From bbf4288e62e6d0120c6919f90abc43fbcfb6033b Mon Sep 17 00:00:00 2001 From: Morten Hjorth-Jensen Date: Tue, 16 Apr 2024 06:13:34 +0200 Subject: [PATCH] Update week13.do.txt --- doc/src/week13/week13.do.txt | 29 ++++++++++++++++++++++++++++- 1 file changed, 28 insertions(+), 1 deletion(-) diff --git a/doc/src/week13/week13.do.txt b/doc/src/week13/week13.do.txt index 90d61628..61b314ac 100644 --- a/doc/src/week13/week13.do.txt +++ b/doc/src/week13/week13.do.txt @@ -162,7 +162,34 @@ That notebook is based on a recent article by Du and Mordatch, _Implicit generat !split ===== Langevin sampling ===== -_Note_: Notes to be added +Also called Stochastic gradient Langevin dynamics (SGLD), is sampling +technique composed of characteristics from Stochastic gradient descent +(SGD) and Langevin dynamics, a mathematical extension of the Langevin +equation. The SGLD is an iterative +optimization algorithm which uses minibatching to create a stochastic +gradient estimator, as used in SGD to optimize a differentiable +objective function.[1] Unlike traditional SGD, SGLD can be used for +Bayesian learning as a sampling method. SGLD may be viewed as Langevin +dynamics applied to posterior distributions, but the key difference is +that the likelihood gradient terms are minibatched, like in SGD. SGLD, +like Langevin dynamics, produces samples from a posterior distribution +of parameters based on available data. + +!split +===== More on the SGLD ===== + +The SGLD uses the probability $p(\theta)$ (note that we limit +ourselves to just a variable $\theta$) and updates the _log_ of this +probability by initializing it through some random prior distribution. + +The update is given by +!bt +\[ +\theta_{i+1}=\theta_{i}+\eta \nabla_{\theta} \log{p(\theta_{i})}+z_i\sqrt{, +\] +!et +where $z_i$ are normally distributed with mean zero and variance one and $i=0,1,\dots,k$, with $k$ the final number of iterations. + !split ===== Theory of Variational Autoencoders =====