Revisiting the Effects of Stochasticity for Hamiltonian Samplers

We revisit the theoretical properties of Hamiltonian stochastic differential equations (SDES) for Bayesian posterior sampling, and we study the two types of errors that arise from numerical SDE simulation: the discretization error and the error due to noisy gradient estimates in the context of data...

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Bibliographic Details
Published in:arXiv.org 2021-11
Main Authors: Franzese, Giulio, Milios, Dimitrios, Filippone, Maurizio, Michiardi, Pietro
Format: Article
Language:English
Subjects:
Algorithms
Bayesian analysis
Differential equations
Empirical analysis
Integrators
Neural networks
Numerical integration
Operators (mathematics)
Sampling
Online Access:Get full text
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Summary:We revisit the theoretical properties of Hamiltonian stochastic differential equations (SDES) for Bayesian posterior sampling, and we study the two types of errors that arise from numerical SDE simulation: the discretization error and the error due to noisy gradient estimates in the context of data subsampling. Our main result is a novel analysis for the effect of mini-batches through the lens of differential operator splitting, revising previous literature results. The stochastic component of a Hamiltonian SDE is decoupled from the gradient noise, for which we make no normality assumptions. This leads to the identification of a convergence bottleneck: when considering mini-batches, the best achievable error rate is \(\mathcal{O}(\eta^2)\), with \(\eta\) being the integrator step size. Our theoretical results are supported by an empirical study on a variety of regression and classification tasks for Bayesian neural networks.
ISSN:2331-8422