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Single Point-Based Distributed Zeroth-Order Optimization with a Non-Convex Stochastic Objective Function

Elissa Mhanna, Mohamad Assaad

TL;DR

This work introduces a zero-order distributed optimization method based on a one-point estimate of the gradient tracking technique and proves that this new technique converges with a single noisy function query at a time in the non-convex setting.

Abstract

Zero-order (ZO) optimization is a powerful tool for dealing with realistic constraints. On the other hand, the gradient-tracking (GT) technique proved to be an efficient method for distributed optimization aiming to achieve consensus. However, it is a first-order (FO) method that requires knowledge of the gradient, which is not always possible in practice. In this work, we introduce a zero-order distributed optimization method based on a one-point estimate of the gradient tracking technique. We prove that this new technique converges with a single noisy function query at a time in the non-convex setting. We then establish a convergence rate of $O(\frac{1}{\sqrt[3]{K}})$ after a number of iterations K, which competes with that of $O(\frac{1}{\sqrt[4]{K}})$ of its centralized counterparts. Finally, a numerical example validates our theoretical results.

Single Point-Based Distributed Zeroth-Order Optimization with a Non-Convex Stochastic Objective Function

TL;DR

This work introduces a zero-order distributed optimization method based on a one-point estimate of the gradient tracking technique and proves that this new technique converges with a single noisy function query at a time in the non-convex setting.

Abstract

Zero-order (ZO) optimization is a powerful tool for dealing with realistic constraints. On the other hand, the gradient-tracking (GT) technique proved to be an efficient method for distributed optimization aiming to achieve consensus. However, it is a first-order (FO) method that requires knowledge of the gradient, which is not always possible in practice. In this work, we introduce a zero-order distributed optimization method based on a one-point estimate of the gradient tracking technique. We prove that this new technique converges with a single noisy function query at a time in the non-convex setting. We then establish a convergence rate of after a number of iterations K, which competes with that of of its centralized counterparts. Finally, a numerical example validates our theoretical results.
Paper Structure (18 sections, 7 theorems, 60 equations, 10 figures, 1 table, 1 algorithm)

This paper contains 18 sections, 7 theorems, 60 equations, 10 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Challenges and Contribution
  4. Notation
  5. Problem Assumptions
  6. Algorithm Description
  7. Convergence Result
  8. Convergence Rate
  9. Numerical Example
  10. Simulation Setup
  11. Simulation Results
  12. Conclusion
  13. Estimated gradient
  14. Gradient Estimate Norm Squared Bound
  15. Proof of Convergence
  16. ...and 3 more sections

Key Result

Proposition 3.3

1M-A Let Assumptions noise and perturbation hold. Then, $g_{i,k}$ is a biased estimator of the agent's gradient $\nabla F_i(x_{i,k})$, $\forall i\in\mathcal{N}$, and where $b_{i,k}$ denotes the bias with respect to the true gradient. Refer to Appendix gradient for details.

Figures (10)

  • Figure 1: The evolution of the expected loss function for the digits $6$ and $7$.
  • Figure 2: The evolution of the consensus error for the digits $6$ and $7$.
  • Figure 3: The evolution of the consensus error for the digits $6$ and $7$ as compared with the rate $O(\frac{1}{K+1})$.
  • Figure 4: The evolution of the gradient tracking error for the digits $6$ and $7$.
  • Figure 5: The evolution of the accuracy for the digits $6$ and $7$.
  • ...and 5 more figures

Theorems & Definitions (8)

  • Remark 2.1
  • Proposition 3.3
  • Lemma 3.4
  • Lemma 3.5
  • Lemma 3.6
  • Theorem 3.7
  • Theorem 4.1
  • Lemma B.1