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arXiv:1408.2037

Quantum Annealing for Variational Bayes Inference

Issei Sato, Kenichi Kurihara, Shu Tanaka, Hiroshi Nakagawa, Seiji Miyashita

Published Aug 9, 2014• 8 citations
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Abstract

This paper presents studies on a deterministic annealing algorithm based on quantum annealing for variational Bayes (QAVB) inference, which can be seen as an extension of the simulated annealing for variational Bayes (SAVB) inference. QAVB is as easy as SAVB to implement. Experiments revealed QAVB finds a better local optimum than SAVB in terms of the variational free energy in latent Dirichlet allocation (LDA).

DOI
10.48550/arxiv.1408.2037

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This paper presents studies on a deterministic annealing algorithm based on quantum annealing for variational Bayes (QAVB) inference, which can be seen as an extension of the simulated annealing for variational Bayes (SAVB) inference. QAVB is as easy as SAVB to implement. Experiments revealed QAVB finds a better local optimum than SAVB in terms of the variational free energy in latent Dirichlet allocation (LDA).
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 VB inference algorithm based on a nontraditional 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 quantum mechanics concepts: i) The ground state of the Hamiltonian of a quantum system—defined from the given data—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 ⌈log K ⌉ qubits and 𝒪( K ) operations per step, where K is the number of values hidden categorical variables can take. Thus, QAVB can match the time complexity of existing VB algorithms, while delivering higher performance.
Variational Bayes (VB) inference is one of the most important algorithms in machine learning and is widely used in engineering and industry. However, VB is known to suffer from the problem of local optima. In this paper, we generalize VB by using quantum mechanics and propose an algorithm, which we call quantum annealing variational Bayes (QAVB) inference. We then show that QAVB drastically improves the performance of VB in a clustering problem described by a Gaussian mixture model, which is essentially important from the viewpoint of optimization. Finally, we discuss an intuitive understanding of how QAVB works well.
Abstract Maximum likelihood estimation (MLE) is one of the most important methods in machine learning, and the expectation-maximization (EM) algorithm is often used to obtain maximum likelihood estimates. However, EM heavily depends on initial configurations and fails to find the global optimum. On the other hand, in the field of physics, quantum annealing (QA) was proposed as a novel optimization approach. Motivated by QA, we propose a quantum annealing extension of EM, which we call the deterministic quantum annealing expectation-maximization (DQAEM) algorithm. We also discuss its advantage in terms of the path integral formulation. Furthermore, by employing numerical simulations, we illustrate how DQAEM works in MLE and show that DQAEM moderate the problem of local optima in EM.