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**Supervised learning** is the machine learning task of learning a function that maps an input to an output based on example input-output pairs.^{ [1] } It infers a function from *labeled training data * consisting of a set of *training examples*.^{ [2] } In supervised learning, each example is a *pair* consisting of an input object (typically a vector) and a desired output value (also called the *supervisory signal*). A supervised learning algorithm analyzes the training data and produces an inferred function, which can be used for mapping new examples. An optimal scenario will allow for the algorithm to correctly determine the class labels for unseen instances. This requires the learning algorithm to generalize from the training data to unseen situations in a "reasonable" way (see inductive bias).

- Steps
- Algorithm choice
- Bias-variance tradeoff
- Function complexity and amount of training data
- Dimensionality of the input space
- Noise in the output values
- Other factors to consider
- Algorithms
- How supervised learning algorithms work
- Empirical risk minimization
- Structural risk minimization
- Generative training
- Generalizations
- Approaches and algorithms
- Applications
- General issues
- See also
- References
- External links

The parallel task in human and animal psychology is often referred to as concept learning.

In order to solve a given problem of supervised learning, one has to perform the following steps:

- Determine the type of training examples. Before doing anything else, the user should decide what kind of data is to be used as a training set. In the case of handwriting analysis, for example, this might be a single handwritten character, an entire handwritten word, or an entire line of handwriting.
- Gather a training set. The training set needs to be representative of the real-world use of the function. Thus, a set of input objects is gathered and corresponding outputs are also gathered, either from human experts or from measurements.
- Determine the input feature representation of the learned function. The accuracy of the learned function depends strongly on how the input object is represented. Typically, the input object is transformed into a feature vector, which contains a number of features that are descriptive of the object. The number of features should not be too large, because of the curse of dimensionality; but should contain enough information to accurately predict the output.
- Determine the structure of the learned function and corresponding learning algorithm. For example, the engineer may choose to use support vector machines or decision trees.
- Complete the design. Run the learning algorithm on the gathered training set. Some supervised learning algorithms require the user to determine certain control parameters. These parameters may be adjusted by optimizing performance on a subset (called a
*validation*set) of the training set, or via cross-validation. - Evaluate the accuracy of the learned function. After parameter adjustment and learning, the performance of the resulting function should be measured on a test set that is separate from the training set.

A wide range of supervised learning algorithms are available, each with its strengths and weaknesses. There is no single learning algorithm that works best on all supervised learning problems (see the No free lunch theorem).

There are four major issues to consider in supervised learning:

A first issue is the tradeoff between *bias* and *variance*.^{ [3] } Imagine that we have available several different, but equally good, training data sets. A learning algorithm is biased for a particular input if, when trained on each of these data sets, it is systematically incorrect when predicting the correct output for . A learning algorithm has high variance for a particular input if it predicts different output values when trained on different training sets. The prediction error of a learned classifier is related to the sum of the bias and the variance of the learning algorithm.^{ [4] } Generally, there is a tradeoff between bias and variance. A learning algorithm with low bias must be "flexible" so that it can fit the data well. But if the learning algorithm is too flexible, it will fit each training data set differently, and hence have high variance. A key aspect of many supervised learning methods is that they are able to adjust this tradeoff between bias and variance (either automatically or by providing a bias/variance parameter that the user can adjust).

The second issue is the amount of training data available relative to the complexity of the "true" function (classifier or regression function). If the true function is simple, then an "inflexible" learning algorithm with high bias and low variance will be able to learn it from a small amount of data. But if the true function is highly complex (e.g., because it involves complex interactions among many different input features and behaves differently in different parts of the input space), then the function will only be able to learn from a very large amount of training data and using a "flexible" learning algorithm with low bias and high variance.

A third issue is the dimensionality of the input space. If the input feature vectors have very high dimension, the learning problem can be difficult even if the true function only depends on a small number of those features. This is because the many "extra" dimensions can confuse the learning algorithm and cause it to have high variance. Hence, high input dimensional typically requires tuning the classifier to have low variance and high bias. In practice, if the engineer can manually remove irrelevant features from the input data, this is likely to improve the accuracy of the learned function. In addition, there are many algorithms for feature selection that seek to identify the relevant features and discard the irrelevant ones. This is an instance of the more general strategy of dimensionality reduction, which seeks to map the input data into a lower-dimensional space prior to running the supervised learning algorithm.

A fourth issue is the degree of noise in the desired output values (the supervisory target variables). If the desired output values are often incorrect (because of human error or sensor errors), then the learning algorithm should not attempt to find a function that exactly matches the training examples. Attempting to fit the data too carefully leads to overfitting. You can overfit even when there are no measurement errors (stochastic noise) if the function you are trying to learn is too complex for your learning model. In such a situation, the part of the target function that cannot be modeled "corrupts" your training data - this phenomenon has been called deterministic noise. When either type of noise is present, it is better to go with a higher bias, lower variance estimator.

In practice, there are several approaches to alleviate noise in the output values such as early stopping to prevent overfitting as well as detecting and removing the noisy training examples prior to training the supervised learning algorithm. There are several algorithms that identify noisy training examples and removing the suspected noisy training examples prior to training has decreased generalization error with statistical significance.^{ [5] }^{ [6] }

Other factors to consider when choosing and applying a learning algorithm include the following:

- Heterogeneity of the data. If the feature vectors include features of many different kinds (discrete, discrete ordered, counts, continuous values), some algorithms are easier to apply than others. Many algorithms, including Support Vector Machines, linear regression, logistic regression, neural networks, and nearest neighbor methods, require that the input features be numerical and scaled to similar ranges (e.g., to the [-1,1] interval). Methods that employ a distance function, such as nearest neighbor methods and support vector machines with Gaussian kernels, are particularly sensitive to this. An advantage of decision trees is that they easily handle heterogeneous data.
- Redundancy in the data. If the input features contain redundant information (e.g., highly correlated features), some learning algorithms (e.g., linear regression, logistic regression, and distance based methods) will perform poorly because of numerical instabilities. These problems can often be solved by imposing some form of regularization.
- Presence of interactions and non-linearities. If each of the features makes an independent contribution to the output, then algorithms based on linear functions (e.g., linear regression, logistic regression, Support Vector Machines, naive Bayes) and distance functions (e.g., nearest neighbor methods, support vector machines with Gaussian kernels) generally perform well. However, if there are complex interactions among features, then algorithms such as decision trees and neural networks work better, because they are specifically designed to discover these interactions. Linear methods can also be applied, but the engineer must manually specify the interactions when using them.

When considering a new application, the engineer can compare multiple learning algorithms and experimentally determine which one works best on the problem at hand (see cross validation). Tuning the performance of a learning algorithm can be very time-consuming. Given fixed resources, it is often better to spend more time collecting additional training data and more informative features than it is to spend extra time tuning the learning algorithms.

The most widely used learning algorithms are:

Given a set of training examples of the form such that is the feature vector of the i-th example and is its label (i.e., class), a learning algorithm seeks a function , where is the input space and is the output space. The function is an element of some space of possible functions , usually called the *hypothesis space*. It is sometimes convenient to represent using a scoring function such that is defined as returning the value that gives the highest score: . Let denote the space of scoring functions.

Although and can be any space of functions, many learning algorithms are probabilistic models where takes the form of a conditional probability model , or takes the form of a joint probability model . For example, naive Bayes and linear discriminant analysis are joint probability models, whereas logistic regression is a conditional probability model.

There are two basic approaches to choosing or : empirical risk minimization and structural risk minimization.^{ [7] } Empirical risk minimization seeks the function that best fits the training data. Structural risk minimization includes a *penalty function* that controls the bias/variance tradeoff.

In both cases, it is assumed that the training set consists of a sample of independent and identically distributed pairs, . In order to measure how well a function fits the training data, a loss function is defined. For training example , the loss of predicting the value is .

The *risk* of function is defined as the expected loss of . This can be estimated from the training data as

- .

In empirical risk minimization, the supervised learning algorithm seeks the function that minimizes . Hence, a supervised learning algorithm can be constructed by applying an optimization algorithm to find .

When is a conditional probability distribution and the loss function is the negative log likelihood: , then empirical risk minimization is equivalent to maximum likelihood estimation.

When contains many candidate functions or the training set is not sufficiently large, empirical risk minimization leads to high variance and poor generalization. The learning algorithm is able to memorize the training examples without generalizing well. This is called overfitting.

Structural risk minimization seeks to prevent overfitting by incorporating a regularization penalty into the optimization. The regularization penalty can be viewed as implementing a form of Occam's razor that prefers simpler functions over more complex ones.

A wide variety of penalties have been employed that correspond to different definitions of complexity. For example, consider the case where the function is a linear function of the form

- .

A popular regularization penalty is , which is the squared Euclidean norm of the weights, also known as the norm. Other norms include the norm, , and the norm, which is the number of non-zero s. The penalty will be denoted by .

The supervised learning optimization problem is to find the function that minimizes

The parameter controls the bias-variance tradeoff. When , this gives empirical risk minimization with low bias and high variance. When is large, the learning algorithm will have high bias and low variance. The value of can be chosen empirically via cross validation.

The complexity penalty has a Bayesian interpretation as the negative log prior probability of , , in which case is the posterior probabability of .

The training methods described above are *discriminative training* methods, because they seek to find a function that discriminates well between the different output values (see discriminative model). For the special case where is a joint probability distribution and the loss function is the negative log likelihood a risk minimization algorithm is said to perform *generative training*, because can be regarded as a generative model that explains how the data were generated. Generative training algorithms are often simpler and more computationally efficient than discriminative training algorithms. In some cases, the solution can be computed in closed form as in naive Bayes and linear discriminant analysis.

There are several ways in which the standard supervised learning problem can be generalized:

- Semi-supervised learning: In this setting, the desired output values are provided only for a subset of the training data. The remaining data is unlabeled.
- Weak supervision: In this setting, noisy, limited, or imprecise sources are used to provide supervision signal for labeling training data.
- Active learning: Instead of assuming that all of the training examples are given at the start, active learning algorithms interactively collect new examples, typically by making queries to a human user. Often, the queries are based on unlabeled data, which is a scenario that combines semi-supervised learning with active learning.
- Structured prediction: When the desired output value is a complex object, such as a parse tree or a labeled graph, then standard methods must be extended.
- Learning to rank: When the input is a set of objects and the desired output is a ranking of those objects, then again the standard methods must be extended.

- Analytical learning
- Artificial neural network
- Backpropagation
- Boosting (meta-algorithm)
- Bayesian statistics
- Case-based reasoning
- Decision tree learning
- Inductive logic programming
- Gaussian process regression
- Genetic Programming
- Group method of data handling
- Kernel estimators
- Learning Automata
- Learning Classifier Systems
- Minimum message length (decision trees, decision graphs, etc.)
- Multilinear subspace learning
- Naive Bayes classifier
- Maximum entropy classifier
- Conditional random field
- Nearest Neighbor Algorithm
- Probably approximately correct learning (PAC) learning
- Ripple down rules, a knowledge acquisition methodology
- Symbolic machine learning algorithms
- Subsymbolic machine learning algorithms
- Support vector machines
- Minimum Complexity Machines (MCM)
- Random Forests
- Ensembles of Classifiers
- Ordinal classification
- Data Pre-processing
- Handling imbalanced datasets
- Statistical relational learning
- Proaftn, a multicriteria classification algorithm

- Bioinformatics
- Cheminformatics
- Database marketing
- Handwriting recognition
- Information retrieval
- Information extraction
- Object recognition in computer vision
- Optical character recognition
- Spam detection
- Pattern recognition
- Speech recognition
- Supervised learning is a special case of Downward causation in biological systems

In machine learning, **support-vector machines** are supervised learning models with associated learning algorithms that analyze data used for classification and regression analysis. The Support Vector Machine (SVM) algorithm is a popular machine learning tool that offers solutions for both classification and regression problems. Developed at AT&T Bell Laboratories by Vapnik with colleagues, it presents one of the most robust prediction methods, based on the statistical learning framework or VC theory proposed by Vapnik and Chervonekis (1974) and Vapnik. Given a set of training examples, each marked as belonging to one or the other 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. An SVM model is a representation of the examples as points in space, mapped so that the examples of the separate categories are divided by a clear gap that is as wide as possible. New examples are then mapped into that same space and predicted to belong to a category based on the side of the gap on which they fall.

In the field of machine learning, the goal of statistical classification is to use an object's characteristics to identify which class it belongs to. A **linear classifier** achieves this by making a classification decision based on the value of a linear combination of the characteristics. An object's characteristics are also known as feature values and are typically presented to the machine in a vector called a feature vector. Such classifiers work well for practical problems such as document classification, and more generally for problems with many variables (features), reaching accuracy levels comparable to non-linear classifiers while taking less time to train and use.

**Pattern recognition** is the automated recognition of patterns and regularities in data. It has applications in statistical data analysis, signal processing, image analysis, information retrieval, bioinformatics, data compression, computer graphics and machine learning. Pattern recognition has its origins in statistics and engineering; some modern approaches to pattern recognition include the use of machine learning, due to the increased availability of big data and a new abundance of processing power. However, these activities can be viewed as two facets of the same field of application, and together they have undergone substantial development over the past few decades. A modern definition of pattern recognition is:

The field of pattern recognition is concerned with the automatic discovery of regularities in data through the use of computer algorithms and with the use of these regularities to take actions such as classifying the data into different categories.

In machine learning, the **perceptron** is an algorithm for supervised learning of binary classifiers. A binary classifier is a function which can decide whether or not an input, represented by a vector of numbers, belongs to some specific class. It is a type of linear classifier, i.e. a classification algorithm that makes its predictions based on a linear predictor function combining a set of weights with the feature vector.

In statistics, **overfitting** is "the production of an analysis that corresponds too closely or exactly to a particular set of data, and may therefore fail to fit additional data or predict future observations reliably". An **overfitted model** is a statistical model that contains more parameters than can be justified by the data. The essence of overfitting is to have unknowingly extracted some of the residual variation as if that variation represented underlying model structure.

In machine learning, **early stopping** is a form of regularization used to avoid overfitting when training a learner with an iterative method, such as gradient descent. Such methods update the learner so as to make it better fit the training data with each iteration. Up to a point, this improves the learner's performance on data outside of the training set. Past that point, however, improving the learner's fit to the training data comes at the expense of increased generalization error. Early stopping rules provide guidance as to how many iterations can be run before the learner begins to over-fit. Early stopping rules have been employed in many different machine learning methods, with varying amounts of theoretical foundation.

In statistics, **nonlinear regression** is a form of regression analysis in which observational data are modeled by a function which is a nonlinear combination of the model parameters and depends on one or more independent variables. The data are fitted by a method of successive approximations.

**Statistical learning theory** is a framework for machine learning drawing from the fields of statistics and functional analysis. Statistical learning theory deals with the 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.

In machine learning, **backpropagation** is a widely used algorithm in training feedforward neural networks for supervised learning. Generalizations of backpropagation exist for other artificial neural networks (ANNs), and for functions generally – a class of algorithms referred to generically as "backpropagation". In fitting a neural network, backpropagation computes the gradient of the loss function with respect to the weights of the network for a single input–output example, and does so efficiently, unlike a naive direct computation of the gradient with respect to each weight individually. This efficiency makes it feasible to use gradient methods for training multilayer networks, updating weights to minimize loss; gradient descent, or variants such as stochastic gradient descent, are commonly used. The backpropagation algorithm works by computing the gradient of the loss function with respect to each weight by the chain rule, computing the gradient one layer at a time, iterating backward from the last layer to avoid redundant calculations of intermediate terms in the chain rule; this is an example of dynamic programming.

In mathematics, statistics, and computer science, particularly in machine learning and inverse problems, **regularization** is the process of adding information in order to solve an ill-posed problem or to prevent overfitting.

A **multilayer perceptron** (MLP) is a class of feedforward artificial neural network (ANN). The term MLP is used ambiguously, sometimes loosely to *any* feedforward ANN, sometimes strictly to refer to networks composed of multiple layers of perceptrons ; see § Terminology. Multilayer perceptrons are sometimes colloquially referred to as "vanilla" neural networks, especially when they have a single hidden layer.

In supervised learning applications in machine learning and statistical learning theory, **generalization error** is a measure of how accurately an algorithm is able to predict outcome values for previously unseen data. Because learning algorithms are evaluated on finite samples, the evaluation of a learning algorithm may be sensitive to sampling error. As a result, measurements of prediction error on the current data may not provide much information about predictive ability on new data. Generalization error can be minimized by avoiding overfitting in the learning algorithm. The performance of a machine learning algorithm is measured by plots of the generalization error values through the learning process, which are called learning curves.

**Semi-supervised learning** is an approach to machine learning that combines a small amount of labeled data with a large amount of unlabeled data during training. Semi-supervised learning falls between unsupervised learning and supervised learning.

In computer science, **online machine learning** is a method of machine learning in which data becomes available in a sequential order and is used to update the best predictor for future data at each step, as opposed to batch learning techniques which generate the best predictor by learning on the entire training data set at once. Online learning is a common technique used in areas of machine learning where it is computationally infeasible to train over the entire dataset, requiring the need of out-of-core algorithms. It is also used in situations where it is necessary for the algorithm to dynamically adapt to new patterns in the data, or when the data itself is generated as a function of time, e.g., stock price prediction. Online learning algorithms may be prone to catastrophic interference, a problem that can be addressed by incremental learning approaches.

In statistics, **projection pursuit regression (PPR)** is a statistical model developed by Jerome H. Friedman and Werner Stuetzle which is an extension of additive models. This model adapts the additive models in that it first projects the data matrix of explanatory variables in the optimal direction before applying smoothing functions to these explanatory variables.

**Gradient boosting** is a machine learning technique for regression and classification problems, which produces a prediction model in the form of an ensemble of weak prediction models, typically decision trees. It builds the model in a stage-wise fashion like other boosting methods do, and it generalizes them by allowing optimization of an arbitrary differentiable loss function.

In statistics and machine learning, the **bias–variance tradeoff** is the property of a set of predictive models whereby models with a lower bias in parameter estimation have a higher variance of the parameter estimates across samples, and vice versa. The **bias–variance dilemma** or **bias–variance problem** is the conflict in trying to simultaneously minimize these two sources of error that prevent supervised learning algorithms from generalizing beyond their training set:

In machine learning, a **probabilistic classifier** is a classifier that is able to predict, given an observation of an input, a probability distribution over a set of classes, rather than only outputting the most likely class that the observation should belong to. Probabilistic classifiers provide classification that can be useful in its own right or when combining classifiers into ensembles.

**Multiple kernel learning** refers to a set of machine learning methods that use a predefined set of kernels and learn an optimal linear or non-linear combination of kernels as part of the algorithm. Reasons to use multiple kernel learning include a) the ability to select for an optimal kernel and parameters from a larger set of kernels, reducing bias due to kernel selection while allowing for more automated machine learning methods, and b) combining data from different sources that have different notions of similarity and thus require different kernels. Instead of creating a new kernel, multiple kernel algorithms can be used to combine kernels already established for each individual data source.

In statistics, **linear regression** is a linear approach to modeling the relationship between a scalar response and one or more explanatory variables. The case of one explanatory variable is called simple linear regression. For more than one explanatory variable, the process is called **multiple linear regression**. This term is distinct from multivariate linear regression, where multiple correlated dependent variables are predicted, rather than a single scalar variable.

- ↑ Stuart J. Russell, Peter Norvig (2010)
*Artificial Intelligence: A Modern Approach, Third Edition*, Prentice Hall ISBN 9780136042594. - ↑ Mehryar Mohri, Afshin Rostamizadeh, Ameet Talwalkar (2012)
*Foundations of Machine Learning*, The MIT Press ISBN 9780262018258. - ↑ S. Geman, E. Bienenstock, and R. Doursat (1992). Neural networks and the bias/variance dilemma. Neural Computation 4, 1–58.
- ↑ G. James (2003) Variance and Bias for General Loss Functions, Machine Learning 51, 115-135. (http://www-bcf.usc.edu/~gareth/research/bv.pdf)
- ↑ C.E. Brodely and M.A. Friedl (1999). Identifying and Eliminating Mislabeled Training Instances, Journal of Artificial Intelligence Research 11, 131-167. (http://jair.org/media/606/live-606-1803-jair.pdf)
- ↑ M.R. Smith and T. Martinez (2011). "Improving Classification Accuracy by Identifying and Removing Instances that Should Be Misclassified".
*Proceedings of International Joint Conference on Neural Networks (IJCNN 2011)*. pp. 2690–2697. CiteSeerX 10.1.1.221.1371 . doi:10.1109/IJCNN.2011.6033571. - ↑ Vapnik, V. N. The Nature of Statistical Learning Theory (2nd Ed.), Springer Verlag, 2000.

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Images, videos and audio are available under their respective licenses.