We consider multi-agent, convex optimization programs subject to separable constraints, where the constraint function of each agent involves only its local decision vector, while the decision vectors of all agents are coupled via a common objective function. We focus on a regularized variant of the so called Jacobi algorithm for decentralized computation in such problems. We first consider the case where the objective function is quadratic, and provide a fixed-point theoretic analysis showing that the algorithm converges to a minimizer of the centralized problem. Moreover, we quantify the potential benefits of such an iterative scheme by comparing it against a scaled projected gradient algorithm. We then consider the general case and show that all limit points of the proposed iteration are optimal solutions of the centralized problem. The efficacy of the proposed algorithm is illustrated by applying it to the problem of optimal charging of electric vehicles, where, as opposed to earlier approaches, we show convergence to an optimal charging scheme for a finite, possibly large, number of vehicles.
The second direction for decentralized optimization involves mainly the so called Jacobi algorithm, which serves as an alternative to gradient algorithms. The Gauss-Seidel algorithm exhibits similarities with the Jacobi one, but is not of parallelizable nature @cite_27 , unless a coloring scheme is adopted (see Section 1.2.4 in @cite_3 ). Under the Jacobi algorithmic setup, at every iteration, instead of performing a gradient step, each agent minimizes the common objective function subject to its local constraints, while keeping the decision vectors of all other agents fixed to their values at the previous iteration. A regularized version of the Jacobi algorithm has been proposed in @cite_13 @cite_30 , and more recently in @cite_16 @cite_15 . Other parallelizable iterative methods are proposed in @cite_29 @cite_31 @cite_0 , where, however, partially separable cost functions are considered.
@cite_30 @cite_31 @cite_29 @cite_3 @cite_0 @cite_27 @cite_15 @cite_16 @cite_13
1974661392 1553365534 2808102858 2769394111
We propose a novel decomposition framework for the distributed optimization of general nonconvex sum-utility functions arising naturally in the system design of wireless multi-user interfering systems. Our main contributions are i) the development of the first class of (inexact) Jacobi best-response algorithms with provable convergence, where all the users simultaneously and iteratively solve a suitably convexified version of the original sum-utility optimization problem; ii) the derivation of a general dynamic pricing mechanism that provides a unified view of existing pricing schemes that are based, instead, on heuristics; and iii) a framework that can be easily particularized to well-known applications, giving rise to very efficient practical (Jacobi or Gauss-Seidel) algorithms that outperform existing ad hoc methods proposed for very specific problems. Interestingly, our framework contains as special cases well-known gradient algorithms for nonconvex sum-utility problems, and many block-coordinate descent schemes for convex functions. This paper develops the Decentralized Linearized Alternating Direction Method of Multipliers (DLM) that minimizes a sum of local cost functions in a multiagent network. The algorithm mimics operation of the decentralized alternating direction method of multipliers (DADMM) except that it linearizes the optimization objective at each iteration. This results in iterations that, instead of successive minimizations, implement steps whose cost is akin to the much lower cost of the gradient descent step used in the distributed gradient method (DGM). The algorithm is proven to converge to the optimal solution when the local cost functions have Lipschitz continuous gradients. Its rate of convergence is shown to be linear if the local cost functions are further assumed to be strongly convex. Numerical experiments in least squares and logistic regression problems show that the number of iterations to achieve equivalent optimality gaps are similar for DLM and ADMM and both much smaller than those of DGM. In that sense, DLM combines the rapid convergence of ADMM with the low computational burden of DGM. We propose a very simple modification of gradient descent and stochastic gradient descent. We show that when applied to a variety of machine learning models including softmax regression, convolutional neural nets, generative adversarial nets, and deep reinforcement learning, this very simple surrogate can dramatically reduce the variance and improve the accuracy of the generalization. The new algorithm, (which depends on one nonnegative parameter) when applied to non-convex minimization, tends to avoid sharp local minima. Instead it seeks somewhat flatter local (and often global) minima. The method only involves preconditioning the gradient by the inverse of a tri-diagonal matrix that is positive definite. The motivation comes from the theory of Hamilton-Jacobi partial differential equations. This theory demonstrates that the new algorithm is almost the same as doing gradient descent on a new function which (a) has the same global minima as the original function and (b) is "more convex". Again, the programming effort in doing this is minimal, in cost, complexity and effort. We implement our algorithm into both PyTorch and Tensorflow platforms, which will be made publicly available. Nesterov's accelerated gradient descent (AGD), an instance of the general family of "momentum methods", provably achieves faster convergence rate than gradient descent (GD) in the convex setting. However, whether these methods are superior to GD in the nonconvex setting remains open. This paper studies a simple variant of AGD, and shows that it escapes saddle points and finds a second-order stationary point in @math iterations, faster than the @math iterations required by GD. To the best of our knowledge, this is the first Hessian-free algorithm to find a second-order stationary point faster than GD, and also the first single-loop algorithm with a faster rate than GD even in the setting of finding a first-order stationary point. Our analysis is based on two key ideas: (1) the use of a simple Hamiltonian function, inspired by a continuous-time perspective, which AGD monotonically decreases per step even for nonconvex functions, and (2) a novel framework called improve or localize, which is useful for tracking the long-term behavior of gradient-based optimization algorithms. We believe that these techniques may deepen our understanding of both acceleration algorithms and nonconvex optimization.