Consensus-Based Interacting Particle Systems and Mean-field PDEs for Optimization and Sampling
summary
Introduces a consensus-based particle method that uses weights e^{-βF} to compute a weighted average of particle locations, derives a mean-field Fokker–Planck PDE with degenerate noise, and proves under regularity and parameter choices that the particle variance decays (exponentially in time) so the mean converges arbitrarily close to the global minimizer for β large (no explicit rate). Replaces isotropic noise with component-wise noise to remove dimension dependence, uses random-batch updates to cut O(N^2) costs, and shows 98% success on toy problems and a 7,290-parameter MNIST test.
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chapters (15)
- 0:00Introduction
- 1:08Background
- 2:30Collective Behavior
- 3:53Two Examples
- 5:30Simulations
- 8:22Optimization
- 13:07Consensusbased
- 20:13Questions
- 26:05Theorem
- 33:43Laplace
- 36:58First improvement
- 40:27Numerical method
- 41:21Random batch method
- 48:54Inverse problem
- 52:01Consensusbased sampling
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