Paper: Accelerated Sampling with Stacked Restricted Boltzmann Machines (ICLR-2024)

Reviewer comment (Evaluation):
Learning nested RBMs seems costly compared to simple parallel tempering approach where additional models are not required.

Author response:
Although parallel tempering (PT) requires no additional training, it requires several (usually hundreds or thousands) copies of the RBM at different temperatures that exchange configurations. The temperatures are also not easy to set a priori. Stacked tempering (ST) requires additional training of a few models (our experiments were performed with 4 RBMs at most). In addition, training the deeper models is faster because: 1) the landscapes are smoother at deeper levels of the stack, and 2) the layers of the stack shrink and therefore these models are trained on data of smaller dimensionality. Table S3 in the Appendix now reports the training times. As can be seen from table S4, the cumulative training of the deeper RBMs takes a time comparable to the training time of the bottom RBM.

Finally, we note that the total sampling time of stacked tempering can be decomposed as training time, which is a constant offset, plus sampling time, which is proportional to the number of generated samples. The stack needs to be trained once only.

Reviewer comment (Evaluation):
As their approach is intended to speed up sampling of RBMs a figure/table demonstrating how the real world sampling time is approved compared to the parallel tempering and AGS baselines would give the reader a better sense of how this method fares practically.

Author response:
We have added table S4 in Supplementary materials during revision, with sampling times (in minutes) in our experiments for different algorithms and datasets. Overall we find that ST is faster than AGS or PT.

Reviewer comment (Presentation):
Adding figures similar to Fig 3(d-h) for the Ising model results in the main text would give the reader a better understanding of the results on this problem.

Author response:
We have added Figure S8 in supplementary showing example trajectories of the magnetization in the Ising model during sampling with the different algorithms. ST samples more easily configurations of alternate magnetization than AGS or PT.

Reviewer comment (Novelty):
A conceptual comparison to deep Boltzmann machines would be a great addition as it would make clear how their approach differs from sampling schemes for deep Boltzmann machines.

Author response:
We agree that applications to deep Boltzmann machines (DBM) would be interesting and we would like to explore this in future work. A DBM can be regarded as a particular case of an RBM, in  which “visible” and “hidden” layers are made of the even and odd layers of the DBM. Therefore, stacked tempering (ST) should apply without modification. Whether ST can help training DBMs is an interesting open question, which we now briefly mention in Discussion.