  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.
