[Paper Review] Online calibration scheme for training restricted Boltzmann machines with quantum annealing

Takeru Goto, Masayuki Ohzeki|arXiv (Cornell University)|Jul 19, 2023

Machine Learning and ELMComputer Science3 citations

TL;DR

This paper proposes an online calibration scheme for training restricted Boltzmann machines (RBMs) using a D-Wave quantum annealer by simultaneously estimating internal parameters—such as inverse temperature and bias-specific coefficients—using the Contrastive Divergence (CD) algorithm. The method improves sample quality and achieves KL divergence performance on par with classical Gibbs sampling, outperforming standard CD training.

ABSTRACT

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.

Motivation & Objective

• To address the challenge of hardware-imperfection-induced inaccuracies in quantum annealer samples used for RBM training.
• To develop a real-time calibration scheme that estimates internal parameters during RBM training, avoiding post-processing or multiple model retraining.
• To improve sample fidelity beyond standard inverse temperature compensation by estimating multiple parameters, including bias-specific coefficients.
• To evaluate the performance of the proposed method against classical sampling (Gibbs) and standard CD training in terms of KL divergence.

Proposed method

• The method uses samples from the D-Wave quantum annealer as training data and applies the Contrastive Divergence (CD) algorithm to estimate internal parameters in real time.
• It formulates the estimation of effective inverse temperature (β_eff) and additional parameters as a minimization of KL divergence between empirical and model distributions.
• The scheme estimates one or more inverse temperature parameters for weights, visible biases, and hidden biases, or uses a coefficient per bias type to improve flexibility.
• All parameters are updated iteratively using the same update rule based on the CD algorithm, enabling simultaneous model training and calibration.
• The approach avoids costly post-processing or likelihood maximization, reducing computational overhead compared to prior methods.
• The method is evaluated across three parameter configurations: one β, three βs (per weight and bias types), and one β with individual bias coefficients.

Experimental results

Research questions

• RQ1 Can real-time calibration of quantum annealer parameters during RBM training improve sample quality compared to standard CD training?
• RQ2 Does estimating multiple internal parameters (e.g., separate β for weights and biases) lead to better performance than a single β estimate?
• RQ3 How does the convergence behavior of different internal parameters affect training dynamics and final model performance?
• RQ4 Can the proposed scheme achieve KL divergence comparable to classical Gibbs sampling while reducing training time?

Key findings

• The proposed method achieved a minimum KL divergence of 2.78 in the three-parameter configuration (one β per weight and bias type), outperforming the standard CD algorithm.
• The one-and-all-bias configuration (one β for weights, coefficients for all biases) achieved the best performance with a KL divergence of 2.78, indicating improved sample distribution accuracy.
• The method’s performance matched that of classical Gibbs sampling, with KL divergence values below 3.30 across all configurations.
• Training with multiple internal parameters led to continuous improvement beyond 500 epochs, unlike the one-parameter case where β_eff converged early and stagnated.
• The inverse temperature β_eff converged within 500 epochs, but other parameters like β_h continued to evolve, suggesting that more parameters require more samples for stabilization.
• The method reduced training time compared to prior approaches by eliminating post-processing and enabling simultaneous training and calibration.