsizes of each agent only depend on local information. In fact, no prior global coordination is required. Finally, we showcase an application of our algorithm in distributed formation control.
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