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IEEE International Conference on Quantum Computing and Engineering (QCE)

Structural Modifications in Quantum-Assisted Training for General Boltzmann Machines

Jose P. Pinilla, Steven J. E. Wilton

Published Sep 15, 2024• 0 citations
Publisher

Abstract

Quantum-assisted training of machine learning models makes use of the sampling capabilities of quantum computing devices to approximate the probability distribution of classical generative machine learning models. One example of this is the use of quantum annealers to sample from a Boltzmann distribution to assist the computation of gradients required to train Boltzmann machines (BMs). Doing so requires the interactions and weights of a BM to be mapped onto the underlying qubit connectivity built into the quantum device hardware. This opens the possibility of formulating any graph topology as a BM, instead of the more common classical approach of restricting their connectivity; i.e., Restricted Boltz-mann Machine (RBM). This work is focused on simultaneous modifications of the connectivity and qubit mapping of general BMs, which we refer to as structure-aware minor-embedding. We show how minor-embedding can be performed taking into account the nature of the BM. We also evaluate weight “pruning”, which is motivated by the intuition that, when a QAML model is pruned, the resulting qubit chains resulting from the minor-embedding methods can provide better quality samples, at the expense of a smaller model.

DOI
10.1109/qce60285.2024.00196

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Quantum-assisted training of machine learning models makes use of the sampling capabilities of quantum computing devices to approximate the probability distribution of classical generative machine learning models. One example of this is the use of quantum annealers to sample from a Boltzmann distribution to assist the computation of gradients required to train Boltzmann machines (BMs). Doing so requires the interactions and weights of a BM to be mapped onto the underlying qubit connectivity built into the quantum device hardware. This opens the possibility of formulating any graph topology as a BM, instead of the more common classical approach of restricting their connectivity; i.e., Restricted Boltz-mann Machine (RBM). This work is focused on simultaneous modifications of the connectivity and qubit mapping of general BMs, which we refer to as structure-aware minor-embedding. We show how minor-embedding can be performed taking into account the nature of the BM. We also evaluate weight “pruning”, which is motivated by the intuition that, when a QAML model is pruned, the resulting qubit chains resulting from the minor-embedding methods can provide better quality samples, at the expense of a smaller model.
This paper describes the components and configurations available in a new quantum-assisted machine learning (QAML) framework. QAML is an open source package that provides a new and flexible test-bed for algorithms to train and evaluate Boltzmann machines (BMs) using quantum annealers. Quantum annealing processors enable the training capabilities for both restricted (RBM) and general Boltzmann machines (BM). These methods rely on the fidelity of samples from those devices, which approximate the characteristic Boltzmann distribution of the BM models. The models and optimization functions are built on top of the PyTorch and D-Wave Ocean libraries. The goal of this paper is to introduce developers and researchers to a familiar, yet flexible, software development framework, to explore the use of quantum computing in machine learning applications. The framework is open-source and has been made publicly available.
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.
Restricted Boltzmann machines are common machine learning models that can utilize quantum annealing devices in their training processes as quantum samplers. While this approach has shown promise as an alternative to classical sampling methods, the limitations of quantum annealing hardware, such as the number of qubits and the lack of connectivity between the qubits, still pose a barrier to wide-scale adoption. We propose the use of multiple software techniques such as dropout method, passive labeling, and parallelization techniques for addressing these hardware limitations. The study found that using these techniques along with quantum sampling showed comparable results to its classical counterparts in certain contexts, while in others the increased complexity of the sampling process hindered the performance of the trained models. This means that further research into the behavior of quantum sampling needs to be done to apply quantum annealing to training tasks of more complicated RBM models.
Energy-based generative models, such as restricted Boltzmann machines (RBMs), require unbiased Boltzmann samples for effective training. Classical Markov chain Monte Carlo methods, however, converge slowly and yield correlated samples, making large-scale training difficult. We address this bottleneck by applying the analytic relation between annealing schedules and effective inverse temperature in diabatic quantum annealing. By implementing this prescription on a quantum annealer, we obtain temperature-controlled Boltzmann samples that enable RBM training with faster convergence and lower validation error than classical sampling. We further identify a systematic temperature misalignment intrinsic to analog quantum computers and propose an analytical rescaling method that mitigates this hardware noise, thereby enhancing the practicality of quantum annealers as Boltzmann samplers. In our method, the model's connectivity is set directly by the qubit connectivity, transforming the computational complexity inherent in classical sampling into a requirement on quantum hardware. This shift allows the approach to extend naturally from RBMs to fully connected Boltzmann machines, opening opportunities inaccessible to classical training methods.
Boltzmann machines have useful roles in deep learning applications, such as generative data modeling, initializing weights for other types of networks, or extracting efficient representations from high-dimensional data. Most Boltzmann machines use restricted topologies that exclude looping connectivity, as such connectivity creates complex distributions that are difficult to sample. We have used an open-system quantum annealer to sample from complex distributions and implement Boltzmann machines with looping connectivity. Further, we have created policies mapping Boltzmann machine variables to the quantum bits of an annealer. These policies, based on correlation and entropy metrics, dynamically reconfigure the topology of Boltzmann machines during training and improve performance.
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.