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Decentralized Online Convex Optimization with Unknown Feedback Delays

Hao Qiu, Mengxiao Zhang, Juliette Achddou

TL;DR

This work tackles decentralized online convex optimization with unknown, time- and agent-varying feedback delays. It introduces gossip-based, adaptive learning-rate algorithms that do not require knowledge of the total delay and decouple delay effects from network topology, achieving near-optimal regret for general convex losses and improved bounds for strongly convex losses. The results include tight lower bounds, an adaptive method that matches known-delay performance without delay knowledge, and empirical validation across network topologies and delay models. Overall, the paper advances delay-robust, fully decentralized online learning with practical, scalable communication strategies.

Abstract

Decentralized online convex optimization (D-OCO), where multiple agents within a network collaboratively learn optimal decisions in real-time, arises naturally in applications such as federated learning, sensor networks, and multi-agent control. In this paper, we study D-OCO under unknown, time-and agent-varying feedback delays. While recent work has addressed this problem (Nguyen et al., 2024), existing algorithms assume prior knowledge of the total delay over agents and still suffer from suboptimal dependence on both the delay and network parameters. To overcome these limitations, we propose a novel algorithm that achieves an improved regret bound of O N $\sqrt$ d tot + N $\sqrt$ T (1-$σ$2) 1/4 , where T is the total horizon, d tot denotes the average total delay across agents, N is the number of agents, and 1 -$σ$ 2 is the spectral gap of the network. Our approach builds upon recent advances in D-OCO (Wan et al., 2024a), but crucially incorporates an adaptive learning rate mechanism via a decentralized communication protocol. This enables each agent to estimate delays locally using a gossip-based strategy without the prior knowledge of the total delay. We further extend our framework to the strongly convex setting and derive a sharper regret bound of O N $δ$max ln T $α$ , where $α$ is the strong convexity parameter and $δ$ max is the maximum number of missing observations averaged over agents. We also show that our upper bounds for both settings are tight up to logarithmic factors. Experimental results validate the effectiveness of our approach, showing improvements over existing benchmark algorithms.

Decentralized Online Convex Optimization with Unknown Feedback Delays

TL;DR

This work tackles decentralized online convex optimization with unknown, time- and agent-varying feedback delays. It introduces gossip-based, adaptive learning-rate algorithms that do not require knowledge of the total delay and decouple delay effects from network topology, achieving near-optimal regret for general convex losses and improved bounds for strongly convex losses. The results include tight lower bounds, an adaptive method that matches known-delay performance without delay knowledge, and empirical validation across network topologies and delay models. Overall, the paper advances delay-robust, fully decentralized online learning with practical, scalable communication strategies.

Abstract

Decentralized online convex optimization (D-OCO), where multiple agents within a network collaboratively learn optimal decisions in real-time, arises naturally in applications such as federated learning, sensor networks, and multi-agent control. In this paper, we study D-OCO under unknown, time-and agent-varying feedback delays. While recent work has addressed this problem (Nguyen et al., 2024), existing algorithms assume prior knowledge of the total delay over agents and still suffer from suboptimal dependence on both the delay and network parameters. To overcome these limitations, we propose a novel algorithm that achieves an improved regret bound of O N d tot + N T (1-2) 1/4 , where T is the total horizon, d tot denotes the average total delay across agents, N is the number of agents, and 1 - 2 is the spectral gap of the network. Our approach builds upon recent advances in D-OCO (Wan et al., 2024a), but crucially incorporates an adaptive learning rate mechanism via a decentralized communication protocol. This enables each agent to estimate delays locally using a gossip-based strategy without the prior knowledge of the total delay. We further extend our framework to the strongly convex setting and derive a sharper regret bound of O N max ln T , where is the strong convexity parameter and max is the maximum number of missing observations averaged over agents. We also show that our upper bounds for both settings are tight up to logarithmic factors. Experimental results validate the effectiveness of our approach, showing improvements over existing benchmark algorithms.
Paper Structure (26 sections, 21 theorems, 160 equations, 3 figures, 2 algorithms)

This paper contains 26 sections, 21 theorems, 160 equations, 3 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Works
  3. Decentralized online convex optimization
  4. Online learning with delayed feedbacks
  5. D-OCO with delayed feedbacks
  6. Preliminary
  7. Protocol
  8. Other Notations
  9. D-OCO with General Convex Loss Functions
  10. Non-Adaptive Algorithm with Known Total Delay
  11. Proof Sketch
  12. Adaptive Algorithm with Unknown Total Delay
  13. Lower bound
  14. D-OCO with Strongly-Convex Loss Functions
  15. Numerical Experiments
  16. ...and 11 more sections

Key Result

Proposition 4

The iterations of eq:matrix_iterations with $\theta=\left(1+\sqrt{1-\sigma_2^2(W)}\right)^{-1}$ ensure that for any $k\in \mathbb{N}$, where $b = \left(1-(1-1/\sqrt{2})\sqrt{1-\sigma_2(W)}\right)$ and $\|\cdot\|_F$ denotes the Frobenius norm of a matrix.

Figures (3)

  • Figure 1: Uniform delays. Comparison with relevant baselines across three network topologies—complete (left), grid (middle), and cycle (right)—under convex losses (top row) and strongly convex losses (bottom row).
  • Figure 2: Geometric delays. Comparison with relevant baselines across three network topologies—complete (left), grid (middle), and cycle (right)—under convex losses (top row) and strongly convex losses (bottom row).
  • Figure 3: Example of configuration used in the lower bound

Theorems & Definitions (33)

  • Definition 3
  • Proposition 4: Proposition 1 in ye2023multi
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Theorem 9
  • Lemma 10: Lemma A.2 of qiu2025exploiting
  • Lemma 11: Lemma 4.13 in orabona2019modern
  • Lemma 12
  • proof
  • ...and 23 more