Bounded Rationality (Posts about Langevin)http://bjlkeng.github.io/enSat, 23 Dec 2023 02:03:03 GMTNikola (getnikola.com)http://blogs.law.harvard.edu/tech/rssBayesian Learning via Stochastic Gradient Langevin Dynamics and Bayes by Backprophttp://bjlkeng.github.io/posts/bayesian-learning-via-stochastic-gradient-langevin-dynamics-and-bayes-by-backprop/Brian Keng<div><p>After a long digression, I'm finally back to one of the main lines of research
that I wanted to write about. The two main ideas in this post are not that
recent but have been quite impactful (one of the
<a class="reference external" href="https://icml.cc/virtual/2021/test-of-time/11808">papers</a> won a recent ICML
test of time award). They address two of the topics that are near and dear to
my heart: Bayesian learning and scalability. Dare I even ask who wouldn't be
interested in the intersection of these topics?</p>
<p>This post is about two techniques to perform scalable Bayesian inference. They
both address the problem using stochastic gradient descent (SGD) but in very
different ways. One leverages the observation that SGD plus some noise will
converge to Bayesian posterior sampling <a class="citation-reference" href="http://bjlkeng.github.io/posts/bayesian-learning-via-stochastic-gradient-langevin-dynamics-and-bayes-by-backprop/#welling2011" id="id1">[Welling2011]</a>, while the other generalizes the
"reparameterization trick" from variational autoencoders to enable non-Gaussian
posterior approximations <a class="citation-reference" href="http://bjlkeng.github.io/posts/bayesian-learning-via-stochastic-gradient-langevin-dynamics-and-bayes-by-backprop/#blundell2015" id="id2">[Blundell2015]</a>. Both are easily implemented in the modern deep
learning toolkit thus benefit from the massive scalability of that toolchain.
As usual, I will go over the necessary background (or refer you to my previous
posts), intuition, some math, and a couple of toy examples that I implemented.</p>
<p><a href="http://bjlkeng.github.io/posts/bayesian-learning-via-stochastic-gradient-langevin-dynamics-and-bayes-by-backprop/">Read more…</a> (53 min remaining to read)</p></div>Bayes by BackpropBayesianelboHMCLangevinmathjaxrmspropsgdSGLDvariational inferencehttp://bjlkeng.github.io/posts/bayesian-learning-via-stochastic-gradient-langevin-dynamics-and-bayes-by-backprop/Wed, 08 Feb 2023 23:25:40 GMTAn Introduction to Stochastic Calculushttp://bjlkeng.github.io/posts/an-introduction-to-stochastic-calculus/Brian Keng<div><p>Through a couple of different avenues I wandered, yet again, down a rabbit hole
leading to the topic of this post. The first avenue was through my main focus
on a particular machine learning topic that utilized some concepts from
physics, which naturally led me to stochastic calculus. The second avenue was
through some projects at work in the quantitative finance space, which is one
of the main applications of stochastic calculus. Naively, I thought I could
write a brief post on it that would satisfy my curiosity -- that didn't work
out at all! The result is this extra long post.</p>
<p>This post is about stochastic calculus, an extension of regular calculus to
stochastic processes. It's not immediately obvious
but the rigour needed to properly understand some of the key ideas requires
going back to the measure theoretic definition of probability theory, so
that's where I start in the background. From there I quickly move on to
stochastic processes, the Wiener process, a particular flavour of stochastic
calculus called Itô calculus, and finally end with a couple of applications.
As usual, I try to include a mix of intuition, rigour where it helps intuition,
and some simple examples. It's a deep and wide topic so I hope you enjoy my
digest of it.</p>
<p><a href="http://bjlkeng.github.io/posts/an-introduction-to-stochastic-calculus/">Read more…</a> (72 min remaining to read)</p></div>Black-Scholes-MertonBrownian motionLangevinmathjaxmeasure theoryprobabilitysigma algebrastochastic calculusWeiner processwhite noisehttp://bjlkeng.github.io/posts/an-introduction-to-stochastic-calculus/Mon, 12 Sep 2022 01:05:55 GMT