https://www.chenk.top/en/tags/langevin-dynamics/Recent content in Langevin Dynamics on Chen Kai BlogHugoenSat, 15 Jun 2024 09:00:00 +0000https://www.chenk.top/en/pde-ml/04-variational-inference/Sat, 15 Jun 2024 09:00:00 +0000https://www.chenk.top/en/pde-ml/04-variational-inference/<p><figure class="article-figure"> <img src="https://blog-pic-ck.oss-cn-beijing.aliyuncs.com/posts/en/pde-ml/04-Variational-Inference/illustration_1.png" alt="PDE and ML (4): Variational Inference and the Fokker-Planck Equation — Chapter overview" loading="lazy" decoding="async" class="content-image"> </figure> </p> <hr> <p>Why do variational inference (a method that looks purely optimization) and Langevin MCMC (a method that looks purely sampling) end up at the same partial differential equation?</p> <p>That is the heart of this article. In continuous time, they are <strong>two faces of the same Fokker–Planck PDE</strong>: one face is the evolution of a density, the other is the Wasserstein gradient flow of KL divergence. Once you see this, several seemingly unrelated tools — the SVGD particle algorithm, the exponential convergence rate from a log-Sobolev inequality, the training of Bayesian neural networks — all snap onto a single picture.</p>