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On the Global Convergence of Particle Swarm Optimization Methods

Hui Huang, Jinniao Qiu, Konstantin Riedl

TL;DR

The paper develops a rigorous mean-field framework for particle swarm optimization (PSO) and proves global convergence to minimizers of possibly nonconvex and nonsmooth objectives. By formulating PSO as a time-continuous system of stochastic differential equations and analyzing its McKean-Vlasov limit, the authors establish consensus formation and, via the Laplace principle, convergence of the consensus point to near-global minima, under a mild well-preparation condition on initialization. They also derive a dimension-free mean-field approximation result with polynomial complexity for the finite-particle PSO, enabling provable accuracy and complexity guarantees for the numerical method. The work extends to PSO with memory, proving well-posedness and analogous convergence results, though the full MFA-extension remains open. An efficient, parallelizable implementation with mini-batch evaluations demonstrates practical viability on high-dimensional machine-learning problems and benchmark functions such as Rastrigin and MNIST-classification tasks.

Abstract

In this paper we provide a rigorous convergence analysis for the renowned particle swarm optimization method by using tools from stochastic calculus and the analysis of partial differential equations. Based on a time-continuous formulation of the particle dynamics as a system of stochastic differential equations, we establish convergence to a global minimizer of a possibly nonconvex and nonsmooth objective function in two steps. First, we prove consensus formation of an associated mean-field dynamics by analyzing the time-evolution of the variance of the particle distribution. We then show that this consensus is close to a global minimizer by employing the asymptotic Laplace principle and a tractability condition on the energy landscape of the objective function. These results allow for the usage of memory mechanisms, and hold for a rich class of objectives provided certain conditions of well-preparation of the hyperparameters and the initial datum. In a second step, at least for the case without memory effects, we provide a quantitative result about the mean-field approximation of particle swarm optimization, which specifies the convergence of the interacting particle system to the associated mean-field limit. Combining these two results allows for global convergence guarantees of the numerical particle swarm optimization method with provable polynomial complexity. To demonstrate the applicability of the method we propose an efficient and parallelizable implementation, which is tested in particular on a competitive and well-understood high-dimensional benchmark problem in machine learning.

On the Global Convergence of Particle Swarm Optimization Methods

TL;DR

The paper develops a rigorous mean-field framework for particle swarm optimization (PSO) and proves global convergence to minimizers of possibly nonconvex and nonsmooth objectives. By formulating PSO as a time-continuous system of stochastic differential equations and analyzing its McKean-Vlasov limit, the authors establish consensus formation and, via the Laplace principle, convergence of the consensus point to near-global minima, under a mild well-preparation condition on initialization. They also derive a dimension-free mean-field approximation result with polynomial complexity for the finite-particle PSO, enabling provable accuracy and complexity guarantees for the numerical method. The work extends to PSO with memory, proving well-posedness and analogous convergence results, though the full MFA-extension remains open. An efficient, parallelizable implementation with mini-batch evaluations demonstrates practical viability on high-dimensional machine-learning problems and benchmark functions such as Rastrigin and MNIST-classification tasks.

Abstract

In this paper we provide a rigorous convergence analysis for the renowned particle swarm optimization method by using tools from stochastic calculus and the analysis of partial differential equations. Based on a time-continuous formulation of the particle dynamics as a system of stochastic differential equations, we establish convergence to a global minimizer of a possibly nonconvex and nonsmooth objective function in two steps. First, we prove consensus formation of an associated mean-field dynamics by analyzing the time-evolution of the variance of the particle distribution. We then show that this consensus is close to a global minimizer by employing the asymptotic Laplace principle and a tractability condition on the energy landscape of the objective function. These results allow for the usage of memory mechanisms, and hold for a rich class of objectives provided certain conditions of well-preparation of the hyperparameters and the initial datum. In a second step, at least for the case without memory effects, we provide a quantitative result about the mean-field approximation of particle swarm optimization, which specifies the convergence of the interacting particle system to the associated mean-field limit. Combining these two results allows for global convergence guarantees of the numerical particle swarm optimization method with provable polynomial complexity. To demonstrate the applicability of the method we propose an efficient and parallelizable implementation, which is tested in particular on a competitive and well-understood high-dimensional benchmark problem in machine learning.
Paper Structure (18 sections, 11 theorems, 100 equations, 4 figures, 1 algorithm)

This paper contains 18 sections, 11 theorems, 100 equations, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Contribution
  3. Prior Arts
  4. Organization
  5. Mean-Field Analysis of PSO without Memory Effects
  6. Well-Posedness of PSO without Memory Effects
  7. Convergence of PSO without Memory Effects to a Global Minimizer
  8. Mean-Field Analysis of PSO with Memory Effects
  9. Well-Posedness of PSO with Memory Effects
  10. Convergence of PSO with Memory Effects to a Global Minimizer
  11. A Wholistic Convergence Statement of PSO without Memory Effects
  12. On the Mean-Field Approximation of PSO without Memory Effects
  13. Convergence of PSO without Memory Effects in Probability
  14. Implementation of PSO and Numerical Results
  15. An Efficient Implementation of PSO
  16. ...and 3 more sections

Key Result

Theorem 2.3

Let $\mathcal{E}$ satisfy Assumptions asm:zero_global--asm:growth and let $m,\gamma,\lambda,\sigma,\alpha,T>0$. If $(\overline{X}_0,\overline{V}_0)$ is distributed according to $f_0\in \mathcal{P}_4(\mathbb{R}^{2d})$, then the nonlinear SDE eq:PSO_without_memory_mf admits a unique strong solution up for some constant $C>0$ depending only on $m,\gamma,\lambda,\sigma,\alpha, c_\mathcal{E}, R,$ and $

Figures (4)

  • Figure 1: An illustration of the PSO dynamics. Agents with positions $X^1,\dots,X^N$ (yellow dots with their trajectories) explore the energy landscape of $\mathcal{E}$ in search of the global minimizer $x^*$ (green star). The dynamics of each particle is governed by five terms. A local drift term (light blue arrow) imposes a force towards its local best position $Y^i_t$ (indicated by a circle). A global drift term (dark blue arrow) drags the agent towards a momentaneous consensus point $y_\alpha(\widehat{\rho}_{Y,t}^N)$ (orange circle) computed as a weighted (visualized through color opacity) average of the particles' local best positions. Friction (purple arrow) counteracts inertia. The two remaining terms are diffusion terms (light and dark green arrows) associated with a respective drift term.
  • Figure 2: Phase transition diagrams comparing PSO without and with memory effects for different inertia parameters $m$ and noise coefficients $\sigma$ (PSO without memory) and $\sigma_2$ (PSO with memory). The empirical success probability is computed from $25$ runs and depicted by color from zero (blue) to one (yellow).
  • Figure 3: Architectures of the NNs used in the experiments of Section \ref{['sec:MLapplication']}, cf. fornasier2021convergence.
  • Figure 4: Comparison of the performances of a shallow (dashed lines) and convolutional (solid lines) NN with architectures as described in Figure \ref{['fig:architectures']}, when trained with PSO as in Algorithm \ref{['algorithm:PSOadvanced']}. Depicted are the accuracies on a test dataset (orange lines) and the values of the objective function $\mathcal{E}$ (blue lines) evaluated on a random sample of the training set of size $10000$.

Theorems & Definitions (26)

  • Remark 1.1
  • Remark 2.1
  • Theorem 2.3
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Remark 2.4
  • Lemma 2.3
  • proof
  • Theorem 2.5
  • ...and 16 more