Asja Fischer

An Introduction to Restricted Boltzmann Machines

Restricted Boltzmann machines (RBMs) are probabilistic graphical models that can be interpreted as stochastic neural networks. The increase in computational power and the development of faster learning algorithms have made them applicable to relevant machine learning problems. They attracted much attention recently after being proposed as building blocks of multi-layer learning systems called deep belief networks. This tutorial introduces RBMs as undirected graphical models. The basic concepts of graphical models are introduced first, however, basic knowledge in statistics is presumed. Different learning algorithms for RBMs are discussed. As most of them are based on Markov chain Monte Carlo (MCMC) methods, an introduction to Markov chains and the required MCMC techniques is provided.

Empirical analysis of the divergence of Gibbs sampling based learning algorithms for restricted Boltzmann machines

Learning algorithms relying on Gibbs sampling based stochastic approximations of the log-likelihood gradient have become a common way to train Restricted Boltzmann Machines (RBMs). We study three of these methods, Contrastive Divergence (CD) and its refined variants Persistent CD (PCD) and Fast PCD (FPCD). As the approximations are biased, the maximum of the log-likelihood is not necessarily obtained. Recently, it has been shown that CD, PCD, and FPCD can even lead to a steady decrease of the log-likelihood during learning. Taking artificial data sets from the literature we study these divergence effects in more detail. Our results indicate that the log-likelihood seems to diverge especially if the target distribution is difficult to learn for the RBM. The decrease of the likelihood can not be detected by an increase of the reconstruction error, which has been proposed as a stopping criterion for CD learning. Weight-decay with a carefully chosen weight-decay-parameter can prevent divergence.

Difference target propagation

Back-propagation has been the workhorse of recent successes of deep learning but it relies on infinitesimal effects (partial derivatives) in order to perform credit assignment. This could become a serious issue as one considers deeper and more non-linear functions, e.g., consider the extreme case of non-linearity where the relation between parameters and cost is actually discrete. Inspired by the biological implausibility of back-propagation, a few approaches have been proposed in the past that could play a similar credit assignment role. In this spirit, we explore a novel approach to credit assignment in deep networks that we call target propagation. The main idea is to compute targets rather than gradients, at each layer. Like gradients, they are propagated backwards. In a way that is related but different from previously proposed proxies for back-propagation which rely on a backwards network with symmetric weights, target propagation relies on auto-encoders at each layer. Unlike back-propagation, it can be applied even when units exchange stochastic bits rather than real numbers. We show that a linear correction for the imperfectness of the auto-encoders, called difference target propagation, is very effective to make target propagation actually work, leading to results comparable to back-propagation for deep networks with discrete and continuous units and denoising auto-encoders and achieving state of the art for stochastic networks.

Neural Network-based Question Answering over Knowledge Graphs on Word and Character Level

Question Answering (QA) systems over Knowledge Graphs (KG) automatically answer natural language questions using facts contained in a knowledge graph. Simple questions, which can be answered by the extraction of a single fact, constitute a large part of questions asked on the web but still pose challenges to QA systems, especially when asked against a large knowledge resource. Existing QA systems usually rely on various components each specialised in solving different sub-tasks of the problem (such as segmentation, entity recognition, disambiguation, and relation classification etc.). In this work, we follow a quite different approach: We train a neural network for answering simple questions in an end-to-end manner, leaving all decisions to the model. It learns to rank subject-predicate pairs to enable the retrieval of relevant facts given a question. The network contains a nested word/character-level question encoder which allows to handle out-of-vocabulary and rare word problems while still being able to exploit word-level semantics. Our approach achieves results competitive with state-of-the-art end-to-end approaches that rely on an attention mechanism.

Bounding the bias of contrastive divergence learning

Optimization based on k-step contrastive divergence (CD) has become a common way to train restricted Boltzmann machines (RBMs). The k-step CD is a biased estimator of the log-likelihood gradient relying on Gibbs sampling. We derive a new upper bound for this bias. Its magnitude depends on k, the number of variables in the RBM, and the maximum change in energy that can be produced by changing a single variable. The last reflects the dependence on the absolute values of the RBM parameters. The magnitude of the bias is also affected by the distance in variation between the modeled distribution and the starting distribution of the Gibbs chain.

The flip-the-state transition operator for restricted Boltzmann machines

Most learning and sampling algorithms for restricted Boltzmann machines (RMBs) rely on Markov chain Monte Carlo (MCMC) methods using Gibbs sampling. The most prominent examples are Contrastive Divergence learning (CD) and its variants as well as Parallel Tempering (PT). The performance of these methods strongly depends on the mixing properties of the Gibbs chain. We propose a Metropolis-type MCMC algorithm relying on a transition operator maximizing the probability of state changes. It is shown that the operator induces an irreducible, aperiodic, and hence properly converging Markov chain, also for the typically used periodic update schemes. The transition operator can replace Gibbs sampling in RBM learning algorithms without producing computational overhead. It is shown empirically that this leads to faster mixing and in turn to more accurate learning.

Stdp-compatible approximation of backpropagation in an energy-based model

We show that Langevin Markov chain Monte Carlo inference in an energy-based model with latent variables has the property that the early steps of inference, starting from a stationary point, correspond to propagating error gradients into internal layers, similar to backpropagation. The backpropagated error is with respect to output units that have received an outside driving force pushing them away from the stationary point. Backpropagated error gradients correspond to temporal derivatives with respect to the activation of hidden units. These lead to a weight update proportional to the product of the presynaptic firing rate and the temporal rate of change of the postsynaptic firing rate. Simulations and a theoretical argument suggest that this rate-based update rule is consistent with those associated with spike-timing-dependent plasticity. The ideas presented in this article could be an element of a theory for explaining how brains perform credit assignment in deep hierarchies as efficiently as backpropagation does, with neural computation corresponding to both approximate inference in continuous-valued latent variables and error backpropagation, at the same time.

Training Restricted Boltzmann Machines

Restricted Boltzmann Machines (RBMs), two-layered probabilistic graphical models that can also be interpreted as feed forward neural networks, enjoy much popularity for pattern analysis and generation. Training RBMs however is challenging. It is based on likelihood maximization, but the likelihood and its gradient are computationally intractable. Therefore, training algorithms such as Contrastive Divergence (CD) and learning based on Parallel Tempering (PT) rely on Markov chain Monte Carlo methods to approximate the gradient. The presented thesis contributes to understanding RBM training methods by presenting an empirical and theoretical analysis of the bias of the CD approximation and a bound on the mixing rate of PT. Furthermore, the thesis improves RBM training by proposing a new transition operator leading to faster mixing Markov chains, by investigating a different parameterization of the RBM model class referred to as centered RBMs, and by exploring estimation techniques from statistical physics to approximate the likelihood. Finally, an analysis of the representational power of deep belief networks with real-valued visible variables is given.

A bound for the convergence rate of parallel tempering for sampling restricted Boltzmann machines

Sampling from restricted Boltzmann machines (RBMs) is done by Markov chain Monte Carlo (MCMC) methods. The faster the convergence of the Markov chain, the more efficiently can high quality samples be obtained. This is also important for robust training of RBMs, which usually relies on sampling. Parallel tempering (PT), an MCMC method that maintains several replicas of the original chain at higher temperatures, has been successfully applied for RBM training. We present the first analysis of the convergence rate of PT for sampling from binary RBMs. The resulting bound on the rate of convergence of the PT Markov chain shows an exponential dependency on the size of one layer and the absolute values of the RBM parameters. It is minimized by a uniform spacing of the inverse temperatures, which is often used in practice. Similarly as in the derivation of bounds on the approximation error for contrastive divergence learning, our bound on the mixing time implies an upper bound on the error of the gradient approximation when the method is used for RBM training.

Graph-based Predictable Feature Analysis

We propose graph-based predictable feature analysis (GPFA), a new method for unsupervised learning of predictable features from high-dimensional time series, where high predictability is understood very generically as low variance in the distribution of the next data point given the previous ones. We show how this measure of predictability can be understood in terms of graph embedding as well as how it relates to the information-theoretic measure of predictive information in special cases. We confirm the effectiveness of GPFA on different datasets, comparing it to three existing algorithms with similar objectives—namely slow feature analysis, forecastable component analysis, and predictable feature analysis—to which GPFA shows very competitive results.