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Online Distributed Zeroth-Order Optimization With Non-Zero-Mean Adverse Noises

Yanfu Qin, Kaihong Lu

TL;DR

The paper tackles online distributed zeroth-order optimization over time-varying directed graphs with a convex constraint set $\Omega$, addressing non-zero-mean adverse noises in gradient estimates. It develops an online distributed zeroth-order mirror descent algorithm that uses a kernel-based gradient estimator and a clipping mechanism to control heavy-tailed estimation errors, coupled with a consensus step via a time-varying weight matrix $A(t)$. The authors prove a high-probability, sublinear dynamic regret bound, showing sublinear growth whenever the minimizer trajectory changes slowly, and they provide explicit rate expressions under scalable parameter choices. A simulation demonstrates moving-target tracking by six sensors, validating both tracking performance and the sublinear regret behavior, highlighting practical robustness to non-zero-mean disturbances in zeroth-order feedback.

Abstract

In this paper, the problem of online distributed zeroth-order optimization subject to a set constraint is studied via a multi-agent network, where each agent can communicate with its immediate neighbors via a time-varying directed graph. Different from the existing works on online distributed zeroth- order optimization, we consider the case where the estimate on the gradients are influenced by some non-zero-mean adverse noises. To handle this problem, we propose a new online dis- tributed zeroth-order mirror descent algorithm involving a kernel function-based estimator and a clipped strategy. Particularly, in the estimator, the kernel function-based strategy is provided to deal with the adverse noises, and eliminate the low-order terms in the Taylor expansions of the objective functions. Furthermore, the performance of the presented algorithm is measured by employing the dynamic regrets, where the offline benchmarks are to find the optimal point at each time. Under the mild assumptions on the graph and the objective functions, we prove that if the variation in the optimal point sequence grows at a certain rate, then the high probability bound of the dynamic regrets increases sublinearly. Finally, a simulation experiment is worked out to demonstrate the effectiveness of our theoretical results.

Online Distributed Zeroth-Order Optimization With Non-Zero-Mean Adverse Noises

TL;DR

The paper tackles online distributed zeroth-order optimization over time-varying directed graphs with a convex constraint set , addressing non-zero-mean adverse noises in gradient estimates. It develops an online distributed zeroth-order mirror descent algorithm that uses a kernel-based gradient estimator and a clipping mechanism to control heavy-tailed estimation errors, coupled with a consensus step via a time-varying weight matrix . The authors prove a high-probability, sublinear dynamic regret bound, showing sublinear growth whenever the minimizer trajectory changes slowly, and they provide explicit rate expressions under scalable parameter choices. A simulation demonstrates moving-target tracking by six sensors, validating both tracking performance and the sublinear regret behavior, highlighting practical robustness to non-zero-mean disturbances in zeroth-order feedback.

Abstract

In this paper, the problem of online distributed zeroth-order optimization subject to a set constraint is studied via a multi-agent network, where each agent can communicate with its immediate neighbors via a time-varying directed graph. Different from the existing works on online distributed zeroth- order optimization, we consider the case where the estimate on the gradients are influenced by some non-zero-mean adverse noises. To handle this problem, we propose a new online dis- tributed zeroth-order mirror descent algorithm involving a kernel function-based estimator and a clipped strategy. Particularly, in the estimator, the kernel function-based strategy is provided to deal with the adverse noises, and eliminate the low-order terms in the Taylor expansions of the objective functions. Furthermore, the performance of the presented algorithm is measured by employing the dynamic regrets, where the offline benchmarks are to find the optimal point at each time. Under the mild assumptions on the graph and the objective functions, we prove that if the variation in the optimal point sequence grows at a certain rate, then the high probability bound of the dynamic regrets increases sublinearly. Finally, a simulation experiment is worked out to demonstrate the effectiveness of our theoretical results.

Paper Structure

This paper contains 8 sections, 7 theorems, 50 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Graph theory
  4. Distributed optimization
  5. Algorithm design
  6. Main results
  7. A simulation example
  8. Conclusions

Key Result

Lemma 1

4749425 Under Assumption as1, for any $i,j\in \mathcal{V}$, where $A(t,s)=A(t)\cdots A(s)$, $\mathcal{C}=2\frac{1+l^{-(n-1)U}}{1-l^{(n-1)U}}$ and $\lambda=(1-l^{(n-1)U})^{\frac{1}{(n-1)U}}$.

Figures (3)

  • Figure 1: $4$-strongly connected graph. The switching order is given by (a)$\to$(b)$\to$(c)$\to$(d)$\to$(a)$\to\dots$
  • Figure 2: The state of the target and the average state of all sensors.
  • Figure 3: The trajectory of $\mathcal{R}_i^d(t)/t$ under Algorithm 1.

Theorems & Definitions (15)

  • Lemma 1
  • Definition 1
  • Remark 1
  • Definition 2
  • Theorem 1
  • Corollary 1
  • Lemma 2
  • Proof 1
  • Remark 2
  • Lemma 3
  • ...and 5 more