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Convergence Analysis of Continuous-Time Distributed Stochastic Gradient Algorithms

Jianhua Sun, Kaihong Lu, Xin Yu

TL;DR

This work introduces a continuous-time distributed stochastic gradient framework where multiple agents cooperatively minimize a sum of convex functions using consensus dynamics and stochastic gradient feedback modeled by Brownian noise. The convergence analysis leverages Itô calculus, Lyapunov methods, and switching balanced directed graphs to show that agent states converge in expectation to a common minimizer, with explicit rates governed by the step-size decay exponent $a$; the fastest polynomial rate is $\mathcal{O}(t^{-1/4}\sqrt{\ln t})$ at $a=3/4$. Theoretical results are supported by simulations on a six-agent network with a time-varying topology, illustrating convergence of both the agent states and objective value to the optimum. These findings advance the understanding of continuous-time distributed stochastic optimization under realistic communication constraints and stochastic gradient information. Potential future directions include non-convex objectives, constraints, and high-probability convergence analyses.

Abstract

In this paper, we propose a new framework to study distributed optimization problems with stochastic gradients by employing a multi-agent system with continuous-time dynamics. Here the goal of the agents is to cooperatively minimize the sum of convex objective functions. When making decisions, each agent only has access to a stochastic gradient of its own objective function rather than the real gradient, and can exchange local state information with its immediate neighbors via a time-varying directed graph. Particularly, the stochasticity is depicted by the Brownian motion. To handle this problem, we propose a continuous-time distributed stochastic gradient algorithm based on the consensus algorithm and the gradient descent strategy. Under mild assumptions on the connectivity of the graph and objective functions, using convex analysis theory, the Lyapunov theory and Ito formula, we prove that the states of the agents asymptotically reach a common minimizer in expectation. Finally, a simulation example is worked out to demonstrate the effectiveness of our theoretical results.

Convergence Analysis of Continuous-Time Distributed Stochastic Gradient Algorithms

TL;DR

This work introduces a continuous-time distributed stochastic gradient framework where multiple agents cooperatively minimize a sum of convex functions using consensus dynamics and stochastic gradient feedback modeled by Brownian noise. The convergence analysis leverages Itô calculus, Lyapunov methods, and switching balanced directed graphs to show that agent states converge in expectation to a common minimizer, with explicit rates governed by the step-size decay exponent ; the fastest polynomial rate is at . Theoretical results are supported by simulations on a six-agent network with a time-varying topology, illustrating convergence of both the agent states and objective value to the optimum. These findings advance the understanding of continuous-time distributed stochastic optimization under realistic communication constraints and stochastic gradient information. Potential future directions include non-convex objectives, constraints, and high-probability convergence analyses.

Abstract

In this paper, we propose a new framework to study distributed optimization problems with stochastic gradients by employing a multi-agent system with continuous-time dynamics. Here the goal of the agents is to cooperatively minimize the sum of convex objective functions. When making decisions, each agent only has access to a stochastic gradient of its own objective function rather than the real gradient, and can exchange local state information with its immediate neighbors via a time-varying directed graph. Particularly, the stochasticity is depicted by the Brownian motion. To handle this problem, we propose a continuous-time distributed stochastic gradient algorithm based on the consensus algorithm and the gradient descent strategy. Under mild assumptions on the connectivity of the graph and objective functions, using convex analysis theory, the Lyapunov theory and Ito formula, we prove that the states of the agents asymptotically reach a common minimizer in expectation. Finally, a simulation example is worked out to demonstrate the effectiveness of our theoretical results.
Paper Structure (12 sections, 7 theorems, 51 equations, 3 figures)

This paper contains 12 sections, 7 theorems, 51 equations, 3 figures.

Table of Contents

  1. Introduction
  2. preliminaries
  3. Mathematical preliminaries
  4. Graph theory
  5. Problem formulation
  6. Main results
  7. Simulation examples
  8. Conclusions
  9. Appendix
  10. Proof of Lemma \ref{['LE3']}
  11. Proof of Lemma \ref{['LE1']}
  12. Proof of Lemma \ref{['LE2']}

Key Result

Lemma 1

Consider Itô process (eq130). Given function $\mathcal{E}(x,t): \mathbb{R}^m\times \mathbb{R}\rightarrow \mathbb{R}$, if $\mathcal{E}(x,t)$ is twice differentiable with respect to $x$ and differentiable with respect to $t$, then

Figures (3)

  • Figure 1: The time-varying graph.
  • Figure 2: The trajectories of $\mathbb{E}[x_i(t)], i=1,\cdots, 6$ under algorithm (\ref{['eq3_1']}).
  • Figure 3: The trajectories of $\mathbb{E}[f(x_i(t))-f(x^*)], i=1,...,6$ under algorithm (\ref{['eq3_1']}).

Theorems & Definitions (13)

  • Definition 1
  • Lemma 1: Itô formula
  • Lemma 2: C11
  • Remark 1
  • Theorem 1
  • Lemma 3: C16
  • Lemma 4
  • Proof 1
  • Lemma 5
  • Proof 2
  • ...and 3 more