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Hamiltonian system built on the distribution of interest.

Variational Bayes (VB) inference algorithm is used widely to estimate both the parameters and the unobserved hidden variables in generative statistical models. The algorithm -- inspired by variational methods used in computational physics -- is iterative and can get easily stuck in local minima, even when classical techniques, such as deterministic annealing (DA), are used. We study a variational Bayes (VB) inference algorithm based on a non-traditional quantum annealing approach -- referred to as quantum annealing variational Bayes (QAVB) inference -- and show that there is indeed a quantum advantage to QAVB over its classical counterparts. In particular, we show that such better performance is rooted in key concepts from quantum mechanics: (i) the ground state of the Hamiltonian of a quantum system -- defined from the given variational Bayes (VB) problem -- corresponds to an optimal solution for the minimization problem of the variational free energy at very low temperatures; (ii) such a ground state can be achieved by a technique paralleling the quantum annealing process; and (iii) starting from this ground state, the optimal solution to the VB problem can be achieved by increasing the heat bath temperature to unity, and thereby avoiding local minima introduced by spontaneous symmetry breaking observed in classical physics based VB algorithms. We also show that the update equations of QAVB can be potentially implemented using $\lceil \log K \rceil$ qubits and $\mathcal{O} (K)$ operations per step. Thus, QAVB can match the time complexity of existing VB algorithms, while delivering higher performance.

In this paper, we propose a novel sampling method, the thermostat-assisted continuously-tempered Hamiltonian Monte Carlo, for the purpose of multimodal Bayesian learning. It simulates a noisy dynamical system by incorporating both a continuously-varying tempering variable and the Nos\'e-Hoover thermostats. A significant benefit is that it is not only able to efficiently generate i.i.d. samples when the underlying posterior distributions are multimodal, but also capable of adaptively neutralising the noise arising from the use of mini-batches. While the properties of the approach have been studied using synthetic datasets, our experiments on three real datasets have also shown its performance gains over several strong baselines for Bayesian learning with various types of neural networks plunged in.

MCMC (Markov chain Monte Carlo) is a family of methods that are applied in computational physics and chemistry and also widely used in bayesian machine learning. It is used to simulate physical systems with Gibbs canonical distribution: $$ p(\vx) \propto \exp\left( - \frac{U(\vx)}{T} \right) $$ Probability $ p(\vx) $ of a system to be in the state $ \vx $ depends on the energy of the state $U(\vx)$ and temperature $ T $ . This distribution describes positions and velocities of particles in the gas, for instance. In bayesian machine learning, it defines distribution of model parameters (such as weights of a neural network). For example, consider a multivariate normal distribution: $$ p(\vx) \propto \exp\left( - \dfrac{1}{2} (\vx - \mu) T \Sigma {-1} (\vx - \mu) \right) $$ which corresponds to the following potential energy: $$ U(\vx) \dfrac{1}{2} (\vx - \mu) T \Sigma {-1} (\vx - \mu), \qquad T 1. $$ Any distribution can be rewritten as Gibbs canonical distribution, but for many problems such energy-based distributions appear very naturally.

We propose a new Markov Chain Monte Carlo algorithm which is a generalization of the stochastic dynamics method. The algorithm performs exploration of the state space using its intrinsic geometric structure, facilitating efficient sampling of complex distributions. Applied to Bayesian learning in neural networks, our algorithm was found to perform at least as well as the best state-of-the-art method while consuming considerably less time. 1 Introduction

We propose a new Markov Chain Monte Carlo algorithm which is a generalization of the stochastic dynamics method. The algorithm performs exploration of the state space using its intrinsic geometric structure, facilitating efficient sampling of complex distributions. Applied to Bayesian learning in neural networks, our algorithm was found to perform at least as well as the best state-of-the-art method while consuming considerably less time. 1 Introduction

We propose a new Markov Chain Monte Carlo algorithm which is a generalization ofthe stochastic dynamics method. The algorithm performs exploration of the state space using its intrinsic geometric structure, facilitating efficientsampling of complex distributions. Applied to Bayesian learning in neural networks, our algorithm was found to perform at least as well as the best state-of-the-art method while consuming considerably less time. 1 Introduction