Clarify discussion of SAMS

[fjclark]

@snarski has commented on the discussion of SAMS at line 1337 in the Overleaf:

I'm not sure it's fair to say that people refer to the simulated tempering (ST) as self-adjusted mixture sampling (SAMS). ST is one of the original samplers for methods sampling a joint distribution on (x, lambda). A drawback of ST is that one needs an initial guess for the free energy estimates.

SAMS uses the ST sampler and a stochastic approximation algorithm to estimate the free energies on the fly. In addition, I believe Tan was the first to use the expression "self-adjustment", for which we did not give any mathematical definition; it is just an observation about the form of the recursions.

It may also be worthwhile to change the language of "optimal weights". The original ST sampler made no discussion of optimal lambda schedules (in the sense of Gelman and Meng's 1998 paper), and SAMS does not do this either.

Any comments @mrshirts / others? Thanks.

[mrshirts]

I think it means to refer to the fact that expanded ensemble borrows from simulated tempering (which is named for (x,T) sampling. I don't know that it's accurate to call sampling in (x,lambda), simulated tempering though it was certainly INSPIRED by simulated tempering. SAMS is a particular version of expanded ensemble that uses a weighting scheme that is optimal under certain common assumptions. I don't know that any of these are necessarily "recommended". All of the expanded ensemble methods are rather brittle, in that the can fail to converge the weights. I think that all of these methods should get a brief call out, but not in depth discussion. Hamiltonian replica exchange is RELATIVELY safe. I will try to clean up these points when I go over this after the semester is over.

[snarski]

I don't know that it's accurate to call sampling in (x,lambda), simulated tempering though it was certainly INSPIRED by simulated tempering.

Agreed.

SAMS is a particular version of expanded ensemble that uses a weighting scheme that is optimal under certain common assumptions.

My comment is mainly that, if one wishes to mention the optimality, one should clarify what is being optimized. If possible, could you please elaborate a little?

Here's why I'm confused: the title of Tan's SAMS paper is "Optimally Adjusted Mixture Sampling and Locally Weighted Histogram Analysis", where the "optimality" refers to minimizing the asymptotic variance of the stochastic recursion through the choice of gain matrix. The gain matrix plays a role analogous to the factor of $1/f'(x_0)$ in Newton's method.

When I hear "a weighting scheme that is optimal" in the context of free energy calculations, I think about optimizing either a $\lambda$-schedule (in terms of path space optimization -- like equation (42) in [1] for one possible objective), or the allocation of sampling resources to individual discrete $\lambda$'s (the well-known equation (40) in [1]). To my knowledge, neither notion of optimality appears in Marinari and Parisi's 1992 paper, nor Tan's 2017 paper. If I'm not missing anything, I would either qualify the optmality, or avoid mentioning it altogether, as it might not be the point of the paragraph.

[1] Simulating Normalizing Constants: From Importance Sampling to Bridge Sampling to Path Sampling , Gelman and Meng, 1998 , https://projecteuclid.org/journalArticle/Download?urlId=10.1214%2Fss%2F1028905934
[2] Optimally Adjusted Mixture Sampling ..., Tan 2017, https://statweb.rutgers.edu/ztan/Publication/Tan-SAMS.pdf
[3] Simulated Tempering : a new Monte Calro scheme, Marinari and Parisi 1991, https://iopscience.iop.org/article/10.1209/0295-5075/19/6/002

[mrshirts]

Noted for when I get a chance to update that section after finals.

[fjclark]

I'm not sure if @mrshirts will have time to make these updates (please let me know if you will), so could you please make tracked changes to address this @snarski? Thanks.

[mrshirts]

I will try to get a chance to look at this. It's optimal in the sense of giving the lowest variance uncertainty in the estimated weights (and therefore free energies). It's in the papers, I will try to dig that out in the next couple day. Family vacation just cancelled for various reasons, I might have more time than anticipated.

[mrshirts]

OK, I have added some information to clarify things in Overleaf, in terms of what the different methods are for serial exploration of states. I actually added a bit more explanation (sorry to make it longer), though at a certain point just referred to the LiveCoMS enhanced sampling paper for more details. Turns out that "Optimal weights" in this case just means "weights that lead to even sampling". I have avoided any discussion of statistical optimality and convergence, just pointed out that all methods require careful monitoring and require some expertise.

[fjclark]

Thanks @mrshirts.

@snarski, could you please comment/ accept the Overleaf changes as required? Currently this is at line 1412. Thanks.

[fjclark]

@snarski, could you please see the Overleaf changes (line 1411) and comment/ accept as required? Thanks.

[fjclark]

@snarski, could you please review the above? Thanks

[snarski]

Sorry for the delay -- I'm on vacation at the moment but will try to take care of this tomorrow -- from what I remember it was fine.

[snarski]

Sorry once again for the delay. I took a look at the line number but couldn't see the relevant text anymore. I looked around for the relevant citation (tan2017optimally) but the two places I found it, I couldn't find the text in question.

However, while poking around, I found another paragraph that related to the same issue, on line 1637. The discussion seemed to me to imply equal weights were desirable. I added this comment:

I don't know if it's worth starting a whole issue here, but I'll just point out that the notion of "equal weights" can be a little misleading to the casual reader. First, "the very notion of a flat distribution is ambiguous unless a metric is specified.", see https://arxiv.org/html/1808.09519v1 (Riemann metric approach to optimal sampling of multidimensional free-energy landscapes). Second, "equal" sampling is not necessarily optimal. This is probably stated elsewhere.

The only important thing for me is that I don't want readers to come away with the idea that one should take literally the statement that spending an equal fraction of the sampling at each state as optimal or even desirable. I also wouldn't want them to confuse the optimality in SAMS with optimality of these weights (for a given path).

So long as that's clear, I can consider this issue resolved.