Layerwise Systematic Scan: Deep Boltzmann Machines and Beyond

For Markov chain Monte Carlo methods, one of the greatest discrepancies between theory and system is the scan order - while most theoretical development on the mixing time analysis deals with random updates, real-world systems are implemented with systematic scans. We bridge this gap for models that exhibit a bipartite structure, including, most notably, the Restricted/Deep Boltzmann Machine. The de facto implementation for these models scans variables in a layerwise fashion. We show that the Gibbs sampler with a layerwise alternating scan order has its relaxation time (in terms of epochs) no larger than that of a random-update Gibbs sampler (in terms of variable updates). We also construct examples to show that this bound is asymptotically tight. Through standard inequalities, our result also implies a comparison on the mixing times.Comment: v2: typo fixes and improved presentatio

Improving sampling, optimization and feature extraction in Boltzmann machines

L’apprentissage supervisé de réseaux hiérarchiques à grande échelle connaît présentement un succès fulgurant. Malgré cette effervescence, l’apprentissage non-supervisé représente toujours, selon plusieurs chercheurs, un élément clé de l’Intelligence Artificielle, où les agents doivent apprendre à partir d’un nombre potentiellement limité de données. Cette thèse s’inscrit dans cette pensée et aborde divers sujets de recherche liés au problème d’estimation de densité par l’entremise des machines de Boltzmann (BM), modèles graphiques probabilistes au coeur de l’apprentissage profond. Nos contributions touchent les domaines de l’échantillonnage, l’estimation de fonctions de partition, l’optimisation ainsi que l’apprentissage de représentations invariantes. Cette thèse débute par l’exposition d’un nouvel algorithme d'échantillonnage adaptatif, qui ajuste (de fa ̧con automatique) la température des chaînes de Markov sous simulation, afin de maintenir une vitesse de convergence élevée tout au long de l’apprentissage. Lorsqu’utilisé dans le contexte de l’apprentissage par maximum de vraisemblance stochastique (SML), notre algorithme engendre une robustesse accrue face à la sélection du taux d’apprentissage, ainsi qu’une meilleure vitesse de convergence. Nos résultats sont présent ́es dans le domaine des BMs, mais la méthode est générale et applicable à l’apprentissage de tout modèle probabiliste exploitant l’échantillonnage par chaînes de Markov. Tandis que le gradient du maximum de vraisemblance peut-être approximé par échantillonnage, l’évaluation de la log-vraisemblance nécessite un estimé de la fonction de partition. Contrairement aux approches traditionnelles qui considèrent un modèle donné comme une boîte noire, nous proposons plutôt d’exploiter la dynamique de l’apprentissage en estimant les changements successifs de log-partition encourus à chaque mise à jour des paramètres. Le problème d’estimation est reformulé comme un problème d’inférence similaire au filtre de Kalman, mais sur un graphe bi-dimensionnel, où les dimensions correspondent aux axes du temps et au paramètre de température. Sur le thème de l’optimisation, nous présentons également un algorithme permettant d’appliquer, de manière efficace, le gradient naturel à des machines de Boltzmann comportant des milliers d’unités. Jusqu’à présent, son adoption était limitée par son haut coût computationel ainsi que sa demande en mémoire. Notre algorithme, Metric-Free Natural Gradient (MFNG), permet d’éviter le calcul explicite de la matrice d’information de Fisher (et son inverse) en exploitant un solveur linéaire combiné à un produit matrice-vecteur efficace. L’algorithme est prometteur: en terme du nombre d’évaluations de fonctions, MFNG converge plus rapidement que SML. Son implémentation demeure malheureusement inefficace en temps de calcul. Ces travaux explorent également les mécanismes sous-jacents à l’apprentissage de représentations invariantes. À cette fin, nous utilisons la famille de machines de Boltzmann restreintes “spike & slab” (ssRBM), que nous modifions afin de pouvoir modéliser des distributions binaires et parcimonieuses. Les variables latentes binaires de la ssRBM peuvent être rendues invariantes à un sous-espace vectoriel, en associant à chacune d’elles, un vecteur de variables latentes continues (dénommées “slabs”). Ceci se traduit par une invariance accrue au niveau de la représentation et un meilleur taux de classification lorsque peu de données étiquetées sont disponibles. Nous terminons cette thèse sur un sujet ambitieux: l’apprentissage de représentations pouvant séparer les facteurs de variations présents dans le signal d’entrée. Nous proposons une solution à base de ssRBM bilinéaire (avec deux groupes de facteurs latents) et formulons le problème comme l’un de “pooling” dans des sous-espaces vectoriels complémentaires.Despite the current widescale success of deep learning in training large scale hierarchical models through supervised learning, unsupervised learning promises to play a crucial role towards solving general Artificial Intelligence, where agents are expected to learn with little to no supervision. The work presented in this thesis tackles the problem of unsupervised feature learning and density estimation, using a model family at the heart of the deep learning phenomenon: the Boltzmann Machine (BM). We present contributions in the areas of sampling, partition function estimation, optimization and the more general topic of invariant feature learning. With regards to sampling, we present a novel adaptive parallel tempering method which dynamically adjusts the temperatures under simulation to maintain good mixing in the presence of complex multi-modal distributions. When used in the context of stochastic maximum likelihood (SML) training, the improved ergodicity of our sampler translates to increased robustness to learning rates and faster per epoch convergence. Though our application is limited to BM, our method is general and is applicable to sampling from arbitrary probabilistic models using Markov Chain Monte Carlo (MCMC) techniques. While SML gradients can be estimated via sampling, computing data likelihoods requires an estimate of the partition function. Contrary to previous approaches which consider the model as a black box, we provide an efficient algorithm which instead tracks the change in the log partition function incurred by successive parameter updates. Our algorithm frames this estimation problem as one of filtering performed over a 2D lattice, with one dimension representing time and the other temperature. On the topic of optimization, our thesis presents a novel algorithm for applying the natural gradient to large scale Boltzmann Machines. Up until now, its application had been constrained by the computational and memory requirements of computing the Fisher Information Matrix (FIM), which is square in the number of parameters. The Metric-Free Natural Gradient algorithm (MFNG) avoids computing the FIM altogether by combining a linear solver with an efficient matrix-vector operation. The method shows promise in that the resulting updates yield faster per-epoch convergence, despite being slower in terms of wall clock time. Finally, we explore how invariant features can be learnt through modifications to the BM energy function. We study the problem in the context of the spike & slab Restricted Boltzmann Machine (ssRBM), which we extend to handle both binary and sparse input distributions. By associating each spike with several slab variables, latent variables can be made invariant to a rich, high dimensional subspace resulting in increased invariance in the learnt representation. When using the expected model posterior as input to a classifier, increased invariance translates to improved classification accuracy in the low-label data regime. We conclude by showing a connection between invariance and the more powerful concept of disentangling factors of variation. While invariance can be achieved by pooling over subspaces, disentangling can be achieved by learning multiple complementary views of the same subspace. In particular, we show how this can be achieved using third-order BMs featuring multiplicative interactions between pairs of random variables

Training deep convolutional architectures for vision

Les tâches de vision artificielle telles que la reconnaissance d’objets demeurent irrésolues à ce jour. Les algorithmes d’apprentissage tels que les Réseaux de Neurones Artificiels (RNA), représentent une approche prometteuse permettant d’apprendre des caractéristiques utiles pour ces tâches. Ce processus d’optimisation est néanmoins difficile. Les réseaux profonds à base de Machine de Boltzmann Restreintes (RBM) ont récemment été proposés afin de guider l’extraction de représentations intermédiaires, grâce à un algorithme d’apprentissage non-supervisé. Ce mémoire présente, par l’entremise de trois articles, des contributions à ce domaine de recherche. Le premier article traite de la RBM convolutionelle. L’usage de champs réceptifs locaux ainsi que le regroupement d’unités cachées en couches partageant les même paramètres, réduit considérablement le nombre de paramètres à apprendre et engendre des détecteurs de caractéristiques locaux et équivariant aux translations. Ceci mène à des modèles ayant une meilleure vraisemblance, comparativement aux RBMs entraînées sur des segments d’images. Le deuxième article est motivé par des découvertes récentes en neurosciences. Il analyse l’impact d’unités quadratiques sur des tâches de classification visuelles, ainsi que celui d’une nouvelle fonction d’activation. Nous observons que les RNAs à base d’unités quadratiques utilisant la fonction softsign, donnent de meilleures performances de généralisation. Le dernière article quand à lui, offre une vision critique des algorithmes populaires d’entraînement de RBMs. Nous montrons que l’algorithme de Divergence Contrastive (CD) et la CD Persistente ne sont pas robustes : tous deux nécessitent une surface d’énergie relativement plate afin que leur chaîne négative puisse mixer. La PCD à "poids rapides" contourne ce problème en perturbant légèrement le modèle, cependant, ceci génère des échantillons bruités. L’usage de chaînes tempérées dans la phase négative est une façon robuste d’adresser ces problèmes et mène à de meilleurs modèles génératifs.High-level vision tasks such as generic object recognition remain out of reach for modern Artificial Intelligence systems. A promising approach involves learning algorithms, such as the Arficial Neural Network (ANN), which automatically learn to extract useful features for the task at hand. For ANNs, this represents a difficult optimization problem however. Deep Belief Networks have thus been proposed as a way to guide the discovery of intermediate representations, through a greedy unsupervised training of stacked Restricted Boltzmann Machines (RBM). The articles presented here-in represent contributions to this field of research. The first article introduces the convolutional RBM. By mimicking local receptive fields and tying the parameters of hidden units within the same feature map, we considerably reduce the number of parameters to learn and enforce local, shift-equivariant feature detectors. This translates to better likelihood scores, compared to RBMs trained on small image patches. In the second article, recent discoveries in neuroscience motivate an investigation into the impact of higher-order units on visual classification, along with the evaluation of a novel activation function. We show that ANNs with quadratic units using the softsign activation function offer better generalization error across several tasks. Finally, the third article gives a critical look at recently proposed RBM training algorithms. We show that Contrastive Divergence (CD) and Persistent CD are brittle in that they require the energy landscape to be smooth in order for their negative chain to mix well. PCD with fast-weights addresses the issue by performing small model perturbations, but may result in spurious samples. We propose using simulated tempering to draw negative samples. This leads to better generative models and increased robustness to various hyperparameters

階層型神経回路モデルにおける学習力学の幾何学的理論

学位の種別: 課程博士審査委員会委員 : （主査）東京大学教授 岡田 真人, 東京大学教授 津田 宏治, 東京大学教授 能瀬 聡直, 東京大学准教授 國廣 昇, 東京大学講師 佐藤 一誠University of Tokyo(東京大学

Visual scene recognition with biologically relevant generative models

This research focuses on developing visual object categorization methodologies that are based on machine learning techniques and biologically inspired generative models of visual scene recognition. Modelling the statistical variability in visual patterns, in the space of features extracted from them by an appropriate low level signal processing technique, is an important matter of investigation for both humans and machines. To study this problem, we have examined in detail two recent probabilistic models of vision: a simple multivariate Gaussian model as suggested by (Karklin & Lewicki, 2009) and a restricted Boltzmann machine (RBM) proposed by (Hinton, 2002). Both the models have been widely used for visual object classification and scene analysis tasks before. This research highlights that these models on their own are not plausible enough to perform the classification task, and suggests Fisher kernel as a means of inducing discrimination into these models for classification power. Our empirical results on standard benchmark data sets reveal that the classification performance of these generative models could be significantly boosted near to the state of the art performance, by drawing a Fisher kernel from compact generative models that computes the data labels in a fraction of total computation time. We compare the proposed technique with other distance based and kernel based classifiers to show how computationally efficient the Fisher kernels are. To the best of our knowledge, Fisher kernel has not been drawn from the RBM before, so the work presented in the thesis is novel in terms of its idea and application to vision problem

Structured representation learning from complex data

This thesis advances several theoretical and practical aspects of the recently introduced restricted Boltzmann machine - a powerful probabilistic and generative framework for modelling data and learning representations. The contributions of this study represent a systematic and common theme in learning structured representations from complex data

Exploring Probability Measures with Markov Processes

In many domains where mathematical modelling is applied, a deterministic description of the system at hand is insufficient, and so it is useful to model systems as being in some way stochastic. This is often achieved by modeling the state of the system as being drawn from a probability measure, which is usually given algebraically, i.e. as a formula. While this representation can be useful for deriving certain characteristics of the system, it is by now well-appreciated that many questions about stochastic systems are best-answered by looking at samples from the associated probability measure. In this thesis, we seek to develop and analyse efficient techniques for generating samples from a given probability measure, with a focus on algorithms which simulate a Markov process with the desired invariant measure. The first work presented in this thesis considers the use of Piecewise-Deterministic Markov Processes (PDMPs) for generating samples. In contrast to usual approaches, PDMPs are i) defined as continuous-time processes, and ii) are typically non-reversible with respect to their invariant measure. These distinctions pose computational and theoretical challenges for the design, analysis, and implementation of PDMP-based samplers. The key contribution of this work is to develop a transparent characterisation of how one can construct a PDMP (within the class of trajectorially-reversible processes) which admits the desired invariant measure, and to offer actionable recommendations on how these processes should be designed in practice. The second work presented in this thesis considers the task of sampling from a probability measure on a discrete space. While work in recent years has made it possible to apply sampling algorithms to probability measures with differentiable densities on continuous spaces in a reasonably generic way, samplers on discrete spaces are still largely derived on a case-by-case basis. The contention of this work is that this is not necessary, and that one can in fact define quite generally-applicable algorithms which can sample efficiently from discrete probability measures. The contributions are then to propose a small collection of algorithms for this task, and verify their efficiency empirically. Building on the previous chapter’s work, our samplers are again defined in continuous time and non-reversible, each of which offer noticeable benefits in efficiency. The third work presented in this thesis concerns a theoretical study of a particular class of Markov Chain-based sampling algorithms which make use of parallel computing resources. The Markov Chains which are produced by this algorithm are mathematically equivalent to a standard Metropolis-Hastings chain, but their real-time convergence properties are affected nontrivially by the application of parallelism. The contribution of this work is to analyse the convergence behaviour of these chains, and to use the ‘optimal scaling’ framework (as developed by Roberts, Rosenthal, and others) to make recommendations concerning the tuning of such algorithms in practice. The introductory chapters provide a general overview on the task of generating samples from a probability measure, with particular focus on methods involving Markov processes. There is also an interlude on the relative benefits of i) continuous-time and ii) non-reversible Markov processes for sampling, which are intended to provide additional context for the reading of the first two works.PhD Studentship paid for by Cantab Capital Institute for the Mathematics of Informatio

Generative probabilistic models for object segmentation

One of the long-standing open problems in machine vision has been the task of ‘object segmentation’, in which an image is partitioned into two sets of pixels: those that belong to the object of interest, and those that do not. A closely related task is that of ‘parts-based object segmentation’, where additionally each of the object’s pixels are labelled as belonging to one of several predetermined parts. There is broad agreement that segmentation is coupled to the task of object recognition. Knowledge of the object’s class can lead to more accurate segmentations, and in turn accurate segmentations can be used to obtain higher recognition rates. In this thesis we focus on one side of this relationship: given the object’s class and its bounding box, how accurately can we segment it? Segmentation is challenging primarily due to the huge amount of variability one sees in images of natural scenes. A large number of factors combine in complex ways to generate the pixel intensities that make up any given image. In this work we approach the problem by developing generative probabilistic models of the objects in question. Not only does this allow us to express notions of variability and uncertainty in a principled way, but also to separate the problems of model design and inference. The thesis makes the following contributions: First, we demonstrate an explicit probabilistic model of images of objects based on a latent Gaussian model of shape. This can be learned from images in an unsupervised fashion. Through experiments on a variety of datasets we demonstrate the advantages of explicitly modelling shape variability. We then focus on the task of constructing more accurate models of shape. We present a type of layered probabilistic model that we call a Shape Boltzmann Machine (SBM) for the task of modelling foreground/background (binary) and parts-based (categorical) shapes. We demonstrate that it constitutes the state-of-the-art and characterises a ‘strong’ model of shape, in that samples from the model look realistic and that it generalises to generate samples that differ from training examples. Finally, we demonstrate how the SBM can be used in conjunction with an appearance model to form a fully generative model of images of objects. We show how parts-based object segmentations can be obtained simply by performing probabilistic inference in this joint model. We apply the model to several challenging datasets and find that its performance is comparable to the state-of-the-art

Deep generative modelling for amortised variational inference

Probabilistic and statistical modelling are the fundamental frameworks that underlie a large proportion of the modern machine learning (ML) techniques. These frameworks allow for the practitioners to develop tailor-made models for their problems that may include their expert knowledge and can learn from data. Learning from data in the Bayesian framework is referred as inference. In general, model-specific inference methods are hard to derive as they require high level of mathematical and statistical dexterity on the practitioner’s part. As a result, there is a large industry of researchers in ML and statistics that work towards developing automatic methods of inference (Carpenter et al., 2017; Tran et al., 2016; Kucukelbir et al., 2016; Ge et al., 2018; Salvatier et al., 2016; Uber, 2017; Lintusaari et al., 2018). These methods are generally model agnostic and are therefore called black-box inference. Recent work has shown that use of deep learning techniques (Rezende and Mohamed, 2015b; Kingma et al., 2016; Srivastava and Sutton, 2017; Mescheder et al., 2017a) within the framework of variational inference (Jordan et al., 1999) not only allows for automatic and accurate inference but does so in a drastically efficient way. The added efficiency comes from the amortisation of the learning cost by using deep neural networks to leverage the smoothness between data points and their posterior parameters. The field of deep learning based amortised variational inference is relatively new and therefore has numerous challenges and issues to be tackled before it can be established as a standard method of inference. To this end, this thesis presents four pieces of original work in the domain of automatic amortised variational inference in statistical models. We first introduce two sets of techniques for amortising variational inference in Bayesian generative models such as the Latent Dirichlet Allocation (Blei et al., 2003) and Pachinko Allocation Machine (Li and McCallum, 2006). These techniques use deep neural networks and stochastic gradient based first order optimisers for inference and can be generically applied for inference in a large number of Bayesian generative models. Similarly, we also introduce a novel variational framework for implicit generative models of data, called VEEGAN. This framework allows for doing inference in statistical models where unlike the Bayesian generative models, a prescribed likelihood function is not available. It makes use of a discriminator based density ratio estimator (Sugiyama et al., 2012) to deal with the intractability of the likelihood function. Implicit generative models such as the generative adversarial networks (Goodfellow et al., 2014) suffer from learning issues like mode collapse (Srivastava et al., 2017) and training instability (Arjovsky et al., 2017). We tackle the mode collapse in GANs using VEEGAN and propose a new training method for implicit generative models, RB-MMDnet based on an alternative density ratio estimation which provide for stable training and optimisation in implicit models. Our results and analysis clearly show that the application of deep generative modelling in variational inference is a promising direction for improving the state of the black-box inference methods. Not only do these methods perform better than the traditional inference methods for the models in question but they do so in a fraction of the time compared to the traditional methods by utilising the latest in the GPU technology