Transduction (machine learning)

Last updated

In logic, statistical inference, and supervised learning, transduction or transductive inference is reasoning from observed, specific (training) cases to specific (test) cases. In contrast, induction is reasoning from observed training cases to general rules, which are then applied to the test cases. The distinction is most interesting in cases where the predictions of the transductive model are not achievable by any inductive model. Note that this is caused by transductive inference on different test sets producing mutually inconsistent predictions.

Contents

Transduction was introduced by Vladimir Vapnik in the 1990s, motivated by his view that transduction is preferable to induction since, according to him, induction requires solving a more general problem (inferring a function) before solving a more specific problem (computing outputs for new cases): "When solving a problem of interest, do not solve a more general problem as an intermediate step. Try to get the answer that you really need but not a more general one." [1] A similar observation had been made earlier by Bertrand Russell: "we shall reach the conclusion that Socrates is mortal with a greater approach to certainty if we make our argument purely inductive than if we go by way of 'all men are mortal' and then use deduction" (Russell 1912, chap VII).

An example of learning which is not inductive would be in the case of binary classification, where the inputs tend to cluster in two groups. A large set of test inputs may help in finding the clusters, thus providing useful information about the classification labels. The same predictions would not be obtainable from a model which induces a function based only on the training cases. Some people may call this an example of the closely related semi-supervised learning, since Vapnik's motivation is quite different. An example of an algorithm in this category is the Transductive Support Vector Machine (TSVM).

A third possible motivation of transduction arises through the need to approximate. If exact inference is computationally prohibitive, one may at least try to make sure that the approximations are good at the test inputs. In this case, the test inputs could come from an arbitrary distribution (not necessarily related to the distribution of the training inputs), which wouldn't be allowed in semi-supervised learning. An example of an algorithm falling in this category is the Bayesian Committee Machine (BCM).

Example problem

The following example problem contrasts some of the unique properties of transduction against induction.

Labels.png

A collection of points is given, such that some of the points are labeled (A, B, or C), but most of the points are unlabeled (?). The goal is to predict appropriate labels for all of the unlabeled points.

The inductive approach to solving this problem is to use the labeled points to train a supervised learning algorithm, and then have it predict labels for all of the unlabeled points. With this problem, however, the supervised learning algorithm will only have five labeled points to use as a basis for building a predictive model. It will certainly struggle to build a model that captures the structure of this data. For example, if a nearest-neighbor algorithm is used, then the points near the middle will be labeled "A" or "C", even though it is apparent that they belong to the same cluster as the point labeled "B".

Transduction has the advantage of being able to consider all of the points, not just the labeled points, while performing the labeling task. In this case, transductive algorithms would label the unlabeled points according to the clusters to which they naturally belong. The points in the middle, therefore, would most likely be labeled "B", because they are packed very close to that cluster.

An advantage of transduction is that it may be able to make better predictions with fewer labeled points, because it uses the natural breaks found in the unlabeled points. One disadvantage of transduction is that it builds no predictive model. If a previously unknown point is added to the set, the entire transductive algorithm would need to be repeated with all of the points in order to predict a label. This can be computationally expensive if the data is made available incrementally in a stream. Further, this might cause the predictions of some of the old points to change (which may be good or bad, depending on the application). A supervised learning algorithm, on the other hand, can label new points instantly, with very little computational cost.

Transduction algorithms

Transduction algorithms can be broadly divided into two categories: those that seek to assign discrete labels to unlabeled points, and those that seek to regress continuous labels for unlabeled points. Algorithms that seek to predict discrete labels tend to be derived by adding partial supervision to a clustering algorithm. Two classes of algorithms can be used: flat clustering and hierarchical clustering. The latter can be further subdivided into two categories: those that cluster by partitioning, and those that cluster by agglomerating. Algorithms that seek to predict continuous labels tend to be derived by adding partial supervision to a manifold learning algorithm.

Partitioning transduction

Partitioning transduction can be thought of as top-down transduction. It is a semi-supervised extension of partition-based clustering. It is typically performed as follows:

Consider the set of all points to be one large partition. While any partition P contains two points with conflicting labels:   Partition P into smaller partitions. For each partition P:   Assign the same label to all of the points in P.

Of course, any reasonable partitioning technique could be used with this algorithm. Max flow min cut partitioning schemes are very popular for this purpose.

Agglomerative transduction

Agglomerative transduction can be thought of as bottom-up transduction. It is a semi-supervised extension of agglomerative clustering. It is typically performed as follows:

Compute the pair-wise distances, D, between all the points. Sort D in ascending order. Consider each point to be a cluster of size 1. For each pair of points {a,b} in D:   If (a is unlabeled) or (b is unlabeled) or (a and b have the same label)     Merge the two clusters that contain a and b.     Label all points in the merged cluster with the same label.

Manifold transduction

Manifold-learning-based transduction is still a very young field of research.

See also

Related Research Articles

<span class="mw-page-title-main">Supervised learning</span> A paradigm in machine learning

Supervised learning (SL) is a paradigm in machine learning where input objects and a desired output value train a model. The training data is processed, building a function that maps new data on expected output values. An optimal scenario will allow for the algorithm to correctly determine output values for unseen instances. This requires the learning algorithm to generalize from the training data to unseen situations in a "reasonable" way. This statistical quality of an algorithm is measured through the so-called generalization error.

<span class="mw-page-title-main">Support vector machine</span> Set of methods for supervised statistical learning

In machine learning, support vector machines are supervised learning models with associated learning algorithms that analyze data for classification and regression analysis. Developed at AT&T Bell Laboratories by Vladimir Vapnik with colleagues SVMs are one of the most robust prediction methods, being based on statistical learning frameworks or VC theory proposed by Vapnik and Chervonenkis (1974). Given a set of training examples, each marked as belonging to one of two categories, an SVM training algorithm builds a model that assigns new examples to one category or the other, making it a non-probabilistic binary linear classifier. SVM maps training examples to points in space so as to maximise the width of the gap between the two categories. New examples are then mapped into that same space and predicted to belong to a category based on which side of the gap they fall.

<span class="mw-page-title-main">Machine learning</span> Study of algorithms that improve automatically through experience

Machine learning (ML) is a field of study in artificial intelligence concerned with the development and study of statistical algorithms that can effectively generalize and thus perform tasks without explicit instructions. Recently, generative artificial neural networks have been able to surpass many previous approaches in performance. Machine learning approaches have been applied to large language models, computer vision, speech recognition, email filtering, agriculture and medicine, where it is too costly to develop algorithms to perform the needed tasks.

Unsupervised learning is a paradigm in machine learning where, in contrast to supervised learning and semi-supervised learning, algorithms learn patterns exclusively from unlabeled data.

<span class="mw-page-title-main">Computational learning theory</span> Theory of machine learning

In computer science, computational learning theory is a subfield of artificial intelligence devoted to studying the design and analysis of machine learning algorithms.

Ray Solomonoff was the inventor of algorithmic probability, his General Theory of Inductive Inference, and was a founder of algorithmic information theory. He was an originator of the branch of artificial intelligence based on machine learning, prediction and probability. He circulated the first report on non-semantic machine learning in 1956.

Solomonoff's theory of inductive inference is a mathematical theory of induction introduced by Ray Solomonoff, based on probability theory and theoretical computer science. In essence, Solomonoff's induction derives the posterior probability of any computable theory, given a sequence of observed data. This posterior probability is derived from Bayes' rule and some universal prior, that is, a prior that assigns a positive probability to any computable theory.

<span class="mw-page-title-main">Statistical learning theory</span> Framework for machine learning

Statistical learning theory is a framework for machine learning drawing from the fields of statistics and functional analysis. Statistical learning theory deals with the statistical inference problem of finding a predictive function based on data. Statistical learning theory has led to successful applications in fields such as computer vision, speech recognition, and bioinformatics.

Computational epistemology is a subdiscipline of formal epistemology that studies the intrinsic complexity of inductive problems for ideal and computationally bounded agents. In short, computational epistemology is to induction what recursion theory is to deduction. It has been applied to problems in philosophy of science.

<span class="mw-page-title-main">Kernel method</span> Class of algorithms for pattern analysis

In machine learning, kernel machines are a class of algorithms for pattern analysis, whose best known member is the support-vector machine (SVM). These methods involve using linear classifiers to solve nonlinear problems. The general task of pattern analysis is to find and study general types of relations in datasets. For many algorithms that solve these tasks, the data in raw representation have to be explicitly transformed into feature vector representations via a user-specified feature map: in contrast, kernel methods require only a user-specified kernel, i.e., a similarity function over all pairs of data points computed using inner products. The feature map in kernel machines is infinite dimensional but only requires a finite dimensional matrix from user-input according to the Representer theorem. Kernel machines are slow to compute for datasets larger than a couple of thousand examples without parallel processing.

<span class="mw-page-title-main">Conditional random field</span> Class of statistical modeling methods

Conditional random fields (CRFs) are a class of statistical modeling methods often applied in pattern recognition and machine learning and used for structured prediction. Whereas a classifier predicts a label for a single sample without considering "neighbouring" samples, a CRF can take context into account. To do so, the predictions are modelled as a graphical model, which represents the presence of dependencies between the predictions. What kind of graph is used depends on the application. For example, in natural language processing, "linear chain" CRFs are popular, for which each prediction is dependent only on its immediate neighbours. In image processing, the graph typically connects locations to nearby and/or similar locations to enforce that they receive similar predictions.

<span class="mw-page-title-main">Structured prediction</span> Supervised machine learning techniques

Structured prediction or structured (output) learning is an umbrella term for supervised machine learning techniques that involves predicting structured objects, rather than scalar discrete or real values.

A constrained conditional model (CCM) is a machine learning and inference framework that augments the learning of conditional models with declarative constraints. The constraint can be used as a way to incorporate expressive prior knowledge into the model and bias the assignments made by the learned model to satisfy these constraints. The framework can be used to support decisions in an expressive output space while maintaining modularity and tractability of training and inference.

<span class="mw-page-title-main">Active learning (machine learning)</span> Machine learning strategy

Active learning is a special case of machine learning in which a learning algorithm can interactively query a user to label new data points with the desired outputs. In statistics literature, it is sometimes also called optimal experimental design. The information source is also called teacher or oracle.

<span class="mw-page-title-main">Feature learning</span> Set of learning techniques in machine learning

In machine learning, feature learning or representation learning is a set of techniques that allows a system to automatically discover the representations needed for feature detection or classification from raw data. This replaces manual feature engineering and allows a machine to both learn the features and use them to perform a specific task.

<span class="mw-page-title-main">Manifold regularization</span>

In machine learning, Manifold regularization is a technique for using the shape of a dataset to constrain the functions that should be learned on that dataset. In many machine learning problems, the data to be learned do not cover the entire input space. For example, a facial recognition system may not need to classify any possible image, but only the subset of images that contain faces. The technique of manifold learning assumes that the relevant subset of data comes from a manifold, a mathematical structure with useful properties. The technique also assumes that the function to be learned is smooth: data with different labels are not likely to be close together, and so the labeling function should not change quickly in areas where there are likely to be many data points. Because of this assumption, a manifold regularization algorithm can use unlabeled data to inform where the learned function is allowed to change quickly and where it is not, using an extension of the technique of Tikhonov regularization. Manifold regularization algorithms can extend supervised learning algorithms in semi-supervised learning and transductive learning settings, where unlabeled data are available. The technique has been used for applications including medical imaging, geographical imaging, and object recognition.

<span class="mw-page-title-main">Glossary of artificial intelligence</span> List of definitions of terms and concepts commonly used in the study of artificial intelligence

This glossary of artificial intelligence is a list of definitions of terms and concepts relevant to the study of artificial intelligence, its sub-disciplines, and related fields. Related glossaries include Glossary of computer science, Glossary of robotics, and Glossary of machine vision.

<span class="mw-page-title-main">Outline of machine learning</span> Overview of and topical guide to machine learning

The following outline is provided as an overview of and topical guide to machine learning:

Weak supervision, also called semi-supervised learning, is a paradigm in machine learning, the relevance and notability of which increased with the advent of large language models due to large amount of data required to train them. It is characterized by using a combination of a small amount of human-labeled data, followed by a large amount of unlabeled data. In other words, the desired output values are provided only for a subset of the training data. The remaining data is unlabeled or imprecisely labeled. Intuitively, it can be seen as an exam and labeled data as sample problems that the teacher solves for the class as an aid in solving another set of problems. In the transductive setting, these unsolved problems act as exam questions. In the inductive setting, they become practice problems of the sort that will make up the exam. Technically, it could be viewed as performing clustering and then labeling the clusters with the labeled data, pushing the decision boundary away from high-density regions, or learning an underlying one-dimensional manifold where the data reside.

Conformal prediction (CP) is a machine learning framework for uncertainty quantification that can produce prediction regions (prediction intervals) for any underlying point predictor (where statistical, machine or deep learning) only assuming exchangeability of the data. CP works by computing nonconformity scores on previously labeled data, and using these to create prediction sets on a new (unlabeled) test data point. A transductive version of CP was first proposed in 1998 by Gammerman, Vovk, and Vapnik, and since, several variants of conformal prediction have been developed with different computational complexities, formal guarantees, and practical applications.

References

  1. Vapnik, Vladimir (2006). "Estimation of Dependences Based on Empirical Data". Information Science and Statistics: 477. doi:10.1007/0-387-34239-7. ISSN   1613-9011.