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 ${\displaystyle x}$ if, when trained on each of these data sets, it is systematically incorrect when predicting the correct output for ${\displaystyle x}$. A learning algorithm has high variance for a particular input ${\displaystyle x}$ 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.There is a clear demarcation between the input and the desired output.

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:

• Support Vector Machines
• linear regression
• logistic regression
• naive Bayes
• linear discriminant analysis
• decision trees
• k-nearest neighbor algorithm
• Neural Networks (Multilayer perceptron)
• Similarity learning

In empirical risk minimization, the supervised learning algorithm seeks the function ${\displaystyle g}$ that minimizes ${\displaystyle R(g)}$. Hence, a supervised learning algorithm can be constructed by applying an optimization algorithm to find ${\displaystyle g}$.

When ${\displaystyle g}$ is a conditional probability distribution ${\displaystyle P(y|x)}$ and the loss function is the negative log likelihood: ${\displaystyle L(y,{\hat {y}})=-\log P(y|x)}$, then empirical risk minimization is equivalent to maximum likelihood estimation.

When ${\displaystyle G}$ 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 ${\displaystyle g}$ is a linear function of the form

${\displaystyle g(x)=\sum _{j=1}^{d}\beta _{j}x_{j}}$.

A popular regularization penalty is ${\displaystyle \sum _{j}\beta _{j}^{2}}$, which is the squared Euclidean norm of the weights, also known as the ${\displaystyle L_{2}}$ norm. Other norms include the ${\displaystyle L_{1}}$ norm, ${\displaystyle \sum _{j}|\beta _{j}|}$, and the ${\displaystyle L_{0}}$ norm, which is the number of non-zero ${\displaystyle \beta _{j}}$s. The penalty will be denoted by ${\displaystyle C(g)}$.

The supervised learning optimization problem is to find the function ${\displaystyle g}$ that minimizes

${\displaystyle J(g)=R_{emp}(g)+\lambda C(g).}$

The parameter ${\displaystyle \lambda }$ controls the bias-variance tradeoff. When ${\displaystyle \lambda =0}$, this gives empirical risk minimization with low bias and high variance. When ${\displaystyle \lambda }$ is large, the learning algorithm will have high bias and low variance. The value of ${\displaystyle \lambda }$ can be chosen empirically via cross validation.

The complexity penalty has a Bayesian interpretation as the negative log prior probability of ${\displaystyle g}$, ${\displaystyle -\log P(g)}$, in which case ${\displaystyle J(g)}$ is the posterior probabability of ${\displaystyle g}$.