Adaptive hyperparameter updating for training restricted Boltzmann machines on quantum annealers

Abstract

Restricted Boltzmann Machines (RBMs) have been proposed for developing neural networks for a variety of unsupervised machine learning applications such as image recognition, drug discovery, and materials design. The Boltzmann probability distribution is used as a model to identify network parameters by optimizing the likelihood of predicting an output given hidden states trained on available data. Training such networks often requires sampling over a large probability space that must be approximated during gradient based optimization. Quantum annealing has been proposed as a means to search this space more efficiently which has been experimentally investigated on D-Wave hardware. D-Wave implementation requires selection of an effective inverse temperature or hyperparameter ($$\beta $$ β ) within the Boltzmann distribution which can strongly influence optimization. Here, we show how this parameter can be estimated as a hyperparameter applied to D-Wave hardware during neural network training by maximizing the likelihood or minimizing the Shannon entropy. We find both methods improve training RBMs based upon D-Wave hardware experimental validation on an image recognition problem. Neural network image reconstruction errors are evaluated using Bayesian uncertainty analysis which illustrate more than an order magnitude lower image reconstruction error using the maximum likelihood over manually optimizing the hyperparameter. The maximum likelihood method is also shown to out-perform minimizing the Shannon entropy for image reconstruction.

Quantum Boltzmann Machines: Applications in Quantitative Finance

“…There is the possibility to treat β as a learnable parameter rather than having to choose a value empirically, as detailed by Xu and Oates in [22]. The method is based on a log-likelihood maximization approach leading to parameter updates of the form (see Appendix C.4 for how one arrives at this result)…”

“…In recent years, a number of researchers have studied using D-Wave quantum annealers to train Boltzmann machines [22,[26][27][28][29][30][31][32][33]. The most common approach is to train a classical RBM with quantum assistance, i.e., using the annealer to generate the samples in the negative phase rather than using Gibbs sampling.…”

“…This derivation follows along the lines of that laid out in [22]. Suppose the D-Wave annealer samples according to a classical Boltzmann distribution p DW of energies E DW = βE, i.e., which we can use to treat the effective inverse temperature as a learnable parameter.…”

Efficient low temperature Monte Carlo sampling using quantum annealing

“…degenerate states is the introduction of parallel tempering with isoenergetic cluster updates in Monte Carlo methods 16 or the combination with simulated annealing on a quantum annealer 17,18 . We mention that Boltzmann machines, which can serve as a link between machine learning and statistical thermodynamics 19 , are investigated in the context of QA [20][21][22][23] from a computer science perspective, but to the best of our knowledge, the direct application of QA for classical finite temperature modeling for statistical physics and materials science has not yet been accomplished and is the subject of the present paper.…”

Efficient low temperature Monte Carlo sampling using quantum annealing

“…Another possibility for the fair sampling of ground and degenerate states is the introduction of parallel tempering with isoenergetic cluster updates in Monte Carlo methods 29 or the combination with simulated annealing on a quantum annealer 30 , 31 . We mention that Boltzmann machines, which can serve as a link between machine learning and statistical thermodynamics 32 , are investigated in the context of QA 33 – 36 from a computer science perspective, but to the best of our knowledge, the direct application of QA for classical finite temperature modeling for statistical physics and materials science has not yet been accomplished and is the subject of the present paper.…”