RBM training on a D-Wave annealer¶

Implements RBM training on a D-Wave annealer, contrastive gradient ascent whose model expectations are sampled from a quantum annealer instead of MCMC.

A restricted Boltzmann machine with visible units \(v\), hidden units \(h\), weights \(W\), and biases \(b,c\) has energy \(E(v,h) = -b^\top v - c^\top h - v^\top W h\). Maximizing the log-likelihood of the data yields a gradient that is the difference between a data-dependent expectation and a model-dependent expectation, the latter requiring samples from the model's Boltzmann distribution.

The method replaces the expensive Markov-chain Monte Carlo estimate of the model term with samples drawn directly from a D-Wave quantum annealer, whose Ising hardware realizes the RBM Boltzmann distribution at an effective inverse temperature controlled by a scaling hyperparameter \(S\). Parameters are then updated by gradient ascent:

\[ \begin{aligned} \frac{\partial \log p(v)}{\partial W_{ij}} &= \langle v_i h_j \rangle_{\mathrm{data}} - \langle v_i h_j \rangle_{\mathrm{model}} \\ \frac{\partial \log p(v)}{\partial b_i} &= \langle v_i \rangle_{\mathrm{data}} - \langle v_i \rangle_{\mathrm{model}} \\ \frac{\partial \log p(v)}{\partial c_j} &= \langle h_j \rangle_{\mathrm{data}} - \langle h_j \rangle_{\mathrm{model}} \\ \theta_{t+1} &= \theta_t + \alpha\, \frac{\partial \log p(v)}{\partial \theta} \end{aligned} \]

where \(\theta \in \{W, b, c\}\) are the RBM parameters, \(\alpha\) is the learning rate, \(\langle \cdot \rangle_{\mathrm{data}}\) is the expectation over the training data with hidden units inferred, and \(\langle \cdot \rangle_{\mathrm{model}}\) is the expectation over the model distribution estimated from D-Wave annealer samples taken at scale \(S\).

Reference: Vivek Dixit, Raja Selvarajan, Muhammad A. Alam, Travis S. Humble, Sabre Kais, "Training Restricted Boltzmann Machines With a D-Wave Quantum Annealer", Frontiers in Physics 2021. https://www.frontiersin.org/journals/physics/articles/10.3389/fphy.2021.589626/full

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