Black Box Reweighting

In the last post, I wondered whether the black box reweighting method might help with the bias.

The Black Box Reweighting (BBRW) method computes weights using the observed distribution of samples, rather than the targeted distribution. (The author's website contains a link to supplementary material with further discussion.)

I ran a simple test, and the bias was slightly worse with the BBRW method, compared to standard reweighting.

The BBRW method does a fine job of solving the problem the authors intended (correct handling of a distribution that may be incompletely sampled, and lower variance of the result - the lower variance of the result was apparent in my tests), but it doesn't solve the reweighting bias issue.

The search for a method to correct reweighting bias continues...

Reweighting is biased

The reweighting technique, where samples from one probability distribution are adjusted with a weight factor to compute averages from a different (but usually very similar) distribution, is biased for a finite number of samples.

The details can be found here.


Question for the audience: Is this known? I don't recall seeing this mentioned any time the method has been presented.

Some possible routes to fix the problem are listed in the paper, but alas, I haven't been able to turn any of them into a useful solution yet.

VMC optimization papers

A recent comment suggested these papers on VMC optimization:

• Alleviation of the Fermion-sign problem by optimization of many-body wave functions


• Wave function optimization in the variational Monte Carlo method

I'd also add these to the pile of papers to read and understand:

• Optimization of quantum Monte Carlo wave functions by energy minimization


• A simple and efficient approach to the optimization of correlated wave functions

And since this post is alread about VMC optimization, some further questions:

1. What about simultaneous geometry and parameter optimization?


2. The focus seems to be on single geometries - what's likely more interesting in the future is a family of geometries - has anyone parameterized the VMC parameters (by the bond lengths or nuclear coordinates)?

Correct estimators in DMC without forward walking?

Computing properties other than the energy using Diffusion Monte Carlo requires forward walking to get correct answers. That may change, if this paper - Hellman-Feynman operator sampling in Diffusion Monte Carlo calculations - is correct.

Although as I look at the paper more, the technique doesn't look any easier than the forward walking implementation by Casulleras and Boronat (the paper does mention this similarity). It appears to use similar data in a different combination. I would be interested to see how the noise, bias, and stability characteristics compare.

The Hellman-Feynman theorem seems to be getting lots of attention in the QMC world recently. It's also been used to construct Zero-Variance Zero-Bias estimators (most recently example here).

Gravitational N-body and the Art of Computational Science

A rather ambitious project to describe a gravitional N-body code and how to write it in Ruby: The Art of Computational Science

As Greg Wilson points out, something like this would be nice to have in Python.

Efficient Global Optimization

The optimization method mentioned last post is an improvement on Efficient Global Optimization, presented in the paper Efficient Global Optimization of ExpensiveBlack-Box Functions (also available from on one of the author's websites)


Another paper listed in the references, A Taxonomy of Global Optimization Methods Based on Response Surfaces(by D. R. Jones, one of the authors on the EGO paper) is a nice overview of various optimization methods. It presents each method and gives ways in which it can be fooled into choosing the wrong point as the optimum. A fix is given, which then leads to the next method in the list.


One of the core ideas is regression based on Gaussian processes (kriging). To learn more about Gaussian processes, see www.gaussianprocess.org. The two Jones papers listed above also show how the GP based methods fit in with other basis set methods.

Useful paper for VMC optimization?

I ran across this paper, An informational approach to the global optimization of expensive-to-evaluate functions, and wondered if it might be useful for VMC optimization?

The type of problem being solved seems to fit:

1. Function is expensive to evaluate.
   
2. Function evaluation may be noisy.
   

Many optimization methods estimate an optimal point and evaluate the function at that point. The method described in the paper estimates the uncertainty in the knowledge of the function, and evaluates the function where it will best improve our knowledge of the function.

Some potential issues:

• This method doesn't make use of gradient information, which may make it less competitive than methods that do use gradient information.


• How well does it scale with the number of parameters? The authors present a method to keep the cost under control as the dimension increases, but is the method still effective then?