Stochastic Fractional Hamiltonian Monte Carlo

In this paper, we propose a novel stochastic fractional Hamiltonian Monte Carlo approach which generalizes the Hamiltonian Monte Carlo method within the framework of fractional calculus and L\'evy diffusion. Due to the large ``jumps'' introduced by L\'evy noise and momentum term, the proposed dynamics is capable of exploring the parameter space more efficiently and effectively. We have shown that the fractional Hamiltonian Monte Carlo could sample the multi-modal and high-dimensional target distribution more efficiently than the existing methods driven by Brownian diffusion. We further extend our method for optimizing deep neural networks. The experimental results show that the proposed stochastic fractional Hamiltonian Monte Carlo for training deep neural networks could converge faster than other popular optimization schemes and generalize better.