Semisupervised learning
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Semisupervised learning is a class of supervised learning tasks and techniques that also make use of unlabeled data for training  typically a small amount of labeled data with a large amount of unlabeled data. Semisupervised learning falls between unsupervised learning (without any labeled training data) and supervised learning (with completely labeled training data). Many machinelearning researchers have found that unlabeled data, when used in conjunction with a small amount of labeled data, can produce considerable improvement in learning accuracy. The acquisition of labeled data for a learning problem often requires a skilled human agent (e.g. to transcribe an audio segment) or a physical experiment (e.g. determining the 3D structure of a protein or determining whether there is oil at a particular location). The cost associated with the labeling process thus may render a fully labeled training set infeasible, whereas acquisition of unlabeled data is relatively inexpensive. In such situations, semisupervised learning can be of great practical value. Semisupervised learning is also of theoretical interest in machine learning and as a model for human learning.
As in the supervised learning framework, we are given a set of independently identically distributed examples with corresponding labels . Additionally, we are given unlabeled examples . Semisupervised learning attempts to make use of this combined information to surpass the classification performance that could be obtained either by discarding the unlabeled data and doing supervised learning or by discarding the labels and doing unsupervised learning.
Semisupervised learning may refer to either transductive learning or inductive learning. The goal of transductive learning is to infer the correct labels for the given unlabeled data only. The goal of inductive learning is to infer the correct mapping from to .
Intuitively, we can think of the learning problem as an exam and labeled data as the few example problems that the teacher solved in class. The teacher also provides a set of unsolved problems. In the transductive setting, these unsolved problems are a takehome exam and you want to do well on them in particular. In the inductive setting, these are practice problems of the sort you will encounter on the inclass exam.
It is unnecessary (and, according to Vapnik's principle, imprudent) to perform transductive learning by way of inferring a classification rule over the entire input space; however, in practice, algorithms formally designed for transduction or induction are often used interchangeably.
Contents
Assumptions used in semisupervised learning
In order to make any use of unlabeled data, we must assume some structure to the underlying distribution of data. Semisupervised learning algorithms make use of at least one of the following assumptions. ^{[1]}
Smoothness assumption
Points which are close to each other are more likely to share a label. This is also generally assumed in supervised learning and yields a preference for geometrically simple decision boundaries. In the case of semisupervised learning, the smoothness assumption additionally yields a preference for decision boundaries in lowdensity regions, so that there are fewer points close to each other but in different classes.
Cluster assumption
The data tend to form discrete clusters, and points in the same cluster are more likely to share a label (although data sharing a label may be spread across multiple clusters). This is a special case of the smoothness assumption and gives rise to feature learning with clustering algorithms.
Manifold assumption
The data lie approximately on a manifold of much lower dimension than the input space. In this case we can attempt to learn the manifold using both the labeled and unlabeled data to avoid the curse of dimensionality. Then learning can proceed using distances and densities defined on the manifold.
The manifold assumption is practical when highdimensional data are being generated by some process that may be hard to model directly, but which only has a few degrees of freedom. For instance, human voice is controlled by a few vocal folds,^{[2]} and images of various facial expressions are controlled by a few muscles. We would like in these cases to use distances and smoothness in the natural space of the generating problem, rather than in the space of all possible acoustic waves or images respectively.
History
The heuristic approach of selftraining (also known as selflearning or selflabeling) is historically the oldest approach to semisupervised learning,^{[1]} with examples of applications starting in the 1960s (see for instance Scudder (1965)^{[3]}).
The transductive learning framework was formally introduced by Vladimir Vapnik in the 1970s.^{[4]} Interest in inductive learning using generative models also began in the 1970s. A probably approximately correct learning bound for semisupervised learning of a Gaussian mixture was demonstrated by Ratsaby and Venkatesh in 1995 ^{[5]}
Semisupervised learning has recently become more popular and practically relevant due to the variety of problems for which vast quantities of unlabeled data are available—e.g. text on websites, protein sequences, or images. For a review of recent work see a survey article by Zhu (2008).^{[6]}
Methods for semisupervised learning
Generative models
Generative approaches to statistical learning first seek to estimate , the distribution of data points belonging to each class. The probability that a given point has label is then proportional to by Bayes' rule. Semisupervised learning with generative models can be viewed either as an extension of supervised learning (classification plus information about ) or as an extension of unsupervised learning (clustering plus some labels).
Generative models assume that the distributions take some particular form parameterized by the vector . If these assumptions are incorrect, the unlabeled data may actually decrease the accuracy of the solution relative to what would have been obtained from labeled data alone. ^{[7]} However, if the assumptions are correct, then the unlabeled data necessarily improves performance.^{[5]}
The unlabeled data are distributed according to a mixture of individualclass distributions. In order to learn the mixture distribution from the unlabeled data, it must be identifiable, that is, different parameters must yield different summed distributions. Gaussian mixture distributions are identifiable and commonly used for generative models.
The parameterized joint distribution can be written as by using the Chain rule. Each parameter vector is associated with a decision function . The parameter is then chosen based on fit to both the labeled and unlabeled data, weighted by :
^{[8]}
Lowdensity separation
Another major class of methods attempts to place boundaries in regions where there are few data points (labeled or unlabeled). One of the most commonly used algorithms is the transductive support vector machine, or TSVM (which, despite its name, may be used for inductive learning as well). Whereas support vector machines for supervised learning seek a decision boundary with maximal margin over the labeled data, the goal of TSVM is a labeling of the unlabeled data such that the decision boundary has maximal margin over all of the data. In addition to the standard hinge loss for labeled data, a loss function is introduced over the unlabeled data by letting . TSVM then selects from a reproducing kernel Hilbert space by minimizing the regularized empirical risk:
An exact solution is intractable due to the nonconvex term , so research has focused on finding useful approximations.^{[8]}
Other approaches that implement lowdensity separation include Gaussian process models, information regularization, and entropy minimization (of which TSVM is a special case).
Graphbased methods
Graphbased methods for semisupervised learning use a graph representation of the data, with a node for each labeled and unlabeled example. The graph may be constructed using domain knowledge or similarity of examples; two common methods are to connect each data point to its nearest neighbors or to examples within some distance . The weight of an edge between and is then set to .
Within the framework of manifold regularization, ^{[9]} ^{[10]} the graph serves as a proxy for the manifold. A term is added to the standard Tikhonov regularization problem to enforce smoothness of the solution relative to the manifold (in the intrinsic space of the problem) as well as relative to the ambient input space. The minimization problem becomes
 ^{[8]}
where is a reproducing kernel Hilbert space and is the manifold on which the data lie. The regularization parameters and control smoothness in the ambient and intrinsic spaces respectively. The graph is used to approximate the intrinsic regularization term. Defining the graph Laplacian where and the vector , we have
 .
The Laplacian can also be used to extend the supervised learning algorithms： regularized least squares and support vector machines (SVM) to semisupervised versions Laplacian regularized least squares and Laplacian SVM.
Heuristic approaches
Some methods for semisupervised learning are not intrinsically geared to learning from both unlabeled and labeled data, but instead make use of unlabeled data within a supervised learning framework. For instance, the labeled and unlabeled examples may inform a choice of representation, distance metric, or kernel for the data in an unsupervised first step. Then supervised learning proceeds from only the labeled examples.
Selftraining is a wrapper method for semisupervised learning. First a supervised learning algorithm is used to select a classifier based on the labeled data only. This classifier is then applied to the unlabeled data to generate more labeled examples as input for another supervised learning problem. Generally only the labels the classifier is most confident of are added at each step.
Cotraining is an extension of selftraining in which multiple classifiers are trained on different (ideally disjoint) sets of features and generate labeled examples for one another.
Semisupervised learning in human cognition
Human responses to formal semisupervised learning problems have yielded varying conclusions about the degree of influence of the unlabeled data (for a summary see ^{[11]}). More natural learning problems may also be viewed as instances of semisupervised learning. Much of human concept learning involves a small amount of direct instruction (e.g. parental labeling of objects during childhood) combined with large amounts of unlabeled experience (e.g. observation of objects without naming or counting them, or at least without feedback).
Human infants are sensitive to the structure of unlabeled natural categories such as images of dogs and cats or male and female faces.^{[12]} More recent work has shown that infants and children take into account not only the unlabeled examples available, but the sampling process from which labeled examples arise.^{[13]}^{[14]}
See also
References
 ↑ ^{1.0} ^{1.1} Chapelle, Olivier; Schölkopf, Bernhard; Zien, Alexander (2006). Semisupervised learning. Cambridge, Mass.: MIT Press. ISBN 9780262033589.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
 ↑ Stevens, K.N.(2000), Acoustic Phonetics, MIT Press, ISBN 0262692503, 9780262692502
 ↑ Scudder, H.J. Probability of Error of Some Adaptive PatternRecognition Machines. IEEE Transaction on Information Theory, 11:363–371 (1965). Cited in Chapelle et al. 2006, page 3.
 ↑ Vapnik, V. and Chervonenkis, A. Theory of Pattern Recognition [in Russian]. Nauka, Moscow (1974). Cited in Chapelle et al. 2006, page 3.
 ↑ ^{5.0} ^{5.1} Ratsaby, J. and Venkatesh, S. Learning from a mixture of labeled and unlabeled examples with parametric side information. In Proceedings of the Eighth Annual Conference on Computational Learning Theory, pages 412417 (1995). Cited in Chapelle et al. 2006, page 4.
 ↑ Zhu, Xiaojin. Semisupervised learning literature survey. Computer Sciences, University of WisconsinMadison (2008).
 ↑ Cozman, F. and Cohen, I. Risks of semisupervised learning: how unlabeled data can degrade performance of generative classifiers. In: Chapelle et al. (2006).
 ↑ ^{8.0} ^{8.1} ^{8.2} Zhu, Xiaojin. SemiSupervised Learning University of WisconsinMadison.
 ↑ M. Belkin, P. Niyogi (2004). "Semisupervised Learning on Riemannian Manifolds". Machine Learning. 56 (Special Issue on Clustering): 209–239.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
 ↑ M. Belkin, P. Niyogi, V. Sindhwani. On Manifold Regularization. AISTATS 2005.
 ↑ Zhu, Xiaojin; Goldberg, Andrew B. (2009). Introduction to semisupervised learning. Morgan & Claypool. ISBN 9781598295481.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
 ↑ Younger, B. A. and Fearing, D. D. (1999), Parsing Items into Separate Categories: Developmental Change in Infant Categorization. Child Development, 70: 291–303.
 ↑ Xu, F. and Tenenbaum, J. B. (2007). "Sensitivity to sampling in Bayesian word learning. Developmental Science". 10: 288–297. doi:10.1111/j.14677687.2007.00590.x. Cite journal requires
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