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

Coordinate Descent Algorithms

Coordinate descent algorithms solve optimization problems by successively performing approximate minimization along coordinate directions or coordinate hyperplanes. They have been used in applications for many years, and their popularity continues to grow because of their usefulness in data analysis, machine learning, and other areas of current interest. This paper describes the fundamentals of the coordinate descent approach, together with variants and extensions and their convergence properties, mostly with reference to convex objectives. We pay particular attention to a certain problem structure that arises frequently in machine learning applications, showing that efficient implementations of accelerated coordinate descent algorithms are possible for problems of this type. We also present some parallel variants and discuss their convergence properties under several models of parallel execution

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

Markov Chain Block Coordinate Descent

The method of block coordinate gradient descent (BCD) has been a powerful method for large-scale optimization. This paper considers the BCD method that successively updates a series of blocks selected according to a Markov chain. This kind of block selection is neither i.i.d. random nor cyclic. On the other hand, it is a natural choice for some applications in distributed optimization and Markov decision process, where i.i.d. random and cyclic selections are either infeasible or very expensive. By applying mixing-time properties of a Markov chain, we prove convergence of Markov chain BCD for minimizing Lipschitz differentiable functions, which can be nonconvex. When the functions are convex and strongly convex, we establish both sublinear and linear convergence rates, respectively. We also present a method of Markov chain inertial BCD. Finally, we discuss potential applications

Incremental Aggregated Proximal and Augmented Lagrangian Algorithms

We consider minimization of the sum of a large number of convex functions, and we propose an incremental aggregated version of the proximal algorithm, which bears similarity to the incremental aggregated gradient and subgradient methods that have received a lot of recent attention. Under cost function differentiability and strong convexity assumptions, we show linear convergence for a sufficiently small constant stepsize. This result also applies to distributed asynchronous variants of the method, involving bounded interprocessor communication delays. We then consider dual versions of incremental proximal algorithms, which are incremental augmented Lagrangian methods for separable equality-constrained optimization problems. Contrary to the standard augmented Lagrangian method, these methods admit decomposition in the minimization of the augmented Lagrangian, and update the multipliers far more frequently. Our incremental aggregated augmented Lagrangian methods bear similarity to several known decomposition algorithms, including the alternating direction method of multipliers (ADMM) and more recent variations. We compare these methods in terms of their properties, and highlight their potential advantages and limitations. We also address the solution of separable inequality-constrained optimization problems through the use of nonquadratic augmented Lagrangiias such as the exponential, and we dually consider a corresponding incremental aggregated version of the proximal algorithm that uses nonquadratic regularization, such as an entropy function. We finally propose a closely related linearly convergent method for minimization of large differentiable sums subject to an orthant constraint, which may be viewed as an incremental aggregated version of the mirror descent method

Asynchronous parallel primal-dual block coordinate update methods for affinely constrained convex programs

Recent several years have witnessed the surge of asynchronous (async-) parallel computing methods due to the extremely big data involved in many modern applications and also the advancement of multi-core machines and computer clusters. In optimization, most works about async-parallel methods are on unconstrained problems or those with block separable constraints. In this paper, we propose an async-parallel method based on block coordinate update (BCU) for solving convex problems with nonseparable linear constraint. Running on a single node, the method becomes a novel randomized primal-dual BCU with adaptive stepsize for multi-block affinely constrained problems. For these problems, Gauss-Seidel cyclic primal-dual BCU needs strong convexity to have convergence. On the contrary, merely assuming convexity, we show that the objective value sequence generated by the proposed algorithm converges in probability to the optimal value and also the constraint residual to zero. In addition, we establish an ergodic $O(1/k)$ convergence result, where $k$ is the number of iterations. Numerical experiments are performed to demonstrate the efficiency of the proposed method and significantly better speed-up performance than its sync-parallel counterpart

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, i.e., the agents' sum-utility, plus a convex (possibly) nonsmooth regularizer. Our interest is on big-data problems in which 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 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 function, the novel scheme hinges on Successive Convex Approximation (SCA) techniques combined with a novel block-wise perturbed push-sum consensus protocol, which is instrumental to perform local block-averaging operations and tracking of gradient averages. 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\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

More Iterations per Second, Same Quality -- Why Asynchronous Algorithms may Drastically Outperform Traditional Ones

In this paper, we consider the convergence of a very general asynchronous-parallel algorithm called ARock, that takes many well-known asynchronous algorithms as special cases (gradient descent, proximal gradient, Douglas Rachford, ADMM, etc.). In asynchronous-parallel algorithms, the computing nodes simply use the most recent information that they have access to, instead of waiting for a full update from all nodes in the system. This means that nodes do not have to waste time waiting for information, which can be a major bottleneck, especially in distributed systems. When the system has $p$ nodes, asynchronous algorithms may complete $\Theta(\ln(p))$ more iterations than synchronous algorithms in a given time period ("more iterations per second"). Although asynchronous algorithms may compute more iterations per second, there is error associated with using outdated information. How many more iterations in total are needed to compensate for this error is still an open question. The main results of this paper aim to answer this question. We prove, loosely, that as the size of the problem becomes large, the number of additional iterations that asynchronous algorithms need becomes negligible compared to the total number ("same quality" of the iterations). Taking these facts together, our results provide solid evidence of the potential of asynchronous algorithms to vastly speed up certain distributed computations.Comment: 29 page

Pathwise Coordinate Optimization for Sparse Learning: Algorithm and Theory

The pathwise coordinate optimization is one of the most important computational frameworks for high dimensional convex and nonconvex sparse learning problems. It differs from the classical coordinate optimization algorithms in three salient features: {\it warm start initialization}, {\it active set updating}, and {\it strong rule for coordinate preselection}. Such a complex algorithmic structure grants superior empirical performance, but also poses significant challenge to theoretical analysis. To tackle this long lasting problem, we develop a new theory showing that these three features play pivotal roles in guaranteeing the outstanding statistical and computational performance of the pathwise coordinate optimization framework. Particularly, we analyze the existing pathwise coordinate optimization algorithms and provide new theoretical insights into them. The obtained insights further motivate the development of several modifications to improve the pathwise coordinate optimization framework, which guarantees linear convergence to a unique sparse local optimum with optimal statistical properties in parameter estimation and support recovery. This is the first result on the computational and statistical guarantees of the pathwise coordinate optimization framework in high dimensions. Thorough numerical experiments are provided to support our theory.Comment: Accepted by the Annals of Statistics, 2016