Bayesian Semi-supervised Learning Research Articles

Mathematical Foundations of Graph-Based Bayesian Semi-Supervised Learning

In recent decades, science and engineering have been revolutionized by a momentous growth in the amount of available data. However, despite the unprecedented ease with which data are now collected and stored, labeling data by supplementing each feature with an informative tag remains to be challenging. Illustrative tasks where the labeling process requires expert knowledge or is tedious and time-consuming include labeling X-rays with a diagnosis, protein sequences with a protein type, texts by their topic, tweets by their sentiment, or videos by their genre. In these and numerous other examples, only a few features may be manually labeled due to cost and time constraints. How can we best propagate label information from a small number of expensive labeled features to a vast number of unlabeled ones? This is the question addressed by semi-supervised learning (SSL). This article overviews recent foundational developments on graph-based Bayesian SSL, a probabilistic framework for label propagation using similarities between features. SSL is an active research area and a thorough review of the extant literature is beyond the scope of this article. Our focus will be on topics drawn from our own research that illustrate the wide range of mathematical tools and ideas that underlie the rigorous study of the statistical accuracy and computational efficiency of graph-based Bayesian SSL.

Combining deep generative and discriminative models for Bayesian semi-supervised learning

Generative models can be used for a wide range of tasks, and have the appealing ability to learn from both labelled and unlabelled data. In contrast, discriminative models cannot learn from unlabelled data, but tend to outperform their generative counterparts in supervised tasks. We develop a framework to jointly train deep generative and discriminative models, enjoying the benefits of both. The framework allows models to learn from labelled and unlabelled data, as well as naturally account for uncertainty in predictive distributions, providing the first Bayesian approach to semi-supervised learning with deep generative models. We demonstrate that our blended discriminative and generative models outperform purely generative models in both predictive performance and uncertainty calibration in a number of semi-supervised learning tasks.

Large data and zero noise limits of graph-based semi-supervised learning algorithms

Abstract Scalings in which the graph Laplacian approaches a differential operator in the large graph limit are used to develop understanding of a number of algorithms for semi-supervised learning; in particular, the probit algorithm, level set and kriging methods. Both optimization and Bayesian approaches are considered, based around a regularizing quadratic form found from an affine transformation of the Laplacian, raised to a possibly fractional, exponent. Conditions on the parameters defining this quadratic form are identified under which well-defined limiting continuum analogues of the optimization and Bayesian semi-supervised learning problems may be found, thereby shedding light on the design of algorithms in the large graph setting. The large graph limits of the optimization formulations are tackled through Γ-convergence, using the recently introduced T L p metric. The small labeling noise limits of the Bayesian formulations are also identified, and contrasted with pre-existing harmonic function approaches to the problem.

Bayesian semi-supervised learning for uncertainty-calibrated prediction of molecular properties and active learning.

Predicting bioactivity and physical properties of small molecules is a central challenge in drug discovery. Deep learning is becoming the method of choice but studies to date focus on mean accuracy as the main metric. However, to replace costly and mission-critical experiments by models, a high mean accuracy is not enough: outliers can derail a discovery campaign, thus models need to reliably predict when it will fail, even when the training data is biased; experiments are expensive, thus models need to be data-efficient and suggest informative training sets using active learning. We show that uncertainty quantification and active learning can be achieved by Bayesian semi-supervised graph convolutional neural networks. The Bayesian approach estimates uncertainty in a statistically principled way through sampling from the posterior distribution. Semi-supervised learning disentangles representation learning and regression, keeping uncertainty estimates accurate in the low data limit and allowing the model to start active learning from a small initial pool of training data. Our study highlights the promise of Bayesian deep learning for chemistry.

Accuracy of latent-variable estimation in Bayesian semi-supervised learning

Hierarchical probabilistic models, such as Gaussian mixture models, are widely used for unsupervised learning tasks. These models consist of observable and latent variables, which represent the observable data and the underlying data-generation process, respectively. Unsupervised learning tasks, such as cluster analysis, are regarded as estimations of latent variables based on the observable ones. The estimation of latent variables in semi-supervised learning, where some labels are observed, will be more precise than that in unsupervised, and one of the concerns is to clarify the effect of the labeled data. However, there has not been sufficient theoretical analysis of the accuracy of the estimation of latent variables. In a previous study, a distribution-based error function was formulated, and its asymptotic form was calculated for unsupervised learning with generative models. It has been shown that, for the estimation of latent variables, the Bayes method is more accurate than the maximum-likelihood method. The present paper reveals the asymptotic forms of the error function in Bayesian semi-supervised learning for both discriminative and generative models. The results show that the generative model, which uses all of the given data, performs better when the model is well specified.

Bayesian semi-supervised learning with support vector machine

This paper introduces a Bayesian semi-supervised support vector machine (Semi-BSVM) model for binary classification. Our semi-supervised learning has a distinct advantage over supervised or inductive learning since by design it reduces the problem of overfitting. While a traditional support vector machine (SVM) has the widest margin based on the labeled data only, our semi-supervised form of SVM attempts to find the widest margin in both the labeled and unlabeled data space. This enables us to use some information from the unlabeled data and improve the overall prediction performance. The likelihood is constructed using a special type of hinge loss function which also involves the unlabeled data. A penalty term is added for the likelihood part constructed from the unlabeled data. The parameters and penalties are controlled through nearly diffuse priors for objectivity of the analysis. The rate of learning from the unlabeled data is reflected through the posterior distribution of the penalty parameter from the unlabeled data. This formulation provides us with a control on how much information should be extracted from the unlabeled data without hurting the overall performance of our model. We have applied our model on three simulation data sets and five real life data sets. Our simulation study and real life data analysis show considerable improvement in prediction quality for our semi-supervised learning over supervised learning methods when we have a high learning rate from the unlabeled data. This phenomenon is particularly evident in cases when the amount of unlabeled data is very large compared to the available labeled data.

Semi-supervised analysis of gene expression profiles for lineage-specific development in the Caenorhabditis elegans embryo

Gene expression profiling is a powerful approach to identify genes that may be involved in a specific biological process on a global scale. For example, gene expression profiling of mutant animals that lack or contain an excess of certain cell types is a common way to identify genes that are important for the development and maintenance of given cell types. However, it is difficult for traditional computational methods, including unsupervised and supervised learning methods, to detect relevant genes from a large collection of expression profiles with high sensitivity and specificity. Unsupervised methods group similar gene expressions together while ignoring important prior biological knowledge. Supervised methods utilize training data from prior biological knowledge to classify gene expression. However, for many biological problems, little prior knowledge is available, which limits the prediction performance of most supervised methods. We present a Bayesian semi-supervised learning method, called BGEN, that improves upon supervised and unsupervised methods by both capturing relevant expression profiles and using prior biological knowledge from literature and experimental validation. Unlike currently available semi-supervised learning methods, this new method trains a kernel classifier based on labeled and unlabeled gene expression examples. The semi-supervised trained classifier can then be used to efficiently classify the remaining genes in the dataset. Moreover, we model the confidence of microarray probes and probabilistically combine multiple probe predictions into gene predictions. We apply BGEN to identify genes involved in the development of a specific cell lineage in the C. elegans embryo, and to further identify the tissues in which these genes are enriched. Compared to K-means clustering and SVM classification, BGEN achieves higher sensitivity and specificity. We confirm certain predictions by biological experiments. The results are available at http://www.csail.mit.edu/~alanqi/projects/BGEN.html.