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 $\varepsilon$ error in chi-square divergence with a computational cost equivalent to $O\bigl(\kappa^2 d^\frac{1}{2}(\log\frac{1}{\varepsilon})^{\frac{3}{2}}\bigr)$ gradient evaluations in the regime $\kappa \ll \frac{d}{\log d}$ under a warm start assumption, where $\kappa$ is the condition number and $d$ is the dimension

Artificial boundary conditions for random ellitpic systems with correlated coefficient field

We are interested in numerical algorithms for computing the electrical field generated by a charge distribution localized on scale $l$ in an infinite heterogeneous correlated random medium, in a situation where the medium is only known in a box of diameter $L\gg l$ around the support of the charge. We show that the algorithm of Lu, Otto and Wang, suggesting optimal Dirichlet boundary conditions motivated by the multipole expansion of Bella, Giunti and Otto, still performs well in correlated media. With overwhelming probability, we obtain a convergence rate in terms of $l$, $L$ and the size of the correlations for which optimality is supported with numerical simulations. These estimates are provided for ensembles which satisfy a multi-scale logarithmic Sobolev inequality, where our main tool is an extension of the semi-group estimates established by the first author. As part of our strategy, we construct sub-linear second-order correctors in this correlated setting which is of independent interest

On explicit L2L^2-convergence rate estimate for piecewise deterministic Markov processes in MCMC algorithms

We establish $L^2$-exponential convergence rate for three popular piecewise deterministic Markov processes for sampling: the randomized Hamiltonian Monte Carlo method, the zigzag process, and the bouncy particle sampler. Our analysis is based on a variational framework for hypocoercivity, which combines a Poincar\'{e}-type inequality in time-augmented state space and a standard $L^2$ energy estimate. Our analysis provides explicit convergence rate estimates, which are more quantitative than existing results.Comment: Under minor revisio

On explicit L2L^2-convergence rate estimate for underdamped Langevin dynamics

We provide a new explicit estimate of exponential decay rate of underdamped Langevin dynamics in $L^2$ distance. To achieve this, we first prove a Poincar\'{e}-type inequality with Gibbs measure in space and Gaussian measure in momentum. Our new estimate provides a more explicit and simpler expression of decay rate; moreover, when the potential is convex with Poincar\'{e} constant $m \ll 1$, our new estimate offers the decay rate of $\mathcal{O}(\sqrt{m})$ after optimizing the choice of friction coefficient, which is much faster compared to $\mathcal{O}(m)$ for the overdamped Langevin dynamics.Comment: We have fixed the bug

Complexity of randomized algorithms for underdamped Langevin dynamics

We establish an information complexity lower bound of randomized algorithms for simulating underdamped Langevin dynamics. More specifically, we prove that the worst $L^2$ strong error is of order $\Omega(\sqrt{d}\, N^{-3/2})$, for solving a family of $d$-dimensional underdamped Langevin dynamics, by any randomized algorithm with only $N$ queries to $\nabla U$, the driving Brownian motion and its weighted integration, respectively. The lower bound we establish matches the upper bound for the randomized midpoint method recently proposed by Shen and Lee [NIPS 2019], in terms of both parameters $N$ and $d$.Comment: 27 pages; some revision (e.g., Sec 2.1), and new supplementary materials in Appendice

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 $\chi^2$ 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 $\Gamma$-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