## Thursday, September 22, 2016

### Hamiltonian Monte Carlo Acceleration Using Surrogate Functions with Random Bases

We just had a meetup on Stan where Eric mentioned Michael Betancourt's video presentation on Hamiltonian Monte Carlo (see below) and I note the Random Features expansion can speed some of these computations:

Hamiltonian Monte Carlo Acceleration Using Surrogate Functions with Random Bases by Cheng Zhang, Babak Shahbaba, Hongkai Zhao

For big data analysis, high computational cost for Bayesian methods often limits their applications in practice. In recent years, there have been many attempts to improve computational efficiency of Bayesian inference. Here we propose an efficient and scalable computational technique for a state-of-the-art Markov Chain Monte Carlo (MCMC) methods, namely, Hamiltonian Monte Carlo (HMC). The key idea is to explore and exploit the structure and regularity in parameter space for the underlying probabilistic model to construct an effective approximation of its geometric properties. To this end, we build a surrogate function to approximate the target distribution using properly chosen random bases and an efficient optimization process. The resulting method provides a flexible, scalable, and efficient sampling algorithm, which converges to the correct target distribution. We show that by choosing the basis functions and optimization process differently, our method can be related to other approaches for the construction of surrogate functions such as generalized additive models or Gaussian process models. Experiments based on simulated and real data show that our approach leads to substantially more efficient sampling algorithms compared to existing state-of-the art methods.

The Geometric Foundations of Hamiltonian Monte Carlo
M. J. Betancourt, Simon Byrne, Samuel Livingstone, Mark Girolami

Although Hamiltonian Monte Carlo has proven an empirical success, the lack of a rigorous theoretical understanding of the algorithm has in many ways impeded both principled developments of the method and use of the algorithm in practice. In this paper we develop the formal foundations of the algorithm through the construction of measures on smooth manifolds, and demonstrate how the theory naturally identifies efficient implementations and motivates promising generalizations.

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