An Efficient Learning Procedure for Deep Boltzmann Machines

We present a new learning algorithm for Boltzmann Machines that contain many layers of hidden variables. Data-dependent statistics are estimated using a variational approximation that tends to focus on a single mode, and data-independent statistics are estimated using persistent Markov chains. The use of two quite different techniques for estimating the two types of statistic that enter into the gradient of the log likelihood makes it practical to learn Boltzmann Machines with multiple hidden layers and millions of parameters. The learning can be made more efficient by using a layer-by-layer "pre-training" phase that initializes the weights sensibly. The pre-training also allows the variational inference to be initialized sensibly with a single bottom-up pass. We present results on the MNIST and NORB datasets showing that Deep Boltzmann Machines learn very good generative models of hand-written digits and 3-D objects. We also show that the features discovered by Deep Boltzmann Machines are a very effective way to initialize the hidden layers of feed-forward neural nets which are then discriminatively fine-tuned

Metric-Free Natural Gradient for Joint-Training of Boltzmann Machines

This paper introduces the Metric-Free Natural Gradient (MFNG) algorithm for training Boltzmann Machines. Similar in spirit to the Hessian-Free method of Martens [8], our algorithm belongs to the family of truncated Newton methods and exploits an efficient matrix-vector product to avoid explicitely storing the natural gradient metric $L$. This metric is shown to be the expected second derivative of the log-partition function (under the model distribution), or equivalently, the variance of the vector of partial derivatives of the energy function. We evaluate our method on the task of joint-training a 3-layer Deep Boltzmann Machine and show that MFNG does indeed have faster per-epoch convergence compared to Stochastic Maximum Likelihood with centering, though wall-clock performance is currently not competitive