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arXiv:1611.04528

Benchmarking Quantum Hardware for Training of Fully Visible Boltzmann Machines

Dmytro Korenkevych, Yanbo Xue, Zhengbing Bian, Fabián A. Chudak, William G. Macready, Jason Tyler Rolfe, Evgeny Andriyash

Published Nov 14, 2016• 25 citations
arXivHTMLPDF

Abstract

Quantum annealing (QA) is a hardware-based heuristic optimization and sampling method applicable to discrete undirected graphical models. While similar to simulated annealing, QA relies on quantum, rather than thermal, effects to explore complex search spaces. For many classes of problems, QA is known to offer computational advantages over simulated annealing. Here we report on the ability of recent QA hardware to accelerate training of fully visible Boltzmann machines. We characterize the sampling distribution of QA hardware, and show that in many cases, the quantum distributions differ significantly from classical Boltzmann distributions. In spite of this difference, training (which seeks to match data and model statistics) using standard classical gradient updates is still effective. We investigate the use of QA for seeding Markov chains as an alternative to contrastive divergence (CD) and persistent contrastive divergence (PCD). Using $k=50$ Gibbs steps, we show that for problems with high-energy barriers between modes, QA-based seeds can improve upon chains with CD and PCD initializations. For these hard problems, QA gradient estimates are more accurate, and allow for faster learning. Furthermore, and interestingly, even the case of raw QA samples (that is, $k=0$) achieved similar improvements. We argue that this relates to the fact that we are training a quantum rather than classical Boltzmann distribution in this case. The learned parameters give rise to hardware QA distributions closely approximating classical Boltzmann distributions that are hard to train with CD/PCD.

DOI
10.48550/arxiv.1611.04528

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Quantum annealing (QA) is a hardware-based heuristic optimization and sampling method applicable to discrete undirected graphical models. While similar to simulated annealing, QA relies on quantum, rather than thermal, effects to explore complex search spaces. For many classes of problems, QA is known to offer computational advantages over simulated annealing. Here we report on the ability of recent QA hardware to accelerate training of fully visible Boltzmann machines. We characterize the sampling distribution of QA hardware, and show that in many cases, the quantum distributions differ significantly from classical Boltzmann distributions. In spite of this difference, training (which seeks to match data and model statistics) using standard classical gradient updates is still effective. We investigate the use of QA for seeding Markov chains as an alternative to contrastive divergence (CD) and persistent contrastive divergence (PCD). Using k=50k=50k=50 Gibbs steps, we show that for problems with high-energy barriers between modes, QA-based seeds can improve upon chains with CD and PCD initializations. For these hard problems, QA gradient estimates are more accurate, and allow for faster learning. Furthermore, and interestingly, even the case of raw QA samples (that is, k=0k=0k=0) achieved similar improvements. We argue that this relates to the fact that we are training a quantum rather than classical Boltzmann distribution in this case. The learned parameters give rise to hardware QA distributions closely approximating classical Boltzmann distributions that are hard to train with CD/PCD.
A hybrid quantum-classical method for learning Boltzmann machines (BM) for a generative and discriminative task is presented. BM are undirected graphs with a network of visible and hidden nodes where the former is used as the reading site. In contrast, the latter is used to manipulate visible states' probability. In Generative BM, the samples of visible data imitate the probability distribution of a given data set. In contrast, the visible sites of discriminative BM are treated as Input/Output (I/O) reading sites where the conditional probability of output state is optimized for a given set of input states. The cost function for learning BM is defined as a weighted sum of Kullback-Leibler (KL) divergence and Negative conditional Log-likelihood (NCLL), adjusted using a hyper-parameter. Here, the KL Divergence is the cost for generative learning, and NCLL is the cost for discriminative learning. A Stochastic Newton-Raphson optimization scheme is presented. The gradients and the Hessians are approximated using direct samples of BM obtained through quantum annealing. Quantum annealers are hardware representing the physics of the Ising model that operates on low but finite temperatures. This temperature affects the probability distribution of the BM; however, its value is unknown. Previous efforts have focused on estimating this unknown temperature through regression of theoretical Boltzmann energies of sampled states with the probability of states sampled by the actual hardware. These approaches assume that the control parameter change does not affect the system temperature; however, this is usually untrue. Instead of using energies, the probability distribution of samples is employed to estimate the optimal parameter set, ensuring that the optimal set can be obtained from a single set of samples. The KL divergence and NCLL are optimized for the system temperature, and the result is used to rescale the control parameter set. The performance of this approach, as tested against the theoretically expected distributions, shows promising results for Boltzmann training on quantum annealers.
In Deep Learning, a well-known approach for training a Deep Neural Network starts by training a generative Deep Belief Network model, typically using Contrastive Divergence (CD), then fine-tuning the weights using backpropagation or other discriminative techniques. However, the generative training can be time-consuming due to the slow mixing of Gibbs sampling. We investigated an alternative approach that estimates model expectations of Restricted Boltzmann Machines using samples from a D-Wave quantum annealing machine. We tested this method on a coarse-grained version of the MNIST data set. In our tests we found that the quantum sampling-based training approach achieves comparable or better accuracy with significantly fewer iterations of generative training than conventional CD-based training. Further investigation is needed to determine whether similar improvements can be achieved for other data sets, and to what extent these improvements can be attributed to quantum effects.
An increase in the efficiency of sampling from Boltzmann distributions would have a significant impact in deep learning and other machine learning applications. Recently, quantum annealers have been proposed as a potential candidate to speed up this task, but several limitations still bar these state-of-the-art technologies from being used effectively. One of the main limitations is that, while the device may indeed sample from a Boltzmann-like distribution, quantum dynamical arguments suggests it will do so with an instance-dependent effective temperature, different from the physical temperature of the device. Unless this unknown temperature can be unveiled, it might not be possible to effectively use a quantum annealer for Boltzmann sampling. In this talk, we present a strategy to overcome this challenge with a simple effective-temperature estimation algorithm. We provide a systematic study assessing the impact of the effective temperatures in the learning of a kind of restricted Boltzmann machine embedded on quantum hardware, which can serve as a building block for deep learning architectures. We also provide a comparison to k-step contrastive divergence (CD-k) with k up to 100. Although assuming a suitable fixed effective temperature also allows to outperform one step contrastive divergence (CD-1), only when using an instance-dependent effective temperature we find a performance close to that of CD-100 for the case studied here. We discuss generalizations of the algorithm to other more expressive generative models, beyond restricted Boltzmann machines.
We propose a scheme to calibrate the internal parameters of a quantum annealer to obtain well-approximated samples for training a restricted Boltzmann machine (RBM). Empirically, samples from quantum annealers obey the Boltzmann distribution, making them suitable for RBM training. Quantum annealers utilize physical phenomena to generate a large number of samples in a short time. However, hardware imperfections make it challenging to obtain accurate samples. Existing research often estimates the inverse temperature for the compensation. Our scheme efficiently utilizes samples for RBM training also to estimate internal parameters. Furthermore, we consider additional parameters and demonstrate that they improve sample quality. We evaluate our approach by comparing the obtained samples with classical Gibbs sampling, which theoretically generates accurate samples. Our results indicate that our scheme demonstrates performance on par with Gibbs sampling. In addition, the training results with our estimation scheme outperform those of the contrastive divergence algorithm, a standard training algorithm for RBM.
Restricted Boltzmann Machine (RBM) is an energy-based, undirected graphical model. It is commonly used for unsupervised and supervised machine learning. Typically, RBM is trained using contrastive divergence (CD). However, training with CD is slow and does not estimate the exact gradient of the log-likelihood cost function. In this work, the model expectation of gradient learning for RBM has been calculated using a quantum annealer (D-Wave 2000Q), where obtaining samples is faster than Markov chain Monte Carlo (MCMC) used in CD. Training and classification results of RBM trained using quantum annealing are compared with the CD-based method. The performance of the two approaches is compared with respect to the classification accuracies, image reconstruction, and log-likelihood results. The classification accuracy results indicate comparable performances of the two methods. Image reconstruction and log-likelihood results show improved performance of the CD-based method. It is shown that the samples obtained from quantum annealer can be used to train an RBM on a 64-bit “bars and stripes” dataset with classification performance similar to an RBM trained with CD. Though training based on CD showed improved learning performance, training using a quantum annealer could be useful as it eliminates computationally expensive MCMC steps of CD.
We present a Hybrid-Quantum-classical method for learning Boltzmann machines (BM) for generative and discriminative tasks. Boltzmann machines are undirected graphs that form the building block of many learning architectures such as Restricted Boltzmann machines (RBM's) and Deep Boltzmann machines (DBM's). They have a network of visible and hidden nodes where the former are used as the reading sites while the latter are used to manipulate the probability of the visible states. BM's are versatile machines that can be used for both learning distributions as a generative task as well as for performing classification or function approximation as a discriminative task. We show that minimizing KL-divergence works best for training BM for applications of function approximation. In our approach, we use Quantum annealers for sampling Boltzmann states. These states are used to approximate gradients in a stochastic gradient descent scheme. The approach is used to demonstrate logic circuits in the discriminative sense and a specialized two-phase distribution using generative BM.