Semi-supervised learning (SSL) uses unlabeled data for training and has been shown to greatly improve performances when compared to a supervised approach on the labeled data available. This claim depends both on the amount of labeled data available and on the algorithm used. In this paper, we compute analytically the gap between the best fully-supervised approach on labeled data and the best semi-supervised approach using both labeled and unlabeled data. We quantify the best possible increase in performance obtained thanks to the unlabeled data, i.e. we compute the accuracy increase due to the information contained in the unlabeled data. Our work deals with a simple high-dimensional Gaussian mixture model for the data in a Bayesian setting. Our rigorous analysis builds on recent theoretical breakthroughs in high-dimensional inference and a large body of mathematical tools from statistical physics initially developed for spin glasses.
To the best of our knowledge, there are much fewer theoretical works dealing with a semi-supervised setting. @cite_18 studies a mixture model where the estimation problem is essentially reduced to the one of estimating the mixing parameter and shows that the information content of unlabeled examples decreases as classes overlap. More closely related to our work, @cite_3 provides the first information theoretic tight analysis for inference of latent community structure given a dense graph along with high dimensional node covariates, correlated with the same latent communities. @cite_20 studies a class of graph-oriented semi-supervised learning algorithms in the limit of large and numerous data similar to our setting.
@cite_18 @cite_20 @cite_3
2145494108 2883384506 2030962165 2170569305 2560052444 2943865428 2114900710 2942681667 2169847772 2738326163 2032665768 2746413642 2095758845 2048679005 2963558289 2941609697 2962699510 2125592902 2619414012 2158535911
We consider the semi-supervised learning problem, where a decision rule is to be learned from labeled and unlabeled data. In this framework, we motivate minimum entropy regularization, which enables to incorporate unlabeled data in the standard supervised learning. Our approach includes other approaches to the semi-supervised problem as particular or limiting cases. A series of experiments illustrates that the proposed solution benefits from unlabeled data. The method challenges mixture models when the data are sampled from the distribution class spanned by the generative model. The performances are definitely in favor of minimum entropy regularization when generative models are misspecified, and the weighting of unlabeled data provides robustness to the violation of the "cluster assumption". Finally, we also illustrate that the method can also be far superior to manifold learning in high dimension spaces. We provide the first information theoretical tight analysis for inference of latent community structure given a sparse graph along with high dimensional node covariates, correlated with the same latent communities. Our work bridges recent theoretical breakthroughs in detection of latent community structure without nodes covariates and a large body of empirical work using diverse heuristics for combining node covariates with graphs for inference. The tightness of our analysis implies in particular, the information theoretic necessity of combining the different sources of information. Our analysis holds for networks of large degrees as well as for a Gaussian version of the model. We present a new direction for semi-supervised learning where self-adjusting generative models replace fixed ones and unlabeled data can potentially improve learning even when labeled data is only partially-observed. We model each class data by a mixture model and use a hierarchical Dirichlet process (HDP) to model observed as well as unobserved classes. We extend the standard HDP model to accommodate unlabeled samples and introduce a new sharing strategy, within the context of Gaussian mixture models, that restricts sharing with covariance matrices while leaving the mean vectors free. Our research is mainly driven by real-world applications with evolving data-generating mechanisms where obtaining a fully-observed labeled data set is impractical. We demonstrate the feasibility of the proposed approach for semi-supervised learning in two such applications. This paper analyzes the performance of semi-supervised learning of mixture models. We show that unlabeled data can lead to an increase in classification error even in situations where additional labeled data would decrease classification error. We present a mathematical analysis of this "degradation" phenomenon and show that it is due to the fact that bias may be adversely affected by unlabeled data. We discuss the impact of these theoretical results to practical situations. Semi-supervised learning algorithms reduce the high cost of acquiring labeled training data by using both labeled and unlabeled data during learning. Deep Convolutional Networks (DCNs) have achieved great success in supervised tasks and as such have been widely employed in the semi-supervised learning. In this paper we leverage the recently developed Deep Rendering Mixture Model (DRMM), a probabilistic generative model that models latent nuisance variation, and whose inference algorithm yields DCNs. We develop an EM algorithm for the DRMM to learn from both labeled and unlabeled data. Guided by the theory of the DRMM, we introduce a novel non-negativity constraint and a variational inference term. We report state-of-the-art performance on MNIST and SVHN and competitive results on CIFAR10. We also probe deeper into how a DRMM trained in a semi-supervised setting represents latent nuisance variation using synthetically rendered images. Taken together, our work provides a unified framework for supervised, unsupervised, and semi-supervised learning. Semi-supervised learning has proven to be a powerful paradigm for leveraging unlabeled data to mitigate the reliance on large labeled datasets. In this work, we unify the current dominant approaches for semi-supervised learning to produce a new algorithm, MixMatch, that works by guessing low-entropy labels for data-augmented unlabeled examples and mixing labeled and unlabeled data using MixUp. We show that MixMatch obtains state-of-the-art results by a large margin across many datasets and labeled data amounts. For example, on CIFAR-10 with 250 labels, we reduce error rate by a factor of 4 (from 38 to 11 ) and by a factor of 2 on STL-10. We also demonstrate how MixMatch can help achieve a dramatically better accuracy-privacy trade-off for differential privacy. Finally, we perform an ablation study to tease apart which components of MixMatch are most important for its success. Active and semi-supervised learning are important techniques when labeled data are scarce. Recently a method was suggested for combining active learning with a semi-supervised learning algorithm that uses Gaussian fields and harmonic functions. This classifier is relational in nature: it relies on having the data presented as a partially labeled graph (also known as a within-network learning problem). This work showed yet again that empirical risk minimization (ERM) was the best method to find the next instance to label and provided an efficient way to compute ERM with the semi-supervised classifier. The computational problem with ERM is that it relies on computing the risk for all possible instances. If we could limit the candidates that should be investigated, then we can speed up active learning considerably. In the case where the data is graphical in nature, we can leverage the graph structure to rapidly identify instances that are likely to be good candidates for labeling. This paper describes a novel hybrid approach of using of community finding and social network analytic centrality measures to identify good candidates for labeling and then using ERM to find the best instance in this candidate set. We show on real-world data that we can limit the ERM computations to a fraction of instances with comparable performance. Existing popular methods for semi-supervised learning with Graph Neural Networks (such as the Graph Convolutional Network) provably cannot learn a general class of neighborhood mixing relationships. To address this weakness, we propose a new model, MixHop, that can learn these relationships, including difference operators, by repeatedly mixing feature representations of neighbors at various distances. Mixhop requires no additional memory or computational complexity, and outperforms on challenging baselines. In addition, we propose sparsity regularization that allows us to visualize how the network prioritizes neighborhood information across different graph datasets. Our analysis of the learned architectures reveals that neighborhood mixing varies per datasets. This work addresses graph-based semi-supervised classification and betweenness computation in large, sparse, networks (several millions of nodes). The objective of semi-supervised classification is to assign a label to unlabeled nodes using the whole topology of the graph and the labeling at our disposal. Two approaches are developed to avoid explicit computation of pairwise proximity between the nodes of the graph, which would be impractical for graphs containing millions of nodes. The first approach directly computes, for each class, the sum of the similarities between the nodes to classify and the labeled nodes of the class, as suggested initially in [1,2]. Along this approach, two algorithms exploiting different state-of-the-art kernels on a graph are developed. The same strategy can also be used in order to compute a betweenness measure. The second approach works on a trellis structure built from biased random walks on the graph, extending an idea introduced in [3]. These random walks allow to define a biased bounded betweenness for the nodes of interest, defined separately for each class. All the proposed algorithms have a linear computing time in the number of edges while providing good results, and hence are applicable to large sparse networks. They are empirically validated on medium-size standard data sets and are shown to be competitive with state-of-the-art techniques. Finally, we processed a novel data set, which is made available for benchmarking, for multi-class classification in a large network: the U.S. patents citation network containing 3M nodes (of six different classes) and 38M edges. The three proposed algorithms achieve competitive results (around 85 classification rate) on this large network-they classify the unlabeled nodes within a few minutes on a standard workstation. We investigate a family of regression problems in a semi-supervised setting. The task is to assign real-valued labels to a set of @math sample points, provided a small training subset of @math labeled points. A goal of semi-supervised learning is to take advantage of the (geometric) structure provided by the large number of unlabeled data when assigning labels. We consider random geometric graphs, with connection radius @math , to represent the geometry of the data set. Functionals which model the task reward the regularity of the estimator function and impose or reward the agreement with the training data. Here we consider the discrete @math -Laplacian regularization. We investigate asymptotic behavior when the number of unlabeled points increases, while the number of training points remains fixed. We uncover a delicate interplay between the regularizing nature of the functionals considered and the nonlocality inherent to the graph constructions. We rigorously obtain almost optimal ranges on the scaling of @math for the asymptotic consistency to hold. We prove that the minimizers of the discrete functionals in random setting converge uniformly to the desired continuum limit. Furthermore we discover that for the standard model used there is a restrictive upper bound on how quickly @math must converge to zero as @math . We introduce a new model which is as simple as the original model, but overcomes this restriction. In this paper, a novel algorithm to blindly separate an instantaneous linear underdetermined mixture of nonstationary sources is proposed. It means that the number of sources exceeds the number of channels of the available data. The separation is based on the working assumption that the sources are piecewise stationary with a different variance in each block. It proceeds in two steps: 1) estimating the mixing matrix, and 2) computing the optimum beamformer in each block to maximize the signal-to-interference ratio of each separated signal with respect to the remaining signals. Estimating the mixing matrix is accomplished through a specialized tensor decomposition of the set of sample covariance matrices of the received mixture in each block. It utilizes optimum weighting, which allows statistically efficient (CRB attaining) estimation provided that the data obey the assumed Gaussian piecewise stationary model. In simulations, performance of the algorithm is successfully tested on blind separation of 16 speech signals from nine linear instantaneous mixtures of these signals. Community detection in the presence of prior information or preferences on solution properties is called semi-supervised or constrained community detection. The task of embedding such existing kinds of knowledge effectively within a community discovery algorithm is challenging. Indeed existing approaches are not flexible enough to incorporate a variety of background information types. This paper provides a framework for semi-supervised community detection based on constraint programming modelling technology for simultaneously modelling different objective functions such as modularity and a comprehensive range of constraint types including community level, instance level, definition based and complex logic constraints. An advantage of the proposed framework is that, using appropriate solvers, optimality can be established for the solutions found. Experiments on real and benchmark data sets show strong performance and flexibility for our proposed framework. Many real-world classification problems involve the prediction of multiple inter-dependent variables forming some structural dependency. Recent progress in machine learning has mainly focused on supervised classification of such structured variables. In this paper, we investigate structured classification in a semi-supervised setting. We present a discriminative approach that utilizes the intrinsic geometry of input patterns revealed by unlabeled data points and we derive a maximum-margin formulation of semi-supervised learning for structured variables. Unlike transductive algorithms, our formulation naturally extends to new test points. We consider the problem of using a large unlabeled sample to boost performance of a learning algorit,hrn when only a small set of labeled examples is available. In particular, we consider a problem setting motivated by the task of learning to classify web pages, in which the description of each example can be partitioned into two distinct views. For example, the description of a web page can be partitioned into the words occurring on that page, and the words occurring in hyperlinks t,hat point to that page. We assume that either view of the example would be sufficient for learning if we had enough labeled data, but our goal is to use both views together to allow inexpensive unlabeled data to augment, a much smaller set of labeled examples. Specifically, the presence of two distinct views of each example suggests strategies in which two learning algorithms are trained separately on each view, and then each algorithm’s predictions on new unlabeled examples are used to enlarge the training set of the other. Our goal in this paper is to provide a PAC-style analysis for this setting, and, more broadly, a PAC-style framework for the general problem of learning from both labeled and unlabeled data. We also provide empirical results on real web-page data indicating that this use of unlabeled examples can lead to significant improvement of hypotheses in practice. *This research was supported in part by the DARPA HPKB program under contract F30602-97-1-0215 and by NSF National Young investigator grant CCR-9357793. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. TO copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and or a fee. COLT 98 Madison WI USA Copyright ACM 1998 l-58113-057--0 98 7... 5.00 92 Tom Mitchell School of Computer Science Carnegie Mellon University Pittsburgh, PA 15213-3891 mitchell+@cs.cmu.edu The recently proposed self-ensembling methods have achieved promising results in deep semi-supervised learning, which penalize inconsistent predictions of unlabeled data under different perturbations. However, they only consider adding perturbations to each single data point, while ignoring the connections between data samples. In this paper, we propose a novel method, called Smooth Neighbors on Teacher Graphs (SNTG). In SNTG, a graph is constructed based on the predictions of the teacher model, i.e., the implicit self-ensemble of models. Then the graph serves as a similarity measure with respect to which the representations of "similar" neighboring points are learned to be smooth on the low-dimensional manifold. We achieve state-of-the-art results on semi-supervised learning benchmarks. The error rates are 9.89 , 3.99 for CIFAR-10 with 4000 labels, SVHN with 500 labels, respectively. In particular, the improvements are significant when the labels are fewer. For the non-augmented MNIST with only 20 labels, the error rate is reduced from previous 4.81 to 1.36 . Our method also shows robustness to noisy labels. In this work we study the semi-supervised framework of confidence set classification with controlled expected size in minimax settings. We obtain semi-supervised minimax rates of convergence under the margin assumption and a H o lder condition on the regression function. Besides, we show that if no further assumptions are made, there is no supervised method that outperforms the semi-supervised estimator proposed in this work. We establish that the best achievable rate for any supervised method is n^ --1 2 , even if the margin assumption is extremely favorable. On the contrary, semi-supervised estimators can achieve faster rates of convergence provided that sufficiently many unlabeled samples are available. We additionally perform numerical evaluation of the proposed algorithms empirically confirming our theoretical findings. In this paper, we consider the cluster estimation problem under the stochastic block model. We show that the semidefinite programming (SDP) formulation for this problem achieves an error rate that decays exponentially in the signal-to-noise ratio. The error bound implies weak recovery in the sparse graph regime with bounded expected degrees as well as exact recovery in the dense regime. An immediate corollary of our results yields error bounds under the censored block model. Moreover, these error bounds are robust, continuing to hold under heterogeneous edge probabilities and a form of the so-called monotone attack. Significantly, this error rate is achieved by the SDP solution itself without any further pre- or post-processing and improves upon existing polynomially decaying error bounds proved using the Grothendieck’s inequality. Our analysis builds on two key ingredients: 1) showing that the graph has a well-behaved spectrum, even in the sparse regime, after discounting an exponentially small number of edges and 2) an order-statistics argument that governs the final error rate. Both arguments highlight the implicit regularization effect of the SDP formulation. There has been increased interest in devising learning techniques that combine unlabeled data with labeled data -- i.e. semi-supervised learning. However, to the best of our knowledge, no study has been performed across various techniques and different types and amounts of labeled and unlabeled data. Moreover, most of the published work on semi-supervised learning techniques assumes that the labeled and unlabeled data come from the same distribution. It is possible for the labeling process to be associated with a selection bias such that the distributions of data points in the labeled and unlabeled sets are different. Not correcting for such bias can result in biased function approximation with potentially poor performance. In this paper, we present an empirical study of various semi-supervised learning techniques on a variety of datasets. We attempt to answer various questions such as the effect of independence or relevance amongst features, the effect of the size of the labeled and unlabeled sets and the effect of noise. We also investigate the impact of sample-selection bias on the semi -supervised learning techniques under study and implement a bivariate probit technique particularly designed to correct for such bias. Graph-based semi-supervised learning is one of the most popular methods in machine learning. Some of its theoretical properties such as bounds for the generalization error and the convergence of the graph Laplacian regularizer have been studied in computer science and statistics literatures. However, a fundamental statistical property, the consistency of the estimator from this method has not been proved. In this article, we study the consistency problem under a non-parametric framework. We prove the consistency of graph-based learning in the case that the estimated scores are enforced to be equal to the observed responses for the labeled data. The sample sizes of both labeled and unlabeled data are allowed to grow in this result. When the estimated scores are not required to be equal to the observed responses, a tuning parameter is used to balance the loss function and the graph Laplacian regularizer. We give a counterexample demonstrating that the estimator for this case can be inconsistent. The theoretical findings are supported by numerical studies. As the availability and importance of relational data—such as the friendships summarized on a social networking website—increases, it becomes increasingly important to have good models for such data. The kinds of latent structure that have been considered for use in predicting links in such networks have been relatively limited. In particular, the machine learning community has focused on latent class models, adapting Bayesian nonparametric methods to jointly infer how many latent classes there are while learning which entities belong to each class. We pursue a similar approach with a richer kind of latent variable—latent features—using a Bayesian nonparametric approach to simultaneously infer the number of features at the same time we learn which entities have each feature. Our model combines these inferred features with known covariates in order to perform link prediction. We demonstrate that the greater expressiveness of this approach allows us to improve performance on three datasets.