Revisiting the effects of stochasticity for Hamiltonian samplers

Franzese, Giulio; Milios, Dimitrios; Filippone, Maurizio; Michiardi, Pietro
Submitted to ArXiV, 4 November 2021

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 O(η2), with η 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.


Type:
Conférence
Date:
2021-11-04
Department:
Data Science
Eurecom Ref:
6740
Copyright:
© EURECOM. Personal use of this material is permitted. The definitive version of this paper was published in Submitted to ArXiV, 4 November 2021 and is available at :

PERMALINK : https://www.eurecom.fr/publication/6740