NEO: Non Equilibrium Sampling on the Orbits of a Deterministic Transform

Achille Thin, Yazid Janati El Idrissi, Sylvain Le Corff, Charles Ollion, Eric Moulines, Arnaud Doucet, Alain Durmus, Christian P Robert

Keywords: Importance Sampling, Monte Carlo, Markov chain Monte Carlo

TL;DR: Novel Monte Carlo methods building on the orbits of a deterministic transform for computing normalizing constants and sampling.

Abstract:

Sampling from a complex distribution $\pi$ and approximating its intractable normalizing constant $\mathrm{Z}$ are challenging problems.
In this paper, a novel family of importance samplers (IS) and Markov chain Monte Carlo (MCMC) samplers is derived.
Given an invertible map $\mathrm{T}$, these schemes combine (with weights) elements from the forward and backward Orbits through points sampled from a proposal distribution $\rho$. The map $\mathrm{T}$ does not leave the target $\pi$ invariant, hence the name NEO, standing for Non-Equilibrium Orbits.
NEO-IS provides unbiased estimators of the normalizing constant and self-normalized IS estimators of expectations under $\pi$ while NEO-MCMC combines multiple NEO-IS estimates of the normalizing constant and an iterated sampling-importance resampling mechanism to sample from $\pi$.
For $\mathrm{T}$ chosen as a discrete-time integrator of a conformal Hamiltonian system, NEO-IS achieves state-of-the art performance on difficult benchmarks and NEO-MCMC is able to explore highly multimodal targets. Additionally, we provide detailed theoretical results for both methods. In particular, we show that NEO-MCMC is uniformly geometrically ergodic and establish explicit mixing time estimates under mild conditions.

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Code: https://github.com/Achillethin/Non_equilibrium_VAE