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Distributed Stochastic Optimization With Unbounded Subgradients Over Randomly Time-Varying Networks

Yan Chen, Alexander L. Fradkov, Keli Fu, Xiaozheng Fu, Tao Li

TL;DR

The paper tackles distributed stochastic convex optimization over uncertain, time-varying networks with unbounded subgradients and coexisting additive and multiplicative communication noises. It develops a distributed stochastic subgradient algorithm operating on a sequence of random digraphs modeled by conditional balancedness and uniform joint connectivity, and proves almost-sure convergence to a common optimizer under linear growth of subgradients and carefully designed step sizes. The analysis combines stochastic Lyapunov methods, convex analysis, algebraic graph theory, and martingale convergence to derive convergence results and rates, including explicit rates for strongly convex local costs across multiple parameter regimes. Numerical experiments on LASSO demonstrate the theoretical findings, showing how network dynamics and noise influence convergence and rates, and validating the practicality of the proposed approach.

Abstract

Motivated by distributed statistical learning over uncertain communication networks, we study distributed stochastic optimization by networked nodes to cooperatively minimize a sum of convex cost functions. The network is modeled by a sequence of time-varying random digraphs with each node representing a local optimizer and each edge representing a communication link. We consider the distributed subgradient optimization algorithm with noisy measurements of local cost functions' subgradients, additive and multiplicative noises among information exchanging between each pair of nodes. By stochastic Lyapunov method, convex analysis, algebraic graph theory and martingale convergence theory, we prove that if the local subgradient functions grow linearly and the sequence of digraphs is conditionally balanced and uniformly conditionally jointly connected, then proper algorithm step sizes can be designed so that all nodes' states converge to the global optimal solution almost surely.

Distributed Stochastic Optimization With Unbounded Subgradients Over Randomly Time-Varying Networks

TL;DR

The paper tackles distributed stochastic convex optimization over uncertain, time-varying networks with unbounded subgradients and coexisting additive and multiplicative communication noises. It develops a distributed stochastic subgradient algorithm operating on a sequence of random digraphs modeled by conditional balancedness and uniform joint connectivity, and proves almost-sure convergence to a common optimizer under linear growth of subgradients and carefully designed step sizes. The analysis combines stochastic Lyapunov methods, convex analysis, algebraic graph theory, and martingale convergence to derive convergence results and rates, including explicit rates for strongly convex local costs across multiple parameter regimes. Numerical experiments on LASSO demonstrate the theoretical findings, showing how network dynamics and noise influence convergence and rates, and validating the practicality of the proposed approach.

Abstract

Motivated by distributed statistical learning over uncertain communication networks, we study distributed stochastic optimization by networked nodes to cooperatively minimize a sum of convex cost functions. The network is modeled by a sequence of time-varying random digraphs with each node representing a local optimizer and each edge representing a communication link. We consider the distributed subgradient optimization algorithm with noisy measurements of local cost functions' subgradients, additive and multiplicative noises among information exchanging between each pair of nodes. By stochastic Lyapunov method, convex analysis, algebraic graph theory and martingale convergence theory, we prove that if the local subgradient functions grow linearly and the sequence of digraphs is conditionally balanced and uniformly conditionally jointly connected, then proper algorithm step sizes can be designed so that all nodes' states converge to the global optimal solution almost surely.

Paper Structure

This paper contains 17 sections, 19 theorems, 223 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Related work
  3. Main contribution
  4. Notations and symbols
  5. Problem formulation
  6. Main results
  7. The convergence of the algorithm \ref{['algorithm']}-\ref{['algorithm2']}
  8. Special cases
  9. Discussion on the assumptions
  10. Convergence rates for the case with strongly convex local cost functions
  11. Numerical simulation
  12. Conclusions
  13. Proofs of Lemmas in \ref{['sec:main']}
  14. supplementary Lemmas
  15. Proofs of \ref{['theorem3.2']}
  16. ...and 2 more sections

Key Result

Theorem 3.1

For the convex optimization problem model and the algorithm algorithm-algorithm2, assume that (a) Assumptions subgradient-stochasticgraph and Conditions (C1)-(C5) hold; (b) there exists a positive integer $h$, positive constants $\theta$ and $\rho_{0}$, such that (b.1) $\inf_{m\geq0}\lambda_{mh}^h\g

Figures (1)

  • Figure 1: (a) LASSO regression: trajectories of states; (b) LASSO regression: convergence of mean square errors with $c(k)=1/(k+1)^{0.4}$ and $\alpha(k)=3/(k+1)$; (c) LASSO regression: the red dashed line is the trajectory of $k^{0.3}E[\|X(k)-\mathbf{1}_{N}\otimes z^{*}\|^2]$; the blue solid line is the trajectory of $E[\|X(k)-\mathbf{1}_{N}\otimes z^{*}\|^2]$.

Theorems & Definitions (36)

  • Remark 2.1
  • Remark 2.2
  • Theorem 3.1
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Corollary 3.1
  • Proof 1
  • Theorem 3.2
  • Corollary 3.2
  • ...and 26 more