**Are you an EPFL student looking for a semester project?**

Work with us on data science and visualisation projects, and deploy your project as an app on top of GraphSearch.

Lecture# Convergence of Adaptive Langevin using hypocoercivity

Description

This lecture covers the convergence of Adaptive Langevin dynamics, focusing on hypocoercive techniques. It reviews the convergence of Langevin type dynamics, demonstrates convergence rates, and discusses the Central Limit Theorem. The instructor presents the structure of Adaptive Langevin dynamics, the removal of mini-batching bias, and the Hamiltonian and overdamped limits. The lecture also explores Fokker-Planck equations, ergodicity results, and the direct L² approach. It delves into the sharpness of scaling, elements of proof, and the Central Limit Theorem. Some numerical results and the normalization of dynamics for Bayesian inference in the large data context are also discussed.

Login to watch the video

Official source

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related concepts (141)

Related lectures (104)

Langevin equation

In physics, a Langevin equation (named after Paul Langevin) is a stochastic differential equation describing how a system evolves when subjected to a combination of deterministic and fluctuating ("random") forces. The dependent variables in a Langevin equation typically are collective (macroscopic) variables changing only slowly in comparison to the other (microscopic) variables of the system. The fast (microscopic) variables are responsible for the stochastic nature of the Langevin equation.

Stochastic differential equation

A stochastic differential equation (SDE) is a differential equation in which one or more of the terms is a stochastic process, resulting in a solution which is also a stochastic process. SDEs have many applications throughout pure mathematics and are used to model various behaviours of stochastic models such as stock prices, random growth models or physical systems that are subjected to thermal fluctuations. SDEs have a random differential that is in the most basic case random white noise calculated as the derivative of a Brownian motion or more generally a semimartingale.

Supersymmetric theory of stochastic dynamics

Supersymmetric theory of stochastic dynamics or stochastics (STS) is an exact theory of stochastic (partial) differential equations (SDEs), the class of mathematical models with the widest applicability covering, in particular, all continuous time dynamical systems, with and without noise. The main utility of the theory from the physical point of view is a rigorous theoretical explanation of the ubiquitous spontaneous long-range dynamical behavior that manifests itself across disciplines via such phenomena as 1/f, flicker, and crackling noises and the power-law statistics, or Zipf's law, of instantonic processes like earthquakes and neuroavalanches.

Numerical stability

In the mathematical subfield of numerical analysis, numerical stability is a generally desirable property of numerical algorithms. The precise definition of stability depends on the context. One is numerical linear algebra and the other is algorithms for solving ordinary and partial differential equations by discrete approximation. In numerical linear algebra, the principal concern is instabilities caused by proximity to singularities of various kinds, such as very small or nearly colliding eigenvalues.

Numerical analysis

Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods that attempt at finding approximate solutions of problems rather than the exact ones. Numerical analysis finds application in all fields of engineering and the physical sciences, and in the 21st century also the life and social sciences, medicine, business and even the arts.

Covers the joint equidistribution of CM points and the ergodic decomposition theorem in compact abelian groups.

Explores Kolmogorov's 0-1 law, convergence of random variables, tightness, and characteristic functions.

Covers independence between random variables and product measures in probability theory.

Explores the Kirillov paradigm for the Heisenberg group and unitary representations.

Covers the convergence criteria for sequences, including operations on limits and sequences defined by recurrence.