We present a new approximate learning algorithm for Boltzmann Machines, using a systematic expansion of the Gibbs free energy to second order in the weights. The linear response correction to the correlations is given by the Hessian of the Gibbs free energy. The computational complexity of the algorithm is cubic in the number of neurons. We compare the performance of the exact BM learning algorithm with first order (Weiss) mean field theory and second order (TAP) mean field theory. The learning task consists of a fully connected Ising spin glass model on 10 neurons.

Boltzmann machine is a powerful tool for modeling probability distributions that govern the training data. A thermal equilibrium state is typically used for Boltzmann machine learning to obtain a suitable probability distribution. The Boltzmann machine learning consists of calculating the gradient of the loss function given in terms of the thermal average, which is the most time consuming procedure. Here, we propose a method to implement the Boltzmann machine learning by using Noisy Intermediate-Scale Quantum (NISQ) devices. We prepare an initial pure state that contains all possible computational basis states with the same amplitude, and apply a variational imaginary time simulation. Readout of the state after the evolution in the computational basis approximates the probability distribution of the thermal equilibrium state that is used for the Boltzmann machine learning. We actually perform the numerical simulations of our scheme and confirm that the Boltzmann machine learning works well by our scheme.

We expand the item response theory to study the case of "cheating students" for a set of exams, trying to detect them by applying a greedy algorithm of inference. This extended model is closely related to the Boltzmann machine learning. In this paper we aim to infer the correct biases and interactions of our model by considering a relatively small number of sets of training data. Nevertheless, the greedy algorithm that we employed in the present study exhibits good performance with a few number of training data. The key point is the sparseness of the interactions in our problem in the context of the Boltzmann machine learning: the existence of cheating students is expected to be very rare (possibly even in real world). We compare a standard approach to infer the sparse interactions in the Boltzmann machine learning to our greedy algorithm and we find the latter to be superior in several aspects.

We present a new approximate learning algorithm for Boltzmann Machines, using a systematic expansion of the Gibbs free energy to second order in the weights. The linear response correction to the correlations is given by the Hessian of the Gibbs free energy. The computational complexity of the algorithm is cubic in the number of neurons. We compare the performance of the exact BM learning algorithm with first order (Weiss) mean field theory and second order (TAP) mean field theory. The learning task consists of a fully connected Ising spin glass model on 10 neurons. We conclude that 1) the method works well for paramagnetic problems 2) the TAP correction gives a significant improvement over the Weiss mean field theory, both for paramagnetic and spin glass problems and 3) that the inclusion of diagonal weights improves the Weiss approximation for paramagnetic problems, but not for spin glass problems.

We present a new approximate learning algorithm for Boltzmann Machines, using a systematic expansion of the Gibbs free energy to second order in the weights. The linear response correction to the correlations is given by the Hessian of the Gibbs free energy. The computational complexity of the algorithm is cubic in the number of neurons. We compare the performance of the exact BM learning algorithm with first order (Weiss) mean field theory and second order (TAP) mean field theory. The learning task consists of a fully connected Ising spin glass model on 10 neurons. We conclude that 1) the method works well for paramagnetic problems 2) the TAP correction gives a significant improvement over the Weiss mean field theory, both for paramagnetic and spin glass problems and 3) that the inclusion of diagonal weights improves the Weiss approximation for paramagnetic problems, but not for spin glass problems.

We present a new approximate learning algorithm for Boltzmann Machines, using a systematic expansion of the Gibbs free energy to second order in the weights. The linear response correction to the correlations is given by the Hessian of the Gibbs free energy. The computational complexity of the algorithm is cubic in the number of neurons. We compare the performance of the exact BM learning algorithm with first order (Weiss) mean field theory and second order (TAP) mean field theory. The learning task consists of a fully connected Ising spin glass model on 10 neurons. We conclude that 1) the method works well for paramagnetic problems 2) the TAP correction gives a significant improvement over the Weiss mean field theory, both for paramagnetic and spin glass problems and 3) that the inclusion of diagonal weights improves the Weiss approximation for paramagnetic problems, but not for spin glass problems.