Carrillo et al. (2021) — ARXIV-2106.02519
Key takeaways
Introduces Consensus Based Sampling (CBS), a derivative-free, affine- invariant interacting particle method that, via a single parameter choice, performs either Bayesian posterior sampling or MAP optimization, with convergence analysis in Gaussian and near-Gaussian settings.
- CBS unifies sampling and MAP optimization in a single particle system via a single parameter λ
- In the mean-field Gaussian setting, converges to the exact posterior with closed-form rates
- Affine invariance: dynamics is unchanged under reparameterization θ → Bθ+b
- Adaptive β via effective-sample-size criterion gives large practical speedups
- Theorem 3.10 (1D): nonlinear CBS sampling has a Gaussian fixed point O(1/β) close to the Laplace approximation
What the evidence does not cover
- Laplace-approximation convergence result (Theorem 3.10) is proven only in dimension d=1.
- Accuracy is only rigorously characterized for unimodal, near-Gaussian posteriors; multimodal distributions are not handled.
- Analysis of the nonlinear Fokker–Planck equation (2.14) assumes existence/uniqueness of strong solutions without proof.
- No theory for the finite-particle system (only mean-field limit analyzed) or for the adaptive β scheme.
- Optimal convergence rates in the non-Gaussian optimization setting are obtained only for α=0; α ∈ (0,1] yields suboptimal rates.
Standing in the anthology
Read — the reading is NOTE-134. Arrived in the imported batch, which brought in the consensus, synchronization and flocking literature that the decentralized-training results rest on.