A decentralized proximal-gradient method with network independent step-sizes and separated convergence rates

This paper proposes a novel proximal-gradient algorithm for a decentralized optimization problem with a composite objective containing smooth and non-smooth terms. Specifically, the smooth and nonsmooth terms are dealt with by gradient and proximal updates, respectively. The proposed algorithm is closely related to a previous algorithm, PG-EXTRA \cite{shi2015proximal}, but has a few advantages. First of all, agents use uncoordinated step-sizes, and the stable upper bounds on step-sizes are independent of network topologies. The step-sizes depend on local objective functions, and they can be as large as those of the gradient descent. Secondly, for the special case without non-smooth terms, linear convergence can be achieved under the strong convexity assumption. The dependence of the convergence rate on the objective functions and the network are separated, and the convergence rate of the new algorithm is as good as one of the two convergence rates that match the typical rates for the general gradient descent and the consensus averaging. We provide numerical experiments to demonstrate the efficacy of the introduced algorithm and validate our theoretical discoveries

99% of Distributed Optimization is a Waste of Time: The Issue and How to Fix it

Many popular distributed optimization methods for training machine learning models fit the following template: a local gradient estimate is computed independently by each worker, then communicated to a master, which subsequently performs averaging. The average is broadcast back to the workers, which use it to perform a gradient-type step to update the local version of the model. It is also well known that many such methods, including SGD, SAGA, and accelerated SGD for over-parameterized models, do not scale well with the number of parallel workers. In this paper we observe that the above template is fundamentally inefficient in that too much data is unnecessarily communicated by the workers, which slows down the overall system. We propose a fix based on a new update-sparsification method we develop in this work, which we suggest be used on top of existing methods. Namely, we develop a new variant of parallel block coordinate descent based on independent sparsification of the local gradient estimates before communication. We demonstrate that with only $m/n$ blocks sent by each of $n$ workers, where $m$ is the total number of parameter blocks, the theoretical iteration complexity of the underlying distributed methods is essentially unaffected. As an illustration, this means that when $n=100$ parallel workers are used, the communication of $99\%$ blocks is redundant, and hence a waste of time. Our theoretical claims are supported through extensive numerical experiments which demonstrate an almost perfect match with our theory on a number of synthetic and real datasets.Comment: 41 pages, 8 algorithms, 10 theorems, 12 figure

Convergence Analysis of Iterative Methods for Nonsmooth Convex Optimization over Fixed Point Sets of Quasi-Nonexpansive Mappings

This paper considers a networked system with a finite number of users and supposes that each user tries to minimize its own private objective function over its own private constraint set. It is assumed that each user's constraint set can be expressed as a fixed point set of a certain quasi-nonexpansive mapping. This enables us to consider the case in which the projection onto the constraint set cannot be computed efficiently. This paper proposes two methods for solving the problem of minimizing the sum of their nondifferentiable, convex objective functions over the intersection of their fixed point sets of quasi-nonexpansive mappings in a real Hilbert space. One method is a parallel subgradient method that can be implemented under the assumption that each user can communicate with other users. The other is an incremental subgradient method that can be implemented under the assumption that each user can communicate with its neighbors. Investigation of the two methods' convergence properties for a constant step size reveals that, with a small constant step size, they approximate a solution to the problem. Consideration of the case in which the step-size sequence is diminishing demonstrates that the sequence generated by each of the two methods strongly converges to the solution to the problem under certain assumptions. Convergence rate analysis of the two methods under certain situations is provided to illustrate the two methods' efficiency. This paper also discusses nonsmooth convex optimization over sublevel sets of convex functions and provides numerical comparisons that demonstrate the effectiveness of the proposed methods

Achieving Geometric Convergence for Distributed Optimization over Time-Varying Graphs

This paper considers the problem of distributed optimization over time-varying graphs. For the case of undirected graphs, we introduce a distributed algorithm, referred to as DIGing, based on a combination of a distributed inexact gradient method and a gradient tracking technique. The DIGing algorithm uses doubly stochastic mixing matrices and employs fixed step-sizes and, yet, drives all the agents' iterates to a global and consensual minimizer. When the graphs are directed, in which case the implementation of doubly stochastic mixing matrices is unrealistic, we construct an algorithm that incorporates the push-sum protocol into the DIGing structure, thus obtaining Push-DIGing algorithm. The Push-DIGing uses column stochastic matrices and fixed step-sizes, but it still converges to a global and consensual minimizer. Under the strong convexity assumption, we prove that the algorithms converge at R-linear (geometric) rates as long as the step-sizes do not exceed some upper bounds. We establish explicit estimates for the convergence rates. When the graph is undirected it shows that DIGing scales polynomially in the number of agents. We also provide some numerical experiments to demonstrate the efficacy of the proposed algorithms and to validate our theoretical findings

Distributed optimization in wireless sensor networks: an island-model framework

Wireless Sensor Networks (WSNs) is an emerging technology in several application domains, ranging from urban surveillance to environmental and structural monitoring. Computational Intelligence (CI) techniques are particularly suitable for enhancing these systems. However, when embedding CI into wireless sensors, severe hardware limitations must be taken into account. In this paper we investigate the possibility to perform an online, distributed optimization process within a WSN. Such a system might be used, for example, to implement advanced network features like distributed modelling, self-optimizing protocols, and anomaly detection, to name a few. The proposed approach, called DOWSN (Distributed Optimization for WSN) is an island-model infrastructure in which each node executes a simple, computationally cheap (both in terms of CPU and memory) optimization algorithm, and shares promising solutions with its neighbors. We perform extensive tests of different DOWSN configurations on a benchmark made up of continuous optimization problems; we analyze the influence of the network parameters (number of nodes, inter-node communication period and probability of accepting incoming solutions) on the optimization performance. Finally, we profile energy and memory consumption of DOWSN to show the efficient usage of the limited hardware resources available on the sensor nodes

Proximal Point Algorithms for Nonsmooth Convex Optimization with Fixed Point Constraints

The problem of minimizing the sum of nonsmooth, convex objective functions defined on a real Hilbert space over the intersection of fixed point sets of nonexpansive mappings, onto which the projections cannot be efficiently computed, is considered. The use of proximal point algorithms that use the proximity operators of the objective functions and incremental optimization techniques is proposed for solving the problem. With the focus on fixed point approximation techniques, two algorithms are devised for solving the problem. One blends an incremental subgradient method, which is a useful algorithm for nonsmooth convex optimization, with a Halpern-type fixed point iteration algorithm. The other is based on an incremental subgradient method and the Krasnosel'ski\u\i-Mann fixed point algorithm. It is shown that any weak sequential cluster point of the sequence generated by the Halpern-type algorithm belongs to the solution set of the problem and that there exists a weak sequential cluster point of the sequence generated by the Krasnosel'ski\u\i-Mann-type algorithm, which also belongs to the solution set. Numerical comparisons of the two proposed algorithms with existing subgradient methods for concrete nonsmooth convex optimization show that the proposed algorithms achieve faster convergence

Two Stochastic Optimization Algorithms for Convex Optimization with Fixed Point Constraints

Two optimization algorithms are proposed for solving a stochastic programming problem for which the objective function is given in the form of the expectation of convex functions and the constraint set is defined by the intersection of fixed point sets of nonexpansive mappings in a real Hilbert space. This setting of fixed point constraints enables consideration of the case in which the projection onto each of the constraint sets cannot be computed efficiently. Both algorithms use a convex function and a nonexpansive mapping determined by a certain probabilistic process at each iteration. One algorithm blends a stochastic gradient method with the Halpern fixed point algorithm. The other is based on a stochastic proximal point algorithm and the Halpern fixed point algorithm; it can be applied to nonsmooth convex optimization. Convergence analysis showed that, under certain assumptions, any weak sequential cluster point of the sequence generated by either algorithm almost surely belongs to the solution set of the problem. Convergence rate analysis illustrated their efficiency, and the numerical results of convex optimization over fixed point sets demonstrated their effectiveness

On the Acceleration of L-BFGS with Second-Order Information and Stochastic Batches

This paper proposes a framework of L-BFGS based on the (approximate) second-order information with stochastic batches, as a novel approach to the finite-sum minimization problems. Different from the classical L-BFGS where stochastic batches lead to instability, we use a smooth estimate for the evaluations of the gradient differences while achieving acceleration by well-scaling the initial Hessians. We provide theoretical analyses for both convex and nonconvex cases. In addition, we demonstrate that within the popular applications of least-square and cross-entropy losses, the algorithm admits a simple implementation in the distributed environment. Numerical experiments support the efficiency of our algorithms

Adaptive Communication Strategies to Achieve the Best Error-Runtime Trade-off in Local-Update SGD

Large-scale machine learning training, in particular distributed stochastic gradient descent, needs to be robust to inherent system variability such as node straggling and random communication delays. This work considers a distributed training framework where each worker node is allowed to perform local model updates and the resulting models are averaged periodically. We analyze the true speed of error convergence with respect to wall-clock time (instead of the number of iterations), and analyze how it is affected by the frequency of averaging. The main contribution is the design of AdaComm, an adaptive communication strategy that starts with infrequent averaging to save communication delay and improve convergence speed, and then increases the communication frequency in order to achieve a low error floor. Rigorous experiments on training deep neural networks show that AdaComm can take $3 \times$ less time than fully synchronous SGD, and still reach the same final training loss.Comment: Accepted to SysML 201

Almost Sure Convergence of Random Projected Proximal and Subgradient Algorithms for Distributed Nonsmooth Convex Optimization

Two distributed algorithms are described that enable all users connected over a network to cooperatively solve the problem of minimizing the sum of all users' objective functions over the intersection of all users' constraint sets, where each user has its own private nonsmooth convex objective function and closed convex constraint set, which is the intersection of a number of simple, closed convex sets. One algorithm enables each user to adjust its estimate by using a proximity operator of its objective function and the metric projection onto one set randomly selected from the simple, closed convex sets. The other is a distributed random projection algorithm that determines each user's estimate by using a subgradient of its objective function instead of the proximity operator. Investigation of the two algorithms' convergence properties for a diminishing step-size rule revealed that, under certain assumptions, the sequences of all users generated by each of the two algorithms converge almost surely to the same solution. Moreover, convergence rate analysis of the two algorithms is provided, and desired choices of the step size sequences such that the two algorithms have fast convergence are discussed. Numerical comparisons for concrete nonsmooth convex optimization support the convergence analysis and demonstrate the effectiveness of the two algorithms