Continuous-time stochastic Mirror Descent on a network: Variance reduction, consensus, convergence

Raginsky, Maxim; Bouvrie, Jake

doi:10.1109/cdc.2012.6426639

Cited by 48 publications

(55 citation statements)

References 13 publications

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“…corresponding respectively to the long-run average (also known as the "ergodic average" in optimization) and the "best value" of X up to time t. The results of [46] and the analysis of Section 4.2 indicate that X(t) is concentrated around interior solutions of X (in the long run and in probability), provided that (SMD) is run with sufficiently small η. That said, in a black-box setting where 13 "Generic linear program" means here that X is a polytope, f : X → R is affine, and f is constant only along the zero-dimensional faces of X [45].…”

Section: 4mentioning

confidence: 93%

On the Convergence of Gradient-Like Flows with Noisy Gradient Input

Mertikopoulos¹,

Staudigl²

2018

SIAM J. Optim.

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In view of solving convex optimization problems with noisy gradient input, we analyze the asymptotic behavior of gradient-like flows under stochastic disturbances. Specifically, we focus on the widely studied class of mirror descent schemes for convex programs with compact feasible regions, and we examine the dynamics' convergence and concentration properties in the presence of noise. In the vanishing noise limit, we show that the dynamics converge to the solution set of the underlying problem (a.s.). Otherwise, when the noise is persistent, we show that the dynamics are concentrated around interior solutions in the long run, and they converge to boundary solutions that are sufficiently "sharp". Finally, we show that a suitably rectified variant of the method converges irrespective of the magnitude of the noise (or the structure of the underlying convex program), and we derive an explicit estimate for its rate of convergence.

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Section: 4mentioning

confidence: 93%

On the Convergence of Gradient-Like Flows with Noisy Gradient Input

Mertikopoulos¹,

Staudigl²

2018

SIAM J. Optim.

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“…, N , we obtain an upper bound on the right hand side of (22). Utilizing that bound and (21) in (20), we get…”

Section: Combination Of the Above Two Inequalities Then Provides The mentioning

confidence: 99%

Convergence of the Iterates in Mirror Descent Methods

Doan

Bose

Nguyễn

et al. 2019

IEEE Control Syst. Lett.

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We consider centralized and distributed mirror descent algorithms over a finite-dimensional Hilbert space, and prove that the problem variables converge to an optimizer of a possibly nonsmooth function when the step sizes are square summable but not summable. Prior literature has focused on the convergence of the function value to its optimum. However, applications from distributed optimization and learning in games require the convergence of the variables to an optimizer, which is generally not guaranteed without assuming strong convexity of the objective function. We provide numerical simulations comparing entropic mirror descent and standard subgradient methods for the robust regression problem. √k) for strongly convex or convex objective functions, respectively, [15], [17].

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“…Definition 4 (Distance-generating function [29]) A function φ : X → R is called a distant-generating function modulus α > 0 with respect to • if φ is convex and continuous on X , the set X o = {x ∈ X | ∇φ(x) = ∅} is convex (note that X o contains the relative interior of X ) and φ restricted to X o is continuously differentiable and strongly convex with parameter α with respect to • in the sense that (yx) T…”

Section: General Theory On Mirror Descentmentioning

confidence: 99%

“…The original mirror ascent/descent method was proposed by Nemirovski and Yudin [23] and later evolved into a series of papers [24,25]. However, most of these algorithms were discrete-time algorithms [20,[26][27][28], with relatively few putting attention on the case of continuous time [29]. As for distributed mirror descent algorithm, the continuous-time case, compared with discrete-time case, is more attractive since it can facilitate the use of the elegant Lyapunov argument [30] to aid the convergence analysis and allow the tool of differential geometry to be used in optimizing constrained problem [31].…”

Section: Introductionmentioning

confidence: 99%

Distributed constrained optimization via continuous-time mirror design

Sheng

2018

Adv Differ Equ

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Recently, distributed convex optimization using a multiagent system has received much attention by many researchers. This problem is frequently approached by combing the consensus algorithms in the multiagent literature and the gradient algorithms in the convex optimization literature. Compared with unconstrained distributed optimization, the constrained case is more challenging, and it is usually tackled by the projected gradient method. However, the projected gradient algorithm involves projection nonlinearity and thus is hard to analyze. To avoid gradient projection, in this paper, we present a novel distributed convex optimization algorithm in continuous time by using mirror design. The resulting optimization dynamics is smooth without using gradient projection and is designed in a primal-dual framework, where the primal and dual dynamics are respectively aided by the mirror descent and the mirror ascent. As for the merit of mirror design in our paper, it avoids gradient projection in the optimization dynamics design and removes the difficulty of analyzing projection nonlinearity. Furthermore, the mirror base primal-dual optimization dynamics facilitates more convenience construction of Lyapunov functions in the stability analysis.

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Continuous-time stochastic Mirror Descent on a network: Variance reduction, consensus, convergence

Cited by 48 publications

References 13 publications

On the Convergence of Gradient-Like Flows with Noisy Gradient Input

On the Convergence of Gradient-Like Flows with Noisy Gradient Input

Convergence of the Iterates in Mirror Descent Methods

Distributed constrained optimization via continuous-time mirror design

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