Efficient Informed Proposals for Discrete Distributions via Newton's Series Approximation
This repository contains code for the paper Efficient Informed Proposals for Discrete Distributions via Newton's Series Approximation, accepted by the 26th International Conference on Artificial Intelligence and Statistics (AISTATS), 2023.
Introduction
Gradients have been exploited in proposal distributions to accelerate the convergence of Markov chain Monte Carlo algorithms on discrete distributions. However, these methods require a natural differentiable extension of the target discrete distribution, which often does not exist or does not provide an effective gradient guidance. In this paper, we develop a gradient-like proposal for any discrete distribution without this strong requirement. Built upon a locally-balanced proposal, our method efficiently approximates the discrete likelihood ratio via a Newton's series expansion % , as a discrete analog of the continuous Taylor series expansion, to enable a large and efficient exploration in discrete spaces. We show that our method can also be viewed as a multilinear extension, thus inheriting its desired properties. We prove that our method has a guaranteed convergence rate with or without the Metropolis-Hastings step. Furthermore, our method outperforms a number of popular alternatives in several different experiments, including the facility location problem, extractive text summarization, and image retrieval.
| Natural differentiable extension available | Natural differentiable extension unavailable | |
|---|---|---|
| Update one coordinate in a step | Gibbs with gradient proposal | Locally-balanced proposal |
| Update multiple coordinates in a step | Discrete Langevin proposal | Newton proposal (Ours) |
Usage
Facility Location Model
run & eval
python location_sample.py
Image Retrieval
run
bash run_all.sh
or
python do_rerank.py
evaluation
python eval_holidays/holidays_map.py XXX.dat
Text Summarisation
run & eval
python main.py
evaluation
python eval_holidays/holidays_map.py XXX.dat
Acknowledgements
This code is based on discrete langevin, Submodular_Summ and Submodular Reranking.