probsys / optimal-approximate-sampling

Optimal approximate sampling from discrete probability distributions

Optimal Approximate Sampling From Discrete Probability Distributions

This repository contains a prototype implementation of the optimal sampling algorithms from:

Feras A. Saad, Cameron E. Freer, Martin C. Rinard, and Vikash K. Mansinghka. Optimal Approximate Sampling From Discrete Probability Distributions. Proc. ACM Program. Lang. 4, POPL, Article 36 (January 2020), 33 pages.

Installing

The Python 3 library can be installed via pip:

pip install optas

The C code for the main sampler is in the c/ directory and the Python 3 libraries are in the src/ directory.

Only Python 3 is required to build and use the software from source.

$ git clone https://github.com/probcomp/optimal-approximate-sampling
$ cd optimal-approximate-sampling
$ python setup.py install

To build the C sampler

$ cd c && make all

Usage

Please refer to the examples in the examples directory. Given a fixed target distribution and error measure:

1. ./examples/sampling.py shows an example of how to find an optimal distribution and sample from it, given a user-specified precision.

2. ./examples/maxerror.py shows an example of how to find an optimal distribution that uses the least possible precision and obtains an error that is less than a user-specified maximum allowable error.

These examples can be run directly as follows:

$ ./pythenv.sh python examples/sampling.py
$ ./pythenv.sh python examples/maxerror.py

Tests

To test the Python library and run a crash test in C (requires pytest and scipy):

$ ./check.sh

Experiments

The code for experiments in the POPL publication is available in a tarball on the ACM Digital Library. Please refer to the online supplementary material at https://doi.org/10.1145/3371104.

Citing

Please use the following BibTeX to cite this work.

@article{saad2020sampling,
title          = {Optimal approximate sampling from discrete probability distributions},
author         = {Saad, Feras A. and Freer, Cameron E. and Rinard, Martin C. and Mansinghka, Vikash K.},
journal        = {Proc. ACM Program. Lang.},
volume         = 4,
number         = {POPL},
month          = jan,
year           = 2020,
pages          = {36:1--36:31},
numpages       = 31,
publisher      = {ACM},
doi            = {10.1145/3371104},
abstract       = {This paper addresses a fundamental problem in random variate generation: given access to a random source that emits a stream of independent fair bits, what is the most accurate and entropy-efficient algorithm for sampling from a discrete probability distribution $(p_1, \dots, p_n)$, where the output distribution $(\hat{p}_1, \dots, \hat{p}_n)$ of the sampling algorithm can be specified with a given level of bit precision? We present a theoretical framework for formulating this problem and provide new techniques for finding sampling algorithms that are optimal both statistically (in the sense of sampling accuracy) and information-theoretically (in the sense of entropy consumption). We leverage these results to build a system that, for a broad family of measures of statistical accuracy, delivers a sampling algorithm whose expected entropy usage is minimal among those that induce the same distribution (i.e., is ``entropy-optimal'') and whose output distribution $(\hat{p}_1, \dots, \hat{p}_n)$ is a closest approximation to the target distribution $(p_1, \dots, p_n)$ among all entropy-optimal sampling algorithms that operate within the specified precision budget. This optimal approximate sampler is also a closer approximation than any (possibly entropy-suboptimal) sampler that consumes a bounded amount of entropy with the specified precision, a class which includes floating-point implementations of inversion sampling and related methods found in many standard software libraries. We evaluate the accuracy, entropy consumption, precision requirements, and wall-clock runtime of our optimal approximate sampling algorithms on a broad set of probability distributions, demonstrating the ways that they are superior to existing approximate samplers and establishing that they often consume significantly fewer resources than are needed by exact samplers.},
}

Related Repositories

For a near-optimal exact dice rolling algorithm see https://github.com/probcomp/fast-loaded-dice-roller.

optimal-approximate-sampling's Issues

Include an example for finding optimal weights to reach a divergence goal

Include an example for finding the smallest set of weights (that is, the "precision" or the smallest sum of weights) that approximate a target probability distribution to a maximum tolerable divergence. Without an example, this is not exactly trivial to do because of the lack of documentation.

• Input: Target distribution; maximum divergence tolerable (e.g., 2^-64); kind of divergence; option to limit the sum of weights to a power of 2. The target distribution can be a list of ratios, but it would be preferred if it can also be a probability mass function that supports mpmath numbers (but I admit that mpmath numbers are ultimately ratios too).
• Output: Optimal probabilities in the form of integer weights.

EDIT: Clarification.