Revisiting the Effects of Stochasticity for Hamiltonian Samplers

Giulio Franzese, Dimitrios Milios, Maurizio Filippone, Pietro Michiardi
Proceedings of the 39th International Conference on Machine Learning, PMLR 162:6744-6778, 2022.

Abstract

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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v162-franzese22a, title = {Revisiting the Effects of Stochasticity for {H}amiltonian Samplers}, author = {Franzese, Giulio and Milios, Dimitrios and Filippone, Maurizio and Michiardi, Pietro}, booktitle = {Proceedings of the 39th International Conference on Machine Learning}, pages = {6744--6778}, year = {2022}, editor = {Chaudhuri, Kamalika and Jegelka, Stefanie and Song, Le and Szepesvari, Csaba and Niu, Gang and Sabato, Sivan}, volume = {162}, series = {Proceedings of Machine Learning Research}, month = {17--23 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v162/franzese22a/franzese22a.pdf}, url = {https://proceedings.mlr.press/v162/franzese22a.html}, abstract = {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.} }
Endnote
%0 Conference Paper %T Revisiting the Effects of Stochasticity for Hamiltonian Samplers %A Giulio Franzese %A Dimitrios Milios %A Maurizio Filippone %A Pietro Michiardi %B Proceedings of the 39th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2022 %E Kamalika Chaudhuri %E Stefanie Jegelka %E Le Song %E Csaba Szepesvari %E Gang Niu %E Sivan Sabato %F pmlr-v162-franzese22a %I PMLR %P 6744--6778 %U https://proceedings.mlr.press/v162/franzese22a.html %V 162 %X 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.
APA
Franzese, G., Milios, D., Filippone, M. & Michiardi, P.. (2022). Revisiting the Effects of Stochasticity for Hamiltonian Samplers. Proceedings of the 39th International Conference on Machine Learning, in Proceedings of Machine Learning Research 162:6744-6778 Available from https://proceedings.mlr.press/v162/franzese22a.html.

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