Consensus-Based Optimization Methods Converge Globally

TL;DR

This work develops a rigorous, mean-field analysis of consensus-based optimization (CBO) to establish global convergence to the unique global minimizer ${v^*}$ for a broad class of locally Lipschitz objective functions ${\cal E}$. By revealing that, in the mean-field limit, individual agents effectively follow the gradient flow of the squared distance to ${v^*}$, the authors derive a convexification mechanism that drives the probability mass toward ${v^*}$ and quantify this via a nonasymptotic Laplace principle bounding the consensus point $v_{\alpha}(\rho_t)$. They prove exponential convergence in mean-field law with rate $(2\lambda-d\sigma^2)$ and provide a probabilistic mean-field-approximation result with $O(N^{-1})$ error, yielding a holistic convergence guarantee for the discrete CBO scheme. The results establish that the hardness of a global optimization problem is encoded in the mean-field approximation error, not in the per-se dynamics, and offer a blueprint for analyzing other CBO variants and related metaheuristics. Overall, the paper provide a solid theoretical foundation for the robustness and scalability of CBO in global optimization tasks.

Abstract

In this paper, we study consensus-based optimization (CBO), which is a multi-agent metaheuristic derivative-free optimization method that can globally minimize nonconvex nonsmooth functions and is amenable to theoretical analysis. Based on an experimentally supported intuition that, on average, CBO performs a gradient descent of the squared Euclidean distance to the global minimizer, we devise a novel technique for proving the convergence to the global minimizer in mean-field law for a rich class of objective functions. The result unveils internal mechanisms of CBO that are responsible for the success of the method. In particular, we prove that CBO performs a convexification of a large class of optimization problems as the number of optimizing agents goes to infinity. Furthermore, we improve prior analyses by requiring mild assumptions about the initialization of the method and by covering objectives that are merely locally Lipschitz continuous. As a core component of this analysis, we establish a quantitative nonasymptotic Laplace principle, which may be of independent interest. From the result of CBO convergence in mean-field law, it becomes apparent that the hardness of any global optimization problem is necessarily encoded in the rate of the mean-field approximation, for which we provide a novel probabilistic quantitative estimate. The combination of these results allows to obtain probabilistic global convergence guarantees of the numerical CBO method.

Figures (3)

• Figure 1: An illustration of the internal mechanisms of CBO. We perform $100$ runs of the CBO algorithm \ref{['eq:dyn_micro_discrete']}--\ref{['eq:dyn_micro_discrete_2']}, with parameters $\Delta t=0.01$, $\alpha = 10^{15}$, $\lambda = 1$ and $\sigma = 0.1$, and $N=32000$ agents initialized according to $\rho_0 = {\cal N}((8,8), 20)$. In addition, we add three individual agents with starting locations $(-2,4)$, $(-1.5,-1.5)$ and $(4.5,1.5)$ to the set of agents in each run as shown in (a), and depict each of their $100$ trajectories as well as their mean trajectory in yellow color in (b). With the (mean) trajectories being rather straight lines, we observe that the individual agents take a straight path from their initial positions to the global minimizer $v^*$ and, in particular, disregard the local landscape of the objective function ${\cal E}$. The trajectories of the individual agents become more concentrated as the overall number of agents $N$ grows.
• Figure 2: (a) The Rastrigin function as objective function ${\cal E}$ and the squared Euclidean distance from $v^*$. (b) The evolution of the variance $\mathrm{Var}(\widehat{\rho}_{t}^N)$ and the functional ${\cal V}(\widehat{\rho}_{t}^N)$ for different initial conditions $\rho_0 = {\cal N}(\mu, 0.8)$ with $\mu\in\{1,2,3,4\}$. The measure $\widehat{\rho}_{t}^N$ is the empirical agent density that is evolved using \ref{['eq:dyn_micro_discrete']} with $N = 320000$ agents, discrete time step size $\Delta t=0.01$ and parameters $\alpha = 10^{15}$, $\lambda = 1$ and $\sigma = 0.5$. As we move the mean of the initial configuration $\rho_0$ away from the global optimizer ${v^*} = 0$, and thereby push $v^*$ into the tails of $\rho_0$, $\mathrm{Var}(\widehat{\rho}_{t}^N)$ increases in the starting phase of the dynamics. ${\cal V}(\widehat{\rho}_{t}^N)$ on the other hand always decreases exponentially at a rate $(2\lambda-d\sigma^2)$, independently of the initial condition $\rho_0$.
• Figure 3: Visualization of the decomposition of $\Omega_r$ for different positions of $v_{\alpha}({\rho_t})$ and values of $\sigma$ in the setting $H\equiv1$. In the proof of Proposition \ref{['lem:lower_bound_probability']} we limit the rate of the mass loss induced by both consensus drift and noise term for the set $K_1^c \cap \Omega_r$, which is colored blue. On the set $K_1 \cap K_2^c \cap \Omega_r$, inked orange, the noise term counterbalances any potential mass loss induced by the drift, while on the gray set $K_1 \cap K_2 \cap \Omega_r$ mass can be lost at an exponential rate $-4\lambda^2/((2c-1)\sigma^2)$.

Theorems & Definitions (41)

• Definition 1.1: Convergence in mean-field law
• Remark 1.2: Mean-field approximation
• Remark 1.3
• Example 2.1
• Definition 3.1
• Theorem 3.2: carrillo2018analytical
• Remark 3.3
• Theorem 3.4
• proof : Proof sketch
• Definition 3.5: Assumptions