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Direct-search methods for decentralized blackbox optimization

El Houcine Bergou, Youssef Diouane, Vyacheslav Kungurtsev, Clément W. Royer

TL;DR

The paper addresses decentralized blackbox optimization by adapting direct-search methods to networks of agents that share information to solve a sum of local functions. It introduces two direct-search variants, DDS-L and DDS-F, built on a penalty reformulation and a local-descent framework, respectively, and proves convergence to stationary points of the penalized objective for DDS-L and asymptotic consensus for DDS-F. Through numerical experiments on toy and Moré–Wild-inspired problems, the authors show that direct-search methods can outperform gradient-approximation-based decentralized strategies in practice, with DDS-F (Vanishing) often yielding the best balance between objective progress and consensus. The work thereby broadens derivative-free optimization tools for decentralized environments, with implications for privacy-preserving and computation-expensive settings, and suggests directions for integrating direct-search with gradient-tracking and stochastic/nonsmooth extensions.

Abstract

Derivative-free optimization algorithms are particularly useful for tackling blackbox optimization problems where the objective function arises from complex and expensive procedures that preclude the use of classical gradient-based methods. In contemporary decentralized environments, such functions are defined locally on different computational nodes due to technical or privacy constraints, introducing additional challenges within the optimization process. In this paper, we adapt direct-search methods, a classical technique in derivative-free optimization, to the decentralized setting. In contrast with zeroth-order algorithms, our algorithms rely on positive spanning sets to define suitable search directions, while still possessing global convergence guarantees thanks to carefully chosen stepsizes. Numerical experiments highlight the advantages of direct-search techniques over gradient-approximation-based strategies.

Direct-search methods for decentralized blackbox optimization

TL;DR

The paper addresses decentralized blackbox optimization by adapting direct-search methods to networks of agents that share information to solve a sum of local functions. It introduces two direct-search variants, DDS-L and DDS-F, built on a penalty reformulation and a local-descent framework, respectively, and proves convergence to stationary points of the penalized objective for DDS-L and asymptotic consensus for DDS-F. Through numerical experiments on toy and Moré–Wild-inspired problems, the authors show that direct-search methods can outperform gradient-approximation-based decentralized strategies in practice, with DDS-F (Vanishing) often yielding the best balance between objective progress and consensus. The work thereby broadens derivative-free optimization tools for decentralized environments, with implications for privacy-preserving and computation-expensive settings, and suggests directions for integrating direct-search with gradient-tracking and stochastic/nonsmooth extensions.

Abstract

Derivative-free optimization algorithms are particularly useful for tackling blackbox optimization problems where the objective function arises from complex and expensive procedures that preclude the use of classical gradient-based methods. In contemporary decentralized environments, such functions are defined locally on different computational nodes due to technical or privacy constraints, introducing additional challenges within the optimization process. In this paper, we adapt direct-search methods, a classical technique in derivative-free optimization, to the decentralized setting. In contrast with zeroth-order algorithms, our algorithms rely on positive spanning sets to define suitable search directions, while still possessing global convergence guarantees thanks to carefully chosen stepsizes. Numerical experiments highlight the advantages of direct-search techniques over gradient-approximation-based strategies.

Paper Structure

This paper contains 17 sections, 6 theorems, 52 equations, 4 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Notations
  3. Background on decentralized optimization
  4. Penalty function and reformulation
  5. Decentralized gradient descent
  6. Decentralized direct-search frameworks
  7. Algorithm based on Lyapunov function decrease
  8. Algorithm based on local function decrease
  9. Convergence results
  10. Generic assumptions
  11. Convergence analysis of Algorithm \ref{['alg:ddsL']} (DDS-L)
  12. Convergence analysis of Algorithm \ref{['alg:ddsF']} (DDS-F)
  13. Numerical Results
  14. Implementation details
  15. A separable problem
  16. ...and 2 more sections

Key Result

Lemma 4.1

Let Assumptions as:lip and as:pss hold. Suppose that the $k$-th iteration of Algorithm alg:ddsL is unsuccessful for agent $i$. Then, one has where $M_i:=L_i+\frac{1-w_{ii}}{\gamma}$.

Figures (4)

  • Figure 1: Convergence plots for problem \ref{['eq:toypb']} in dimension $n=5$.
  • Figure 2: Convergence plots for problem \ref{['eq:toypb']} with $\gamma=1$.
  • Figure 3: Performance profiles using three different optimality metrics.
  • Figure 4: Data profiles using three different optimality metrics.

Theorems & Definitions (6)

  • Lemma 4.1
  • Theorem 4.1
  • Proposition 4.1
  • Lemma 4.2
  • Proposition 4.2
  • Theorem 4.2