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

Distributed Big-Data Optimization via Block Communications

We study distributed multi-agent large-scale optimization problems, wherein the cost function is composed of a smooth possibly nonconvex sum-utility plus a DC (Difference-of-Convex) regularizer. We consider the scenario where the dimension of the optimization variables is so large that optimizing and/or transmitting the entire set of variables could cause unaffordable computation and communication overhead. To address this issue, we propose the first distributed algorithm whereby agents optimize and communicate only a portion of their local variables. The scheme hinges on successive convex approximation (SCA) to handle the nonconvexity of the objective function, coupled with a novel block-signal tracking scheme, aiming at locally estimating the average of the agents' gradients. Asymptotic convergence to stationary solutions of the nonconvex problem is established. Numerical results on a sparse regression problem show the effectiveness of the proposed algorithm and the impact of the block size on its practical convergence speed and communication cost

Distributed big-data optimization via block communications

We study distributed multi-agent large-scale optimization problems, wherein the cost function is composed of a smooth possibly nonconvex sum-utility plus a DC (Difference-of-Convex) regularizer. We consider the scenario where the dimension of the optimization variables is so large that optimizing and/or transmitting the entire set of variables could cause unaffordable computation and communication overhead. To address this issue, we propose the first distributed algorithm whereby agents optimize and communicate only a portion of their local variables. The scheme hinges on successive convex approximation (SCA) to handle the nonconvexity of the objective function, coupled with a novel block- signal tracking scheme, aiming at locally estimating the average of the agents\u2019 gradients. Asymptotic convergence to stationary solutions of the nonconvex problem is established. Numerical results on a sparse regression problem show the effectiveness of the proposed algorithm and the impact of the block size on its practical convergence speed and communication cost

A Distributed Asynchronous Method of Multipliers for Constrained Nonconvex Optimization

This paper presents a fully asynchronous and distributed approach for tackling optimization problems in which both the objective function and the constraints may be nonconvex. In the considered network setting each node is active upon triggering of a local timer and has access only to a portion of the objective function and to a subset of the constraints. In the proposed technique, based on the method of multipliers, each node performs, when it wakes up, either a descent step on a local augmented Lagrangian or an ascent step on the local multiplier vector. Nodes realize when to switch from the descent step to the ascent one through an asynchronous distributed logic-AND, which detects when all the nodes have reached a predefined tolerance in the minimization of the augmented Lagrangian. It is shown that the resulting distributed algorithm is equivalent to a block coordinate descent for the minimization of the global augmented Lagrangian. This allows one to extend the properties of the centralized method of multipliers to the considered distributed framework. Two application examples are presented to validate the proposed approach: a distributed source localization problem and the parameter estimation of a neural network.Comment: arXiv admin note: substantial text overlap with arXiv:1803.0648

Distributed Big-Data Optimization via Block-wise Gradient Tracking

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 plus a convex (possibly) nonsmooth regularizer. Our interest is on big-data problems in which the number of optimization variables is large. 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 where, at each iteration, agents update in an uncoordinated fashion only one block of the entire decision vector. To deal with the nonconvexity of the cost, our scheme hinges on Successive Convex Approximation (SCA) techniques combined with a novel block-wise perturbed push-sum consensus protocol, instrumental for local block-averaging operations and tracking of gradient averages. Asymptotic convergence to stationary solutions of the nonconvex problem is proved. Numerical results show the effectiveness of the proposed algorithm and highlight how the block dimension impacts on the communication overhead and practical convergence speed

Supervised Learning Under Distributed Features

This work studies the problem of learning under both large datasets and large-dimensional feature space scenarios. The feature information is assumed to be spread across agents in a network, where each agent observes some of the features. Through local cooperation, the agents are supposed to interact with each other to solve an inference problem and converge towards the global minimizer of an empirical risk. We study this problem exclusively in the primal domain, and propose new and effective distributed solutions with guaranteed convergence to the minimizer with linear rate under strong convexity. This is achieved by combining a dynamic diffusion construction, a pipeline strategy, and variance-reduced techniques. Simulation results illustrate the conclusions

Recent Advances in Randomized Methods for Big Data Optimization

In this thesis, we discuss and develop randomized algorithms for big data problems. In particular, we study the finite-sum optimization with newly emerged variance- reduction optimization methods (Chapter 2), explore the efficiency of second-order information applied to both convex and non-convex finite-sum objectives (Chapter 3) and employ the fast first-order method in power system problems (Chapter 4).In Chapter 2, we propose two variance-reduced gradient algorithms – mS2GD and SARAH. mS2GD incorporates a mini-batching scheme for improving the theoretical complexity and practical performance of SVRG/S2GD, aiming to minimize a strongly convex function represented as the sum of an average of a large number of smooth con- vex functions and a simple non-smooth convex regularizer. While SARAH, short for StochAstic Recursive grAdient algoritHm and using a stochastic recursive gradient, targets at minimizing the average of a large number of smooth functions for both con- vex and non-convex cases. Both methods fall into the category of variance-reduction optimization, and obtain a total complexity of O((n+κ)log(1/ε)) to achieve an ε-accuracy solution for strongly convex objectives, while SARAH also maintains a sub-linear convergence for non-convex problems. Meanwhile, SARAH has a practical variant SARAH+ due to its linear convergence of the expected stochastic gradients in inner loops.In Chapter 3, we declare that randomized batches can be applied with second- order information, as to improve upon convergence in both theory and practice, with a framework of L-BFGS as a novel approach to finite-sum optimization problems. We provide theoretical analyses for both convex and non-convex objectives. Meanwhile, we propose LBFGS-F as a variant where Fisher information matrix is used instead of Hessian information, and prove it applicable to a distributed environment within the popular applications of least-square and cross-entropy losses.In Chapter 4, we develop fast randomized algorithms for solving polynomial optimization problems on the applications of alternating-current optimal power flows (ACOPF) in power system field. The traditional research on power system problem focuses on solvers using second-order method, while no randomized algorithms have been developed. First, we propose a coordinate-descent algorithm as an online solver, applied for solving time-varying optimization problems in power systems. We bound the difference between the current approximate optimal cost generated by our algorithm and the optimal cost for a relaxation using the most recent data from above by a function of the properties of the instance and the rate of change to the instance over time. Second, we focus on a steady-state problem in power systems, and study means of switching from solving a convex relaxation to Newton method working on a non-convex (augmented) Lagrangian of the problem