17 research outputs found
Non-reversible Parallel Tempering for Deep Posterior Approximation
Parallel tempering (PT), also known as replica exchange, is the go-to
workhorse for simulations of multi-modal distributions. The key to the success
of PT is to adopt efficient swap schemes. The popular deterministic even-odd
(DEO) scheme exploits the non-reversibility property and has successfully
reduced the communication cost from to given sufficiently many
chains. However, such an innovation largely disappears in big data due to
the limited chains and few bias-corrected swaps. To handle this issue, we
generalize the DEO scheme to promote non-reversibility and propose a few
solutions to tackle the underlying bias caused by the geometric stopping time.
Notably, in big data scenarios, we obtain an appealing communication cost
based on the optimal window size. In addition, we also adopt
stochastic gradient descent (SGD) with large and constant learning rates as
exploration kernels. Such a user-friendly nature enables us to conduct
approximation tasks for complex posteriors without much tuning costs.Comment: Accepted by AAAI 202
Subsampling Error in Stochastic Gradient Langevin Diffusions
The Stochastic Gradient Langevin Dynamics (SGLD) are popularly used to
approximate Bayesian posterior distributions in statistical learning procedures
with large-scale data. As opposed to many usual Markov chain Monte Carlo (MCMC)
algorithms, SGLD is not stationary with respect to the posterior distribution;
two sources of error appear: The first error is introduced by an
Euler--Maruyama discretisation of a Langevin diffusion process, the second
error comes from the data subsampling that enables its use in large-scale data
settings. In this work, we consider an idealised version of SGLD to analyse the
method's pure subsampling error that we then see as a best-case error for
diffusion-based subsampling MCMC methods. Indeed, we introduce and study the
Stochastic Gradient Langevin Diffusion (SGLDiff), a continuous-time Markov
process that follows the Langevin diffusion corresponding to a data subset and
switches this data subset after exponential waiting times. There, we show that
the Wasserstein distance between the posterior and the limiting distribution of
SGLDiff is bounded above by a fractional power of the mean waiting time.
Importantly, this fractional power does not depend on the dimension of the
state space. We bring our results into context with other analyses of SGLD
Birth-death dynamics for sampling: Global convergence, approximations and their asymptotics
Motivated by the challenge of sampling Gibbs measures with nonconvex
potentials, we study a continuum birth-death dynamics. We improve results in
previous works [51,57] and provide weaker hypotheses under which the
probability density of the birth-death governed by Kullback-Leibler divergence
or by divergence converge exponentially fast to the Gibbs equilibrium
measure, with a universal rate that is independent of the potential barrier. To
build a practical numerical sampler based on the pure birth-death dynamics, we
consider an interacting particle system, which is inspired by the gradient flow
structure and the classical Fokker-Planck equation and relies on kernel-based
approximations of the measure. Using the technique of -convergence of
gradient flows, we show that on the torus, smooth and bounded positive
solutions of the kernelized dynamics converge on finite time intervals, to the
pure birth-death dynamics as the kernel bandwidth shrinks to zero. Moreover we
provide quantitative estimates on the bias of minimizers of the energy
corresponding to the kernelized dynamics. Finally we prove the long-time
asymptotic results on the convergence of the asymptotic states of the
kernelized dynamics towards the Gibbs measure.Comment: significant mathematical changes with more rigor on gradient flow
MCMC-driven learning
This paper is intended to appear as a chapter for the Handbook of Markov
Chain Monte Carlo. The goal of this chapter is to unify various problems at the
intersection of Markov chain Monte Carlo (MCMC) and machine
learning\unicode{x2014}which includes black-box variational inference,
adaptive MCMC, normalizing flow construction and transport-assisted MCMC,
surrogate-likelihood MCMC, coreset construction for MCMC with big data, Markov
chain gradient descent, Markovian score climbing, and
more\unicode{x2014}within one common framework. By doing so, the theory and
methods developed for each may be translated and generalized