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Distributed Online Randomized Gradient-Free Optimization with Compressed Communication

Longkang Zhu, Xinli Shi, Xiangping Xu, Jinde Cao, Xiangyong Chen

TL;DR

This work tackles distributed online convex optimization under limited gradient feedback and communication constraints by introducing Online Compressed Gradient Tracking (OCGT). The framework unifies two settings—one-point bandit feedback and stochastic gradient feedback—within a single gradient-tracking plus compression scheme that employs error compensation to mitigate bias from communication reduction. The authors establish dynamic regret bounds that capture environmental variation, gradient heterogeneity, and quantization effects, and they demonstrate favorable scaling compared with uncompressed, full-gradient baselines. Empirical results on logistic regression tasks with real datasets confirm substantial communication savings while maintaining competitive dynamic performance and accuracy. Overall, OCGT offers a practical, theoretically grounded approach to scalable distributed optimization in non-stationary environments.

Abstract

This paper addresses two fundamental challenges in distributed online convex optimization: communication efficiency and optimization under limited feedback. We propose a unified framework named Online Compressed Gradient Tracking (OCGT), which includes two variants: One-point Bandit Feedback (OCGT-BF) and Stochastic Gradient Feedback (OCSGT). The proposed algorithms harness data compression and either gradient-free or stochastic gradient optimization techniques within distributed networks. The proposed framework incorporates a compression scheme with error compensation mechanisms to reduce communication overhead while maintaining convergence guarantees. Unlike traditional approaches that assume perfect communication and full gradient access, OCGT operates effectively under practical constraints by combining gradient-like tracking with one-point or stochastic gradient feedback estimation. We provide a theoretical analysis demonstrating dynamic regret bounds for both variants. Finally, extensive experiments validate that OCGT achieves low dynamic regret while significantly reducing communication requirements.

Distributed Online Randomized Gradient-Free Optimization with Compressed Communication

TL;DR

This work tackles distributed online convex optimization under limited gradient feedback and communication constraints by introducing Online Compressed Gradient Tracking (OCGT). The framework unifies two settings—one-point bandit feedback and stochastic gradient feedback—within a single gradient-tracking plus compression scheme that employs error compensation to mitigate bias from communication reduction. The authors establish dynamic regret bounds that capture environmental variation, gradient heterogeneity, and quantization effects, and they demonstrate favorable scaling compared with uncompressed, full-gradient baselines. Empirical results on logistic regression tasks with real datasets confirm substantial communication savings while maintaining competitive dynamic performance and accuracy. Overall, OCGT offers a practical, theoretically grounded approach to scalable distributed optimization in non-stationary environments.

Abstract

This paper addresses two fundamental challenges in distributed online convex optimization: communication efficiency and optimization under limited feedback. We propose a unified framework named Online Compressed Gradient Tracking (OCGT), which includes two variants: One-point Bandit Feedback (OCGT-BF) and Stochastic Gradient Feedback (OCSGT). The proposed algorithms harness data compression and either gradient-free or stochastic gradient optimization techniques within distributed networks. The proposed framework incorporates a compression scheme with error compensation mechanisms to reduce communication overhead while maintaining convergence guarantees. Unlike traditional approaches that assume perfect communication and full gradient access, OCGT operates effectively under practical constraints by combining gradient-like tracking with one-point or stochastic gradient feedback estimation. We provide a theoretical analysis demonstrating dynamic regret bounds for both variants. Finally, extensive experiments validate that OCGT achieves low dynamic regret while significantly reducing communication requirements.

Paper Structure

This paper contains 10 sections, 14 theorems, 115 equations, 4 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Problem Formulation
  4. Compression Methods
  5. Random Gradient-Free Optimization
  6. Online Compressed Gradient Tracking
  7. Algorithm Description
  8. Dynamic Regret Analysis
  9. Experiments
  10. Conclusions

Key Result

Lemma 1

nesterov2017random If $f_{i, t}(x)$ is $L_i$-smooth, then for any $x \in \mathbb{R}^m$, smoothing parameter $v_t > 0$, and random direction $u_{i,t} \sim \mathcal{N}(0, I_m)$, the one-point gradient estimator $G_{i, t}(x, u_{i,t})$ defined in bandit feed satisfies

Figures (4)

  • Figure 2: Comparison of the maximum regret $\max_{i \in \mathcal{V}} \mathcal{R}_i(T)/T$ versus the number of transfer bits for OGT (uncompressed) and three OCGT-BF variants (quantizer-based, top-k, and random-k).
  • Figure : (a)
  • Figure : (a)
  • Figure : (b)

Theorems & Definitions (34)

  • Definition 1: Dynamic Regret
  • Lemma 1
  • Remark 1
  • Remark 2
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Theorem 1
  • proof
  • ...and 24 more