[Paper Review] Convergence of a first-order consensus-based global optimization algorithm

Seung‐Yeal Ha, Shi Jin+1 more|arXiv (Cornell University)|Oct 18, 2019

Distributed Control Multi-Agent Systems23 references4 citations

TL;DR

This paper provides a direct convergence analysis of a first-order consensus-based optimization (CBO) algorithm for non-convex global optimization, bypassing the mean-field limit used in prior work. It proves almost sure global consensus for any initial data and establishes a dimension-independent condition ensuring the consensus state lies near the global minimum as the inverse temperature β → ∞.

ABSTRACT

Global optimization of a non-convex objective function often appears in large-scale machine-learning and artificial intelligence applications. Recently, consensus-based optimization (in short CBO) methods have been introduced as one of the gradient-free optimization methods. In this paper, we provide a convergence analysis for the first-order CBO method in \cite{C-J-L-Z}. Prior to the current work, the convergence study was carried out for CBO methods on corresponding mean-field limit, a Fokker-Planck equation, which does not imply the convergence of the CBO method {\it per se}. Based on the consensus estimate directly on the first-order CBO model, we provide a convergence analysis of the first-order CBO method \cite{C-J-L-Z} without resorting to the corresponding mean-field model. Our convergence analysis consists of two steps. In the first step, we show that the CBO model exhibits a global consensus time asymptotically for any initial data, and in the second step, we provide a sufficient condition on system parameters--which is dimension independent-- and initial data which guarantee that the converged consensus state lies in a small neighborhood of the global minimum almost surely.

Motivation & Objective

• To address the lack of rigorous convergence analysis for first-order CBO algorithms in non-convex optimization.
• To establish convergence of the CBO algorithm directly at the particle level, without relying on the mean-field Fokker-Planck limit.
• To derive a dimension-independent sufficient condition on system parameters and initial data ensuring the consensus state approximates the global minimum.
• To analyze both continuous and discrete CBO formulations, with rigorous results only for the continuous case due to mathematical constraints.
• To lay the foundation for extending the analysis to less regular objective functions and other metaheuristic algorithms.

Proposed method

• Rewrite the CBO algorithm in a first-order consensus form using a weighted average of particle positions.
• Derive an explicit exact formula for the difference between particle states in the continuous-time model.
• Use the explicit formula to prove almost sure global consensus for any initial configuration.
• Apply Laplace's principle to show that the consensus state concentrates near the global minimum as β → ∞.
• Employ Itô calculus and stochastic analysis for twice differentiable and bounded objective functions in the continuous case.
• Conduct numerical simulations to validate consensus formation and decay rates in both continuous and discrete settings.

Experimental results

Research questions

• RQ1 Does the N-particle CBO system exhibit global consensus almost surely for any initial data?
• RQ2 Under what conditions does the consensus state of the CBO algorithm converge to a neighborhood of the global minimum?
• RQ3 Can convergence be established directly on the particle-level model without using the mean-field Fokker-Planck limit?
• RQ4 How do system parameters like λ, σ, and β affect the convergence speed and accuracy?
• RQ5 Can the analysis be extended to less regular objective functions or discrete-time implementations?

Key findings

• Global consensus emerges almost surely for any initial data in the continuous CBO model, regardless of dimension d.
• The consensus state lies in an O(1/β)-neighborhood of the global minimum under a dimension-independent condition on system parameters and initial data.
• The convergence to the global minimum is guaranteed as β → ∞, leveraging Laplace's principle for asymptotic concentration.
• Numerical results confirm faster consensus with increasing noise intensity σ, even when 2λ < σ².
• The discrete CBO algorithm also exhibits consensus, but rigorous convergence analysis remains open due to lack of suitable stochastic tools.
• The explicit formula for state differences confirms that consensus occurs if λ > 0 or σ > 0, independent of the condition 2λ > σ².