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

Weighted Contrastive Divergence

Learning algorithms for energy based Boltzmann architectures that rely on gradient descent are in general computationally prohibitive, typically due to the exponential number of terms involved in computing the partition function. In this way one has to resort to approximation schemes for the evaluation of the gradient. This is the case of Restricted Boltzmann Machines (RBM) and its learning algorithm Contrastive Divergence (CD). It is well-known that CD has a number of shortcomings, and its approximation to the gradient has several drawbacks. Overcoming these defects has been the basis of much research and new algorithms have been devised, such as persistent CD. In this manuscript we propose a new algorithm that we call Weighted CD (WCD), built from small modifications of the negative phase in standard CD. However small these modifications may be, experimental work reported in this paper suggest that WCD provides a significant improvement over standard CD and persistent CD at a small additional computational cost

bbl: Boltzmann Bayes Learner for High-Dimensional Inference with Discrete Predictors in R

Non-regression-based inferences, such as discriminant analysis, can account for the effect of predictor distributions that may be significant in big data modeling. We describe bbl, an R package for Boltzmann Bayes learning, which enables a comprehensive supervised learning of the association between a large number of categorical predictors and multi-level response variables. Its basic underlying statistical model is a collection of (fully visible) Boltzmann machines inferred for each distinct response level. The algorithm reduces to the naive Bayes learner when interaction is ignored. We illustrate example use cases for various scenarios, ranging from modeling of a relatively small set of factors with heterogeneous levels to those with hundreds or more predictors with uniform levels such as image or genomic data. We show how bbl explicitly quantifies the extra power provided by interactions via higher predictive performance of the model. In comparison to deep learning-based methods such as restricted Boltzmann machines, bbl-trained models can be interpreted directly via their bias and interaction parameters

Foundations and Advances in Deep Learning

Deep neural networks have become increasingly popular under the name of deep learning recently due to their success in challenging machine learning tasks. Although the popularity is mainly due to recent successes, the history of neural networks goes as far back as 1958 when Rosenblatt presented a perceptron learning algorithm. Since then, various kinds of artificial neural networks have been proposed. They include Hopfield networks, self-organizing maps, neural principal component analysis, Boltzmann machines, multi-layer perceptrons, radial-basis function networks, autoencoders, sigmoid belief networks, support vector machines and deep belief networks. The first part of this thesis investigates shallow and deep neural networks in search of principles that explain why deep neural networks work so well across a range of applications. The thesis starts from some of the earlier ideas and models in the field of artificial neural networks and arrive at autoencoders and Boltzmann machines which are two most widely studied neural networks these days. The author thoroughly discusses how those various neural networks are related to each other and how the principles behind those networks form a foundation for autoencoders and Boltzmann machines. The second part is the collection of the ten recent publications by the author. These publications mainly focus on learning and inference algorithms of Boltzmann machines and autoencoders. Especially, Boltzmann machines, which are known to be difficult to train, have been in the main focus. Throughout several publications the author and the co-authors have devised and proposed a new set of learning algorithms which includes the enhanced gradient, adaptive learning rate and parallel tempering. These algorithms are further applied to a restricted Boltzmann machine with Gaussian visible units. In addition to these algorithms for restricted Boltzmann machines the author proposed a two-stage pretraining algorithm that initializes the parameters of a deep Boltzmann machine to match the variational posterior distribution of a similarly structured deep autoencoder. Finally, deep neural networks are applied to image denoising and speech recognition

Learning Graphical Models Using Multiplicative Weights

We give a simple, multiplicative-weight update algorithm for learning undirected graphical models or Markov random fields (MRFs). The approach is new, and for the well-studied case of Ising models or Boltzmann machines, we obtain an algorithm that uses a nearly optimal number of samples and has quadratic running time (up to logarithmic factors), subsuming and improving on all prior work. Additionally, we give the first efficient algorithm for learning Ising models over general alphabets. Our main application is an algorithm for learning the structure of t-wise MRFs with nearly-optimal sample complexity (up to polynomial losses in necessary terms that depend on the weights) and running time that is $n^{O(t)}$. In addition, given $n^{O(t)}$ samples, we can also learn the parameters of the model and generate a hypothesis that is close in statistical distance to the true MRF. All prior work runs in time $n^{\Omega(d)}$ for graphs of bounded degree d and does not generate a hypothesis close in statistical distance even for t=3. We observe that our runtime has the correct dependence on n and t assuming the hardness of learning sparse parities with noise. Our algorithm--the Sparsitron-- is easy to implement (has only one parameter) and holds in the on-line setting. Its analysis applies a regret bound from Freund and Schapire's classic Hedge algorithm. It also gives the first solution to the problem of learning sparse Generalized Linear Models (GLMs)

Boltzmann Machine Learning with the Latent Maximum Entropy Principle

We present a new statistical learning paradigm for Boltzmann machines based on a new inference principle we have proposed: the latent maximum entropy principle (LME). LME is different both from Jaynes maximum entropy principle and from standard maximum likelihood estimation. We demonstrate the LME principle BY deriving new algorithms for Boltzmann machine parameter estimation, and show how robust and fast new variant of the EM algorithm can be developed. Our experiments show that estimation based on LME generally yields better results than maximum likelihood estimation, particularly when inferring hidden units from small amounts of data