Nonequilibrium thermodynamics of restricted Boltzmann machines

In this work, we analyze the nonequilibrium thermodynamics of a class of neural networks known as restricted Boltzmann machines (RBMs) in the context of unsupervised learning. We show how the network is described as a discrete Markov process and how the detailed balance condition and the Maxwell-Boltzmann equilibrium distribution are sufficient conditions for a complete thermodynamics description, including nonequilibrium fluctuation theorems. Numerical simulations in a fully trained RBM are performed and the heat exchange fluctuation theorem is verified with excellent agreement to the theory. We observe how the contrastive divergence functional, mostly used in unsupervised learning of RBMs, is closely related to nonequilibrium thermodynamic quantities. We also use the framework to interpret the estimation of the partition function of RBMs with the annealed importance sampling method from a thermodynamics standpoint. Finally, we argue that unsupervised learning of RBMs is equivalent to a work protocol in a system driven by the laws of thermodynamics in the absence of labeled data.

Figure 1

The figure shows a RBM with $m=4$ units in the visible layer (blue), $v$, and $n=3$ units in the hidden layer (yellow), $h$. Notice that neurons from the same layer (same color) are not connected. The example shows the RBM performing a single step of the Markov chain, initially at state, $s=(v,h)$, generated with weights $\lambda$, transitioning into another state, $s^{\prime }=(v^{\prime },h^{\prime })$, generated with weights ${\lambda }^{\prime }$.

Figure 2

The figure shows the XFT for a RBM trained over a set of handwritten digits (MNIST), with $m=784$ units in the visible layer $\left(v\right)$ and $n=500$ units in the hidden layer $\left(h\right)$. Theoretical predictions are the solid lines. The initial temperature is $T_1=1$ and the final temperatures are $T_2=1.2$ $(\circ )$, $T_2=1.1$ $(\square )$ for heating (in blue) and $T_2=0.9$ $\left(▵\right)$, $T_2=0.8$ $(▿)$ for cooling examples (in red). The energy difference pdf, $P\left(\mathrm{\Delta }E\right)$, is obtained in the nonequilibrium situation of a single update $\left(K=1\right)$ of the discrete Markov chain with $T_2$, from an ensemble of $2\times {10}^7$ RBMs initially prepared at thermal equilibrium with $T_1$.