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Gradient Free Cooperative Seeking of a Moving Source

Elad Michael, Chris Manzie, Tony A. Wood, Daniel Zelazo, Iman Shames

TL;DR

This work tackles gradient-free tracking of a moving source by a network of agents, merging extremum seeking with formation control to bound gradient-estimation error and accelerate convergence. It introduces a composite objective $\hat{f}_k = F_k + \phi$, where $F_k$ sums time-varying local costs and $\phi$ encodes formation objectives, under which the agents follow a distributed update $x^{(i)}_{k+1} = x^{(i)}_k - \alpha(\nabla_{x^{(i)}}\hat{f}_k + \varepsilon_k)$ with a Lipschitz gradient and Polyak-Łojasiewicz ($\mu_f$) structure. A dimension-free gradient-estimation method is developed via neighbor-based sampling, using an ellipse $\mathcal{E}^{(i)}_k$ centered at $g^{(i)}_k$ to bound $||g^{(i)}_k - \nabla f_k(x^{(i)}_k)||$, with a refinement for large neighbor sets that reduces conservatism. Theoretical guarantees show convergence to a bounded neighbourhood of the moving minimisers of $\hat{f}_k$, and simulations in 2D and 3D with different formations validate the approach and illustrate practical performance, including an open-source implementation.

Abstract

In this paper, we consider the optimisation of a time varying scalar field by a network of agents with no gradient information. We propose a composite control law, blending extremum seeking with formation control in order to converge to the extrema faster by minimising the gradient estimation error. By formalising the relationship between the formation and the gradient estimation error, we provide a novel analysis to prove the convergence of the network to a bounded neighbourhood of the field's time varying extrema. We assume the time-varying field satisfies the Polyak Lojasiewicz inequality and the gradient is Lipschitz continuous at each iteration. Numerical studies and comparisons are provided to support the theoretical results.

Gradient Free Cooperative Seeking of a Moving Source

TL;DR

This work tackles gradient-free tracking of a moving source by a network of agents, merging extremum seeking with formation control to bound gradient-estimation error and accelerate convergence. It introduces a composite objective , where sums time-varying local costs and encodes formation objectives, under which the agents follow a distributed update with a Lipschitz gradient and Polyak-Łojasiewicz () structure. A dimension-free gradient-estimation method is developed via neighbor-based sampling, using an ellipse centered at to bound , with a refinement for large neighbor sets that reduces conservatism. Theoretical guarantees show convergence to a bounded neighbourhood of the moving minimisers of , and simulations in 2D and 3D with different formations validate the approach and illustrate practical performance, including an open-source implementation.

Abstract

In this paper, we consider the optimisation of a time varying scalar field by a network of agents with no gradient information. We propose a composite control law, blending extremum seeking with formation control in order to converge to the extrema faster by minimising the gradient estimation error. By formalising the relationship between the formation and the gradient estimation error, we provide a novel analysis to prove the convergence of the network to a bounded neighbourhood of the field's time varying extrema. We assume the time-varying field satisfies the Polyak Lojasiewicz inequality and the gradient is Lipschitz continuous at each iteration. Numerical studies and comparisons are provided to support the theoretical results.
Paper Structure (12 sections, 50 equations, 6 figures, 2 algorithms)

This paper contains 12 sections, 50 equations, 6 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Cooperative Gradient Descent
  4. Gradient Estimation and Error
  5. Bounding Ellipse for large Neighbour Sets
  6. Simulations
  7. Conclusion
  8. Proof of Lemma \ref{['lem:noisyGradDesc']}
  9. Proof of Lemma \ref{['lem:minimiserRelation']}
  10. Proof of Theorem \ref{['thm:compositeConvergence']}
  11. Proof of Lemma \ref{['lem:gradPolyhedron']}
  12. Proof of Theorem \ref{['thm:boundingEllipse']}

Figures (6)

  • Figure 1: Ellipse bounding demonstration of Theorem \ref{['thm:boundingEllipse']}.
  • Figure 2: Comparing the bounds \ref{['eq:ellipseDef']} and \ref{['eq:otherEllipseDef']}.
  • Figure 3: Neighbour topology for six agents in two dimensions.
  • Figure 4: Agent Trajectories using the composite method from Section \ref{['sec:coopGradDesc']}.
  • Figure 5: Comparison of formation distance from the signal source.
  • ...and 1 more figures

Theorems & Definitions (2)

  • Remark 1
  • Remark 2