Unbiased Estimation using the Underdamped Langevin Dynamics

In this work we consider the unbiased estimation of expectations w.r.t.~probability measures that have non-negative Lebesgue density, and which are known point-wise up-to a normalizing constant. We focus upon developing an unbiased method via the underdamped Langevin dynamics, which has proven to be popular of late due to applications in statistics and machine learning. Specifically in continuous-time, the dynamics can be constructed to admit the probability of interest as a stationary measure. We develop a novel scheme based upon doubly randomized estimation, which requires access only to time-discretized versions of the dynamics and are the ones that are used in practical algorithms. We prove, under standard assumptions, that our estimator is of finite variance and either has finite expected cost, or has finite cost with a high probability. To illustrate our theoretical findings we provide numerical experiments that verify our theory, which include challenging examples from Bayesian statistics and statistical physics.

Multilevel Bayesian Deep Neural Networks

In this article we consider Bayesian inference associated to deep neural networks (DNNs) and in particular, trace-class neural network (TNN) priors which were proposed by Sell et al. [39]. Such priors were developed as more robust alternatives to classical architectures in the context of inference problems. For this work we develop multilevel Monte Carlo (MLMC) methods for such models. MLMC is a popular variance reduction technique, with particular applications in Bayesian statistics and uncertainty quantification. We show how a particular advanced MLMC method that was introduced in [4] can be applied to Bayesian inference from DNNs and establish mathematically, that the computational cost to achieve a particular mean square error, associated to posterior expectation computation, can be reduced by several orders, versus more conventional techniques. To verify such results we provide numerous numerical experiments on model problems arising in machine learning. These include Bayesian regression, as well as Bayesian classification and reinforcement learning.

Unbiased Estimation of the Gradient of the Log-Likelihood for a Class of Continuous-Time State-Space Models

In this paper, we consider static parameter estimation for a class of continuous-time state-space models. Our goal is to obtain an unbiased estimate of the gradient of the log-likelihood (score function), which is an estimate that is unbiased even if the stochastic processes involved in the model must be discretized in time. To achieve this goal, we apply a doubly randomized scheme, that involves a novel coupled conditional particle filter (CCPF) on the second level of randomization. Our novel estimate helps facilitate the application of gradient-based estimation algorithms, such as stochastic-gradient Langevin descent. We illustrate our methodology in the context of stochastic gradient descent (SGD) in several numerical examples and compare with the Rhee & Glynn estimator.