Efficient Evaluation of the Partition Function of RBMs with Annealed Importance Sampling
This work addresses a bottleneck in probabilistic modeling for machine learning practitioners, offering an incremental improvement to existing methods.
The paper tackles the computationally expensive problem of evaluating the partition function Z in Restricted Boltzmann Machines (RBMs), which is critical for learning algorithms, by proposing an improved Annealed Importance Sampling (AIS) method that uses a better starting distribution to reduce the number of Monte Carlo steps needed. The result is a good estimation of Z with little computational cost, as demonstrated in both small- and large-sized problems.
Probabilistic models based on Restricted Boltzmann Machines (RBMs) imply the evaluation of normalized Boltzmann factors, which in turn require from the evaluation of the partition function Z. The exact evaluation of Z, though, becomes a forbiddingly expensive task as the system size increases. This even worsens when one considers most usual learning algorithms for RBMs, where the exact evaluation of the gradient of the log-likelihood of the empirical distribution of the data includes the computation of Z at each iteration. The Annealed Importance Sampling (AIS) method provides a tool to stochastically estimate the partition function of the system. So far, the standard use of the AIS algorithm in the Machine Learning context has been done using a large number of Monte Carlo steps. In this work we show that this may not be required if a proper starting probability distribution is employed as the initialization of the AIS algorithm. We analyze the performance of AIS in both small- and large-sized problems, and show that in both cases a good estimation of Z can be obtained with little computational cost.