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IEEE International Conference on Quantum Computing and Engineering (QCE)

Fast training of fully-connected Boltzmann Machines on an Adiabatic Quantum Computer

Lorenzo Rocutto, Davide Noè, Lorenzo Del Moro, Enrico Prati

Published Sep 17, 2023• 2 citations
Publisher

Abstract

We demonstrate how to leverage thermal noise within an Adiabatic Quantum Computer (AQC) to train a complete Boltzmann Machine (BM). In contrast to previous implementations, our fully-connected BM includes hidden units and we sample both negative and positive statistics from the AQC. By training the network on a standard benchmark Bars and Stripes dataset, the maximally parallelized classical algorithm takes 11.4 times longer to reach the same likelihood as the quantum algorithm, indicating that AQCs may expedite BM training. The ability of AQCs to sample Boltzmann-distributed configurations quickly has the potential to impact BMs training as well as influence the field of stochastic machine learning.

DOI
10.1109/qce57702.2023.00077

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We demonstrate how to leverage thermal noise within an Adiabatic Quantum Computer (AQC) to train a complete Boltzmann Machine (BM). In contrast to previous implementations, our fully-connected BM includes hidden units and we sample both negative and positive statistics from the AQC. By training the network on a standard benchmark Bars and Stripes dataset, the maximally parallelized classical algorithm takes 11.4 times longer to reach the same likelihood as the quantum algorithm, indicating that AQCs may expedite BM training. The ability of AQCs to sample Boltzmann-distributed configurations quickly has the potential to impact BMs training as well as influence the field of stochastic machine learning.
Abstract Despite the anticipated speed‐up of quantum computing, the achievement of a measurable advantage remains subject to ongoing debate. Adiabatic Quantum Computers (AQCs) are quantum devices designed to solve quadratic uncostrained binary optimization (QUBO) problems, but their intrinsic thermal noise can be leveraged to train computationally demanding machine learning algorithms such as the Boltzmann Machine (BM). Despite an asymptotic advantage is expected only for large networks, a limited quantum speed up can be already achieved on a small BM is shown, by exploiting parallel adiabatic computation. This approach exhibits a 8.6‐fold improvement in wall time on the Bars and Stripes dataset when compared to a parallelized classical Gibbs sampling method, which has never been outperformed before by quantum approaches.
Boltzmann Machines constitute a paramount class of neural networks for unsupervised learning and recommendation systems. Their bipartite version, called Restricted Boltzmann Machine (RBM), is the most developed because of its satisfactory trade-off between computability on classical computers and computational power. Though the diffusion of RBMs is quite limited as their training remains hard. Recently, a renewed interest has emerged as Adiabatic Quantum Computers (AQCs), which suggest a potential increase of the training speed with respect to conventional hardware. Due to the limited number of connections among the qubits forming the graph of existing hardware, associating one qubit per node of the neural network implies an incomplete graph. Thanks to embedding techniques, we developed a complete graph connecting nodes constituted by virtual qubits. The complete graph outperforms previous implementations based on incomplete graphs. Despite the fact that the learning rate per epoch is still slower with respect to a classical machine, the advantage is expected by the increase of number of nodes which impacts on the classical computational time but not on the quantum hardware based computation.
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.
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.
Abstract Restricted Boltzmann machines (RBMs) constitute a class of neural networks for unsupervised learning with applications ranging from pattern classification to quantum state reconstruction. Despite the potential representative power, the diffusion of RBMs is quite limited since their training process proves to be hard. The advent of commercial adiabatic quantum computers (AQCs) raised the expectation that the implementations of RBMs on such quantum devices can increase the training speed with respect to conventional hardware. Here, the feasibility of a complete RBM on AQCs is demonstrated, thanks to an embedding that associates the nodes of the neural networks to virtual qubits. A semantic quantum search is implemented thanks to a reverse annealing schedule. Such an approach exploits more information from the training data, mimicking the behavior of the classical Gibbs sampling algorithm. The semantic training is shown to quickly raise the sampling probability of a subset of the set of the configurations. Even without a proper optimization of the annealing schedule, the RBM semantically trained achieves good scores on reconstruction tasks. The development of such techniques paves the way toward the establishment of a quantum advantage of adiabatic quantum computers, especially given the foreseen improvement of such hardware.
Quantum-assisted sampling is a promising technique to enable training probabilistic ML models, which otherwise depend on slow-mixing classical sampling methods; such as, the use of Quantum Annealing Processors (QAP) to train Boltzmann Machines (BMs). Previous work has shown that QAPs can sample from a Boltzmann distribution, although, at an unknown instance-dependent temperature. Due to this distribution divergence, existing training algorithms have resorted to negative-phase temperature scaling. This method, although effective under arduous tuning, introduces unwanted noise to the sampleset due to the quantization errors caused by the underutilization of the QAP bias ranges; and is prone to bias overflow. We introduce a change in the training algorithm to allow positive-phase temperature scaling; an approach that reduces the impact of quantization noise, while still incorporating temperature scaling. As a result, we see an overall improvement in the convergence rate and testing accuracy, when compared to the state-of-the-art approach.