Layerwise Systematic Scan: Deep Boltzmann Machines and Beyond
Abstract
For Markov chain Monte Carlo methods, one of the greatest discrepancies between theory and system is the scan order — while most theoretical development on the mixing time analysis deals with random updates, real-world systems are implemented with systematic scans. We bridge this gap for models that exhibit a bipartite structure, including, most notably, the Restricted/Deep Boltzmann Machine. The de facto implementation for these models scans variables in a layer-wise fashion. We show that the Gibbs sampler with a layerwise alternating scan order has its relaxation time (in terms of epochs) no larger than that of a random-update Gibbs sampler (in terms of variable updates). We also construct examples to show that this bound is asymptotically tight. Through standard inequalities, our result also implies a comparison on the mixing times.
BibTeX
@InProceedings{pmlr-v84-guo18a,
title = {Layerwise Systematic Scan: Deep Boltzmann Machines and Beyond},
author = {Guo, Heng and Kara, Kaan and Zhang, Ce},
booktitle = {Proceedings of the Twenty-First International Conference on Artificial Intelligence and Statistics},
pages = {178--187},
year = {2018},
editor = {Storkey, Amos and Perez-Cruz, Fernando},
volume = {84},
series = {Proceedings of Machine Learning Research},
month = {09--11 Apr},
publisher = {PMLR},
pdf = {http://proceedings.mlr.press/v84/guo18a/guo18a.pdf},
url = {https://proceedings.mlr.press/v84/guo18a.html},
abstract = {For Markov chain Monte Carlo methods, one of the greatest discrepancies between theory and system is the scan order — while most theoretical development on the mixing time analysis deals with random updates, real-world systems are implemented with systematic scans. We bridge this gap for models that exhibit a bipartite structure, including, most notably, the Restricted/Deep Boltzmann Machine. The de facto implementation for these models scans variables in a layer-wise fashion. We show that the Gibbs sampler with a layerwise alternating scan order has its relaxation time (in terms of epochs) no larger than that of a random-update Gibbs sampler (in terms of variable updates). We also construct examples to show that this bound is asymptotically tight. Through standard inequalities, our result also implies a comparison on the mixing times. }
}