An Asynchronous Distributed Proximal Gradient Method for Composite Convex Optimization

Abstract We propose a distributed first-order augmented Lagrangian (DFAL) algorithm to minimize the sum of composite convex functions, where each term in the sum is a private cost function belonging to a node, and only nodes connected by an edge can directly communicate with each other. This optimization model abstracts a number of applications in distributed sensing and machine learning. We show that any limit point of DFAL iterates is optimal; and for any ǫ > 0, an ǫ-optimal and ǫ-feasible solution can be computed within O(log(ǫ −1 )) DFAL iterations, which require O( ψ 1.5 max dmin ǫ −1 ) proximal gradient computations and communications per node in total, where ψ max denotes the largest eigenvalue of the graph Laplacian, and d min is the minimum degree of the graph. We also propose an asynchronous version of DFAL by incorporating randomized block coordinate descent methods; and demonstrate the efficiency of DFAL on large scale sparse-group LASSO problems

A randomized primal distributed algorithm for partitioned and big-data non-convex optimization

In this paper we consider a distributed optimization scenario in which the aggregate objective function to minimize is partitioned, big-data and possibly non-convex. Specifically, we focus on a set-up in which the dimension of the decision variable depends on the network size as well as the number of local functions, but each local function handled by a node depends only on a (small) portion of the entire optimization variable. This problem set-up has been shown to appear in many interesting network application scenarios. As main paper contribution, we develop a simple, primal distributed algorithm to solve the optimization problem, based on a randomized descent approach, which works under asynchronous gossip communication. We prove that the proposed asynchronous algorithm is a proper, ad-hoc version of a coordinate descent method and thus converges to a stationary point. To show the effectiveness of the proposed algorithm, we also present numerical simulations on a non-convex quadratic program, which confirm the theoretical results

Breaking the Nonsmooth Barrier: A Scalable Parallel Method for Composite Optimization

Due to their simplicity and excellent performance, parallel asynchronous variants of stochastic gradient descent have become popular methods to solve a wide range of large-scale optimization problems on multi-core architectures. Yet, despite their practical success, support for nonsmooth objectives is still lacking, making them unsuitable for many problems of interest in machine learning, such as the Lasso, group Lasso or empirical risk minimization with convex constraints. In this work, we propose and analyze ProxASAGA, a fully asynchronous sparse method inspired by SAGA, a variance reduced incremental gradient algorithm. The proposed method is easy to implement and significantly outperforms the state of the art on several nonsmooth, large-scale problems. We prove that our method achieves a theoretical linear speedup with respect to the sequential version under assumptions on the sparsity of gradients and block-separability of the proximal term. Empirical benchmarks on a multi-core architecture illustrate practical speedups of up to 12x on a 20-core machine.Comment: Appears in Advances in Neural Information Processing Systems 30 (NIPS 2017), 28 page

Distributed Big-Data Optimization via Block-Iterative Convexification and Averaging

In this paper, we study distributed big-data nonconvex optimization in multi-agent networks. We consider the (constrained) minimization of the sum of a smooth (possibly) nonconvex function, i.e., the agents' sum-utility, plus a convex (possibly) nonsmooth regularizer. Our interest is in big-data problems wherein there is a large number of variables to optimize. If treated by means of standard distributed optimization algorithms, these large-scale problems may be intractable, due to the prohibitive local computation and communication burden at each node. We propose a novel distributed solution method whereby at each iteration agents optimize and then communicate (in an uncoordinated fashion) only a subset of their decision variables. To deal with non-convexity of the cost function, the novel scheme hinges on Successive Convex Approximation (SCA) techniques coupled with i) a tracking mechanism instrumental to locally estimate gradient averages; and ii) a novel block-wise consensus-based protocol to perform local block-averaging operations and gradient tacking. Asymptotic convergence to stationary solutions of the nonconvex problem is established. Finally, numerical results show the effectiveness of the proposed algorithm and highlight how the block dimension impacts on the communication overhead and practical convergence speed