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Communication-Efficient Distributed Online Nonconvex Optimization with Time-Varying Constraints

Kunpeng Zhang, Lei Xu, Xinlei Yi, Guanghui Wen, Ming Cao, Karl H. Johansson, Tianyou Chai, Tao Yang

TL;DR

This work tackles distributed online nonconvex optimization with time-varying inequality constraints over time-varying directed graphs. It develops two compressed distributed online primal--dual algorithms tailored to one-point and two-point bandit feedback, together with gradient estimators and a general compression scheme with bounded absolute error. The authors prove sublinear network regret and network cumulative constraint violation bounds, with Slater’s condition providing further reductions; the two-point variant achieves bounds aligned with existing convex-bandit results, while the one-point variant attains sublinear regret in the nonconvex setting. A simulation on distributed online localization demonstrates the theory and highlights communication-efficiency advantages under compression.

Abstract

This paper considers distributed online nonconvex optimization with time-varying inequality constraints over a network of agents, where the nonconvex local loss and convex local constraint functions can vary arbitrarily across iterations. For a time-varying directed graph, we propose two distributed bandit online primal--dual algorithm with compressed communication to efficiently utilize communication resources in the one-point and two-point bandit feedback settings, respectively. To measure the performance of the proposed algorithms, we use a network regret metric grounded in the first-order optimality condition associated with the variational inequality. We show that the compressed algorithms establish sublinear network regret and cumulative constraint violation bounds. Moreover, the network cumulative constraint violation bounds are reduced under Slater's condition. Finally, a simulation example is presented to validate the theoretical results.

Communication-Efficient Distributed Online Nonconvex Optimization with Time-Varying Constraints

TL;DR

This work tackles distributed online nonconvex optimization with time-varying inequality constraints over time-varying directed graphs. It develops two compressed distributed online primal--dual algorithms tailored to one-point and two-point bandit feedback, together with gradient estimators and a general compression scheme with bounded absolute error. The authors prove sublinear network regret and network cumulative constraint violation bounds, with Slater’s condition providing further reductions; the two-point variant achieves bounds aligned with existing convex-bandit results, while the one-point variant attains sublinear regret in the nonconvex setting. A simulation on distributed online localization demonstrates the theory and highlights communication-efficiency advantages under compression.

Abstract

This paper considers distributed online nonconvex optimization with time-varying inequality constraints over a network of agents, where the nonconvex local loss and convex local constraint functions can vary arbitrarily across iterations. For a time-varying directed graph, we propose two distributed bandit online primal--dual algorithm with compressed communication to efficiently utilize communication resources in the one-point and two-point bandit feedback settings, respectively. To measure the performance of the proposed algorithms, we use a network regret metric grounded in the first-order optimality condition associated with the variational inequality. We show that the compressed algorithms establish sublinear network regret and cumulative constraint violation bounds. Moreover, the network cumulative constraint violation bounds are reduced under Slater's condition. Finally, a simulation example is presented to validate the theoretical results.
Paper Structure (10 sections, 12 theorems, 131 equations, 6 figures, 1 table, 2 algorithms)

This paper contains 10 sections, 12 theorems, 131 equations, 6 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Compressed Distributed Online Primal--Dual Algorithm with One-Point Bandit Feedback
  4. Algorithm Description
  5. Performance Analysis
  6. Compressed Distributed Online Primal--Dual Algorithm with Two-Point Bandit Feedback
  7. Algorithm Description
  8. Performance Analysis
  9. Simulation Example
  10. Conclusions

Key Result

Theorem 1

Suppose Assumptions 1--5 hold. For all $i \in [ n ]$, let $\{ {{x_{i,t}}} \}$ be the sequences generated by Algorithm 1 with where ${\alpha _0} > 0$, ${\theta _1} \in ( {0,1} )$, ${\gamma _0} \in ( {0,r{{( \mathbb{X} )}^2}/( {2{p^2}F_2^2} )} ]$, ${\theta _2} \in ( {0,{\theta _1}/3} )$, ${\theta _3} \in ( {{\theta _2},( {{\theta _1} - {\theta _2}} )/2} ]$, ${s_0} > 0$, and ${\theta _4} \ge 1$ are

Figures (6)

  • Figure 1: Evolutions of network regret under different trade-off parameters.
  • Figure 2: Evolutions of network cumulative constraint violation under different trade-off parameters.
  • Figure 3: Evolutions of network regret with and without Slater’s condition.
  • Figure 4: Evolutions of network cumulative constraint violation with and without Slater’s condition.
  • Figure 5: Evolutions of network regret under different quantization levels.
  • ...and 1 more figures

Theorems & Definitions (24)

  • Remark 1
  • Theorem 1
  • proof
  • Remark 2
  • Theorem 2
  • proof
  • Remark 3
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • ...and 14 more