# Efficient Informed Proposals for Discrete Distributions via Newton's Series Approximation

## TL;DR

This paper introduces a novel Newton's series-based approach for creating efficient, gradient-like proposals for discrete distributions, enabling faster convergence in MCMC without requiring differentiable target distributions.

## Contribution

The authors develop a general, gradient-free proposal method for discrete distributions using Newton's series expansion, improving exploration and convergence in MCMC algorithms.

## Key findings

- Outperforms existing methods in various experiments
- Provides guaranteed convergence rates
- Effective in diverse applications like text summarization and image retrieval

## Abstract

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 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 Newton's 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.