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Asynchronous Decentralized Optimization with Constraints: Achievable Speeds of Convergence for Directed Graphs

Firooz Shahriari-Mehr, Ashkan Panahi

TL;DR

This work develops ASY-DAGP, an asynchronous, constraint-enabled decentralized optimization algorithm for directed graphs, accommodating heterogeneous node speeds, communication delays, and message losses. It extends the synchronous DAGP by incorporating local buffers and auxiliary variables to preserve a null condition and achieve consensus without requiring strong convexity. The authors introduce Linear Quadratic Performance Estimation Problems (LQ-PEP) to analyze convergence, yielding $O(1/\sqrt{K})$ optimality gaps and $O(1/K)$ feasibility/consensus gaps under a delay-response parameter $\kappa$, and they demonstrate robustness to substantial communication failures. Comprehensive experiments on directed networks confirm faster wall-clock convergence and resilience to dropped messages compared with several baselines. Overall, the paper provides a principled asynchronous framework with a tractable analysis tool for constrained decentralized optimization on directed graphs.

Abstract

We address a decentralized convex optimization problem, where every agent has its unique local objective function and constraint set. Agents compute at different speeds, and their communication may be delayed and directed. For this setup, we propose an asynchronous double averaging and gradient projection (ASY-DAGP) algorithm. Our algorithm handles difficult scenarios such as message failure, by employing local buffers and utilizing the temporal correlation in the transmitted messages. We guarantee the convergence speed of our algorithm using performance estimation problems (PEP). In particular, we introduce the concept of the linear quadratic (LQ) PEP. This approach simplifies the analysis of smooth convex optimization problems, going beyond Lyapunov function analyses and avoiding restrictive assumptions such as strong-convexity. Numerical experiments validate the effectiveness of our proposed algorithm.

Asynchronous Decentralized Optimization with Constraints: Achievable Speeds of Convergence for Directed Graphs

TL;DR

This work develops ASY-DAGP, an asynchronous, constraint-enabled decentralized optimization algorithm for directed graphs, accommodating heterogeneous node speeds, communication delays, and message losses. It extends the synchronous DAGP by incorporating local buffers and auxiliary variables to preserve a null condition and achieve consensus without requiring strong convexity. The authors introduce Linear Quadratic Performance Estimation Problems (LQ-PEP) to analyze convergence, yielding optimality gaps and feasibility/consensus gaps under a delay-response parameter , and they demonstrate robustness to substantial communication failures. Comprehensive experiments on directed networks confirm faster wall-clock convergence and resilience to dropped messages compared with several baselines. Overall, the paper provides a principled asynchronous framework with a tractable analysis tool for constrained decentralized optimization on directed graphs.

Abstract

We address a decentralized convex optimization problem, where every agent has its unique local objective function and constraint set. Agents compute at different speeds, and their communication may be delayed and directed. For this setup, we propose an asynchronous double averaging and gradient projection (ASY-DAGP) algorithm. Our algorithm handles difficult scenarios such as message failure, by employing local buffers and utilizing the temporal correlation in the transmitted messages. We guarantee the convergence speed of our algorithm using performance estimation problems (PEP). In particular, we introduce the concept of the linear quadratic (LQ) PEP. This approach simplifies the analysis of smooth convex optimization problems, going beyond Lyapunov function analyses and avoiding restrictive assumptions such as strong-convexity. Numerical experiments validate the effectiveness of our proposed algorithm.
Paper Structure (16 sections, 5 theorems, 73 equations, 3 figures, 2 algorithms)

This paper contains 16 sections, 5 theorems, 73 equations, 3 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Methodology
  3. Problem setup
  4. ASY-DAGP algorithm
  5. Convergence guarantees
  6. Proof outline of Theorem \ref{['thm: theorem1']}
  7. Performance Estimation Problem
  8. Linear Quadratic (LQ-)PEP
  9. Establishing Theorem \ref{['thm: theorem1']}
  10. Experimental results
  11. Related works
  12. Conclusion
  13. Mathematical Notation
  14. Analysis of fixed points
  15. Convergence analysis
  16. ...and 1 more sections

Key Result

Theorem 1

Suppose that $(\mathbf{S},\bar{\mathbf{R}},\mathbf{P})$ is proper. There exist a constant $C$ independent of the delays such that for $\kappa<\frac{1}{C}$ and a sufficiently large $K$, ASY-DAGP converges as follows: where $K$ indicates that all nodes have performed $K$ iterations, $\bar{\mathbf{x}}^v_K = \frac{1}{K}\sum_{k=0}^{K-1}\mathbf{x}^v_k$, $\bar{\mathbf{x}}_K = \frac{1}{M}\sum_{v=1}^M \ba

Figures (3)

  • Figure 1: Synchronous setup (left): In each iteration, agents complete their computations and transmit their updated variables to their neighbors. The nodes wait until all messages are delivered to their destination. Then, they simultaneously initiate the next iteration. Asynchronous setup (right): Each agent completes its computation and sends the updated variables to its neighbors. Immediately, it starts its next computation using available information. There is a possibility of receiving multiple messages from one node or sometimes no messages.
  • Figure 2: (a,b) Solving a constrained problem, and drawing a comparison to DAGP and its throttled version. (c) Solving unconstrained logistic regression problem, and comparing to APPG and ASY-SPA. (d) Solving a constrained problem over undirected graphs, and comparing to ASY-PG-EXTRA.
  • Figure 3: Robustness to message losses with the communication failure probability of $p$.

Theorems & Definitions (12)

  • Definition 1: Proper matrices
  • Definition 2: Delay response
  • Theorem 1: Rates of convergence
  • Remark 1
  • Remark 2
  • Proposition 1
  • Lemma 1: Inverse-matrix decomposition
  • proof
  • Lemma 2: Inversely-related roots
  • proof
  • ...and 2 more