125 research outputs found
Dynamics of Langevin Simulation
This chapter [of a supplement to Prog. Theo. Phys.] reviews numerical
simulations of quantum field theories based on stochastic quantization and the
Langevin equation. The topics discussed include renormalization of finite
step-size algorithms, Fourier acceleration, and the relation of the Langevin
equation to hybrid stochastic algorithms and hybrid Monte Carlo.Comment: 20 p
Quasi-symplectic Langevin Variational Autoencoder
Variational autoencoder (VAE) is a very popular and well-investigated
generative model vastly used in neural learning research. To leverage VAE in
practical tasks dealing with a massive dataset of large dimensions it is
required to deal with the difficulty of building low variance evidence lower
bounds (ELBO). Markov ChainMonte Carlo (MCMC) is one of the effective
approaches to tighten the ELBO for approximating the posterior distribution.
Hamiltonian Variational Autoencoder(HVAE) is an effective MCMC inspired
approach for constructing a low-variance ELBO which is also amenable to the
reparameterization trick. In this work, we propose a Quasi-symplectic Langevin
Variational autoencoder (Langevin-VAE) by incorporating the gradients
information in the inference process through the Langevin dynamic. We show the
effectiveness of the proposed approach by toy and real-world examples
Langevin Quasi-Monte Carlo
Langevin Monte Carlo (LMC) and its stochastic gradient versions are powerful
algorithms for sampling from complex high-dimensional distributions. To sample
from a distribution with density , LMC
iteratively generates the next sample by taking a step in the gradient
direction with added Gaussian perturbations. Expectations w.r.t. the
target distribution are estimated by averaging over LMC samples. In
ordinary Monte Carlo, it is well known that the estimation error can be
substantially reduced by replacing independent random samples by quasi-random
samples like low-discrepancy sequences. In this work, we show that the
estimation error of LMC can also be reduced by using quasi-random samples.
Specifically, we propose to use completely uniformly distributed (CUD)
sequences with certain low-discrepancy property to generate the Gaussian
perturbations. Under smoothness and convexity conditions, we prove that LMC
with a low-discrepancy CUD sequence achieves smaller error than standard LMC.
The theoretical analysis is supported by compelling numerical experiments,
which demonstrate the effectiveness of our approach
The shifted ODE method for underdamped Langevin MCMC
In this paper, we consider the underdamped Langevin diffusion (ULD) and
propose a numerical approximation using its associated ordinary differential
equation (ODE). When used as a Markov Chain Monte Carlo (MCMC) algorithm, we
show that the ODE approximation achieves a -Wasserstein error of
in
steps under
the standard smoothness and strong convexity assumptions on the target
distribution. This matches the complexity of the randomized midpoint method
proposed by Shen and Lee [NeurIPS 2019] which was shown to be order optimal by
Cao, Lu and Wang. However, the main feature of the proposed numerical method is
that it can utilize additional smoothness of the target log-density . More
concretely, we show that the ODE approximation achieves a -Wasserstein error
of in
and
steps when Lipschitz
continuity is assumed for the Hessian and third derivative of . By
discretizing this ODE using a third order Runge-Kutta method, we can obtain a
practical MCMC method that uses just two additional gradient evaluations per
step. In our experiment, where the target comes from a logistic regression,
this method shows faster convergence compared to other unadjusted Langevin MCMC
algorithms
Complexity of zigzag sampling algorithm for strongly log-concave distributions
We study the computational complexity of zigzag sampling algorithm for
strongly log-concave distributions. The zigzag process has the advantage of not
requiring time discretization for implementation, and that each proposed
bouncing event requires only one evaluation of partial derivative of the
potential, while its convergence rate is dimension independent. Using these
properties, we prove that the zigzag sampling algorithm achieves
error in chi-square divergence with a computational cost equivalent to
gradient evaluations in the regime under a warm
start assumption, where is the condition number and is the
dimension
NySALT: Nystr\"{o}m-type inference-based schemes adaptive to large time-stepping
Large time-stepping is important for efficient long-time simulations of
deterministic and stochastic Hamiltonian dynamical systems. Conventional
structure-preserving integrators, while being successful for generic systems,
have limited tolerance to time step size due to stability and accuracy
constraints. We propose to use data to innovate classical integrators so that
they can be adaptive to large time-stepping and are tailored to each specific
system. In particular, we introduce NySALT, Nystr\"{o}m-type inference-based
schemes adaptive to large time-stepping. The NySALT has optimal parameters for
each time step learnt from data by minimizing the one-step prediction error.
Thus, it is tailored for each time step size and the specific system to achieve
optimal performance and tolerate large time-stepping in an adaptive fashion. We
prove and numerically verify the convergence of the estimators as data size
increases. Furthermore, analysis and numerical tests on the deterministic and
stochastic Fermi-Pasta-Ulam (FPU) models show that NySALT enlarges the maximal
admissible step size of linear stability, and quadruples the time step size of
the St\"{o}rmer--Verlet and the BAOAB when maintaining similar levels of
accuracy.Comment: 26 pages, 7 figure
- âŠ