In
machine learning
Machine learning (ML) is a field of inquiry devoted to understanding and building methods that 'learn', that is, methods that leverage data to improve performance on some set of tasks. It is seen as a part of artificial intelligence.
Machine ...
, support vector machines (SVMs, also support vector networks
) are
supervised learning
Supervised learning (SL) is a machine learning paradigm for problems where the available data consists of labelled examples, meaning that each data point contains features (covariates) and an associated label. The goal of supervised learning alg ...
models with associated learning
algorithm
In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing ...
s that analyze data for
classification and
regression analysis
In statistical modeling, regression analysis is a set of statistical processes for estimating the relationships between a dependent variable (often called the 'outcome' or 'response' variable, or a 'label' in machine learning parlance) and one ...
. Developed at
AT&T Bell Laboratories by
Vladimir Vapnik
Vladimir Naumovich Vapnik (russian: Владимир Наумович Вапник; born 6 December 1936) is one of the main developers of the Vapnik–Chervonenkis theory of statistical learning, and the co-inventor of the support-vector machin ...
with colleagues (Boser et al., 1992,
Guyon et al., 1993,
Cortes
Cortes, Cortés, Cortês, Corts, or Cortès may refer to:
People
* Cortes (surname), including a list of people with the name
** Hernán Cortés (1485–1547), a Spanish conquistador
Places
* Cortes, Navarre, a village in the South border of ...
and
Vapnik
Vladimir Naumovich Vapnik (russian: Владимир Наумович Вапник; born 6 December 1936) is one of the main developers of the Vapnik–Chervonenkis theory of statistical learning, and the co-inventor of the support-vector machin ...
, 1995,
Vapnik et al., 1997) SVMs are one of the most robust prediction methods, being based on statistical learning frameworks or
VC theory
VC may refer to:
Military decorations
* Victoria Cross, a military decoration awarded by the United Kingdom and also by certain Commonwealth nations
** Victoria Cross for Australia
** Victoria Cross (Canada)
** Victoria Cross for New Zealand
* Vic ...
proposed by Vapnik (1982, 1995) 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
Binary may refer to:
Science and technology Mathematics
* Binary number, a representation of numbers using only two digits (0 and 1)
* Binary function, a function that takes two arguments
* Binary operation, a mathematical operation that ta ...
linear classifier
In the field of machine learning, the goal of statistical classification is to use an object's characteristics to identify which class (or group) it belongs to. A linear classifier achieves this by making a classification decision based on the val ...
(although methods such as
Platt scaling
In machine learning, Platt scaling or Platt calibration is a way of transforming the outputs of a classification model into a probability distribution over classes. The method was invented by John Platt in the context of support vector machine ...
exist to use SVM in a probabilistic classification setting). 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.
In addition to performing
linear classification, SVMs can efficiently perform a non-linear classification using what is called the
kernel trick
In machine learning, kernel machines are a class of algorithms for pattern analysis, whose best known member is the support-vector machine (SVM). The general task of pattern analysis is to find and study general types of relations (for example ...
, implicitly mapping their inputs into high-dimensional feature spaces.
When data are unlabelled, supervised learning is not possible, and an
unsupervised learning
Unsupervised learning is a type of algorithm that learns patterns from untagged data. The hope is that through mimicry, which is an important mode of learning in people, the machine is forced to build a concise representation of its world and t ...
approach is required, which attempts to find natural
clustering of the data to groups, and then map new data to these formed groups. The support vector clustering
algorithm, created by
Hava Siegelmann
Hava Siegelmann is a professor of computer science. Her academic position is in the school of Computer Science and the Program of Neuroscience and Behavior at the University of Massachusetts Amherst; she is the director of the school's Biologica ...
and
Vladimir Vapnik
Vladimir Naumovich Vapnik (russian: Владимир Наумович Вапник; born 6 December 1936) is one of the main developers of the Vapnik–Chervonenkis theory of statistical learning, and the co-inventor of the support-vector machin ...
, applies the statistics of support vectors, developed in the support vector machines algorithm, to categorize unlabeled data.
Motivation
Classifying data is a common task in
machine learning
Machine learning (ML) is a field of inquiry devoted to understanding and building methods that 'learn', that is, methods that leverage data to improve performance on some set of tasks. It is seen as a part of artificial intelligence.
Machine ...
.
Suppose some given data points each belong to one of two classes, and the goal is to decide which class a ''new''
data point
In statistics, a unit of observation is the unit described by the data that one analyzes. A study may treat groups as a unit of observation with a country as the unit of analysis, drawing conclusions on group characteristics from data collected at ...
will be in. In the case of support vector machines, a data point is viewed as a
-dimensional vector (a list of
numbers), and we want to know whether we can separate such points with a
-dimensional
hyperplane. This is called a
linear classifier
In the field of machine learning, the goal of statistical classification is to use an object's characteristics to identify which class (or group) it belongs to. A linear classifier achieves this by making a classification decision based on the val ...
. There are many hyperplanes that might classify the data. One reasonable choice as the best hyperplane is the one that represents the largest separation, or
margin
Margin may refer to:
Physical or graphical edges
*Margin (typography), the white space that surrounds the content of a page
*Continental margin, the zone of the ocean floor that separates the thin oceanic crust from thick continental crust
*Leaf ...
, between the two classes. So we choose the hyperplane so that the distance from it to the nearest data point on each side is maximized. If such a hyperplane exists, it is known as the ''
maximum-margin hyperplane
In geometry, the hyperplane separation theorem is a theorem about disjoint convex sets in ''n''-dimensional Euclidean space. There are several rather similar versions. In one version of the theorem, if both these sets are closed and at least one ...
'' and the linear classifier it defines is known as a ''maximum-
margin classifier In machine learning, a margin classifier is a classifier which is able to give an associated distance from the decision boundary for each example. For instance, if a linear classifier (e.g. perceptron or linear discriminant analysis) is used, the ...
''; or equivalently, the ''
perceptron of optimal stability''.
More formally, a support vector machine constructs a
hyperplane or set of hyperplanes in a high or infinite-dimensional space, which can be used for
classification,
regression, or other tasks like outliers detection. Intuitively, a good separation is achieved by the hyperplane that has the largest distance to the nearest training-data point of any class (so-called functional margin), since in general the larger the margin, the lower the
generalization error of the classifier.
Whereas the original problem may be stated in a finite-dimensional space, it often happens that the sets to discriminate are not
linearly separable
In Euclidean geometry, linear separability is a property of two sets of point (geometry), points. This is most easily visualized in two dimensions (the Euclidean plane) by thinking of one set of points as being colored blue and the other set of poi ...
in that space. For this reason, it was proposed
that the original finite-dimensional space be mapped into a much higher-dimensional space, presumably making the separation easier in that space. To keep the computational load reasonable, the mappings used by SVM schemes are designed to ensure that
dot product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an alge ...
s of pairs of input data vectors may be computed easily in terms of the variables in the original space, by defining them in terms of a
kernel function In operator theory, a branch of mathematics, a positive-definite kernel is a generalization of a positive-definite function or a positive-definite matrix. It was first introduced by James Mercer in the early 20th century, in the context of solving ...
selected to suit the problem. The hyperplanes in the higher-dimensional space are defined as the set of points whose dot product with a vector in that space is constant, where such a set of vectors is an orthogonal (and thus minimal) set of vectors that defines a hyperplane. The vectors defining the hyperplanes can be chosen to be linear combinations with parameters
of images of
feature vector
In machine learning and pattern recognition, a feature is an individual measurable property or characteristic of a phenomenon. Choosing informative, discriminating and independent features is a crucial element of effective algorithms in pattern r ...
s
that occur in the data base. With this choice of a hyperplane, the points
in the
feature space
In machine learning and pattern recognition, a feature is an individual measurable property or characteristic of a phenomenon. Choosing informative, discriminating and independent features is a crucial element of effective algorithms in pattern r ...
that are mapped into the hyperplane are defined by the relation
Note that if
becomes small as
grows further away from
, each term in the sum measures the degree of closeness of the test point
to the corresponding data base point
. In this way, the sum of kernels above can be used to measure the relative nearness of each test point to the data points originating in one or the other of the sets to be discriminated. Note the fact that the set of points
mapped into any hyperplane can be quite convoluted as a result, allowing much more complex discrimination between sets that are not convex at all in the original space.
Applications
SVMs can be used to solve various real-world problems:
* SVMs are helpful in
text and hypertext categorization, as their application can significantly reduce the need for labeled training instances in both the standard inductive and
transductive settings. Some methods for
shallow semantic parsing In natural language processing, semantic role labeling (also called Semantic parsing, shallow semantic parsing or slot-filling) is the process that assigns labels to words or phrases in a sentence that indicates their semantic role in the sentence, ...
are based on support vector machines.
*
Classification of images can also be performed using SVMs. Experimental results show that SVMs achieve significantly higher search accuracy than traditional query refinement schemes after just three to four rounds of relevance feedback. This is also true for
image segmentation
In digital image processing and computer vision, image segmentation is the process of partitioning a digital image into multiple image segments, also known as image regions or image objects ( sets of pixels). The goal of segmentation is to simpli ...
systems, including those using a modified version SVM that uses the privileged approach as suggested by Vapnik.
* Classification of satellite data like
SAR data using supervised SVM.
* Hand-written characters can be
recognized using SVM.
* The SVM algorithm has been widely applied in the biological and other sciences. They have been used to classify proteins with up to 90% of the compounds classified correctly.
Permutation test
A permutation test (also called re-randomization test) is an exact statistical hypothesis test making use of the proof by contradiction.
A permutation test involves two or more samples. The null hypothesis is that all samples come from the same di ...
s based on SVM weights have been suggested as a mechanism for interpretation of SVM models. Support vector machine weights have also been used to interpret SVM models in the past. Posthoc interpretation of support vector machine models in order to identify features used by the model to make predictions is a relatively new area of research with special significance in the biological sciences.
History
The original SVM algorithm was invented by
Vladimir N. Vapnik and
Alexey Ya. Chervonenkis in 1964. In 1992, Bernhard Boser,
Isabelle Guyon and
Vladimir Vapnik
Vladimir Naumovich Vapnik (russian: Владимир Наумович Вапник; born 6 December 1936) is one of the main developers of the Vapnik–Chervonenkis theory of statistical learning, and the co-inventor of the support-vector machin ...
suggested a way to create nonlinear classifiers by applying the
kernel trick
In machine learning, kernel machines are a class of algorithms for pattern analysis, whose best known member is the support-vector machine (SVM). The general task of pattern analysis is to find and study general types of relations (for example ...
to maximum-margin hyperplanes.
The "soft margin" incarnation, as is commonly used in software packages, was proposed by
Corinna Cortes
Corinna Cortes is a Danish computer scientist known for her contributions to machine learning. She is currently the Head of Google Research, New York. Cortes is a recipient of the Paris Kanellakis Theory and Practice Award for her work on theoreti ...
and Vapnik in 1993 and published in 1995.
Linear SVM
We are given a training dataset of
points of the form
where the
are either 1 or −1, each indicating the class to which the point
belongs. Each
is a
-dimensional
real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010) ...
vector. We want to find the "maximum-margin hyperplane" that divides the group of points
for which
from the group of points for which
, which is defined so that the distance between the hyperplane and the nearest point
from either group is maximized.
Any
hyperplane can be written as the set of points
satisfying
where
is the (not necessarily normalized)
normal vector to the hyperplane. This is much like
Hesse normal form
The Hesse normal form named after Otto Hesse, is an equation used in analytic geometry, and describes a line in \mathbb^2 or a plane in Euclidean space \mathbb^3 or a hyperplane in higher dimensions.John Vince: ''Geometry for Computer Graphics''. ...
, except that
is not necessarily a unit vector. The parameter
determines the offset of the hyperplane from the origin along the normal vector
.
Hard-margin
If the training data is
linearly separable
In Euclidean geometry, linear separability is a property of two sets of point (geometry), points. This is most easily visualized in two dimensions (the Euclidean plane) by thinking of one set of points as being colored blue and the other set of poi ...
, we can select two parallel hyperplanes that separate the two classes of data, so that the distance between them is as large as possible. The region bounded by these two hyperplanes is called the "margin", and the maximum-margin hyperplane is the hyperplane that lies halfway between them. With a normalized or standardized dataset, these hyperplanes can be described by the equations
:
(anything on or above this boundary is of one class, with label 1)
and
:
(anything on or below this boundary is of the other class, with label −1).
Geometrically, the distance between these two hyperplanes is
, so to maximize the distance between the planes we want to minimize
. The distance is computed using the
distance from a point to a plane In Euclidean space, the distance from a point to a plane is the distance between a given point and its orthogonal projection on the plane, the perpendicular distance to the nearest point on the plane.
It can be found starting with a change of varia ...
equation. We also have to prevent data points from falling into the margin, we add the following constraint: for each
either
or
These constraints state that each data point must lie on the correct side of the margin.
This can be rewritten as
We can put this together to get the optimization problem:
The
and
that solve this problem determine our classifier,
where
is the
sign function.
An important consequence of this geometric description is that the max-margin hyperplane is completely determined by those
that lie nearest to it. These
are called ''support vectors''.
Soft-margin
To extend SVM to cases in which the data are not linearly separable, the ''
hinge loss
In machine learning, the hinge loss is a loss function used for training classifiers. The hinge loss is used for "maximum-margin" classification, most notably for support vector machines (SVMs).
For an intended output and a classifier score , th ...
'' function is helpful
Note that
is the ''i''-th target (i.e., in this case, 1 or −1), and
is the ''i''-th output.
This function is zero if the constraint in is satisfied, in other words, if
lies on the correct side of the margin. For data on the wrong side of the margin, the function's value is proportional to the distance from the margin.
The goal of the optimization then is to minimize
where the parameter
determines the trade-off between increasing the margin size and ensuring that the
lie on the correct side of the margin. By deconstructing the hinge loss, this optimization problem can be massaged into the following:
Thus, for large values of
, it will behave similar to the hard-margin SVM, if the input data are linearly classifiable, but will still learn if a classification rule is viable or not. (
is inversely related to
, e.g. in ''
LIBSVM
LIBSVM and LIBLINEAR are two popular open source machine learning libraries, both developed at the National Taiwan University and both written in C++ though with a C API. LIBSVM implements the Sequential minimal optimization (SMO) algorithm ...
''.)
Nonlinear Kernels
The original maximum-margin hyperplane algorithm proposed by Vapnik in 1963 constructed a
linear classifier
In the field of machine learning, the goal of statistical classification is to use an object's characteristics to identify which class (or group) it belongs to. A linear classifier achieves this by making a classification decision based on the val ...
. However, in 1992,
Bernhard Boser,
Isabelle Guyon and
Vladimir Vapnik
Vladimir Naumovich Vapnik (russian: Владимир Наумович Вапник; born 6 December 1936) is one of the main developers of the Vapnik–Chervonenkis theory of statistical learning, and the co-inventor of the support-vector machin ...
suggested a way to create nonlinear classifiers by applying the
kernel trick
In machine learning, kernel machines are a class of algorithms for pattern analysis, whose best known member is the support-vector machine (SVM). The general task of pattern analysis is to find and study general types of relations (for example ...
(originally proposed by Aizerman et al.) to maximum-margin hyperplanes.
The resulting algorithm is formally similar, except that every
dot product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an alge ...
is replaced by a nonlinear
kernel
Kernel may refer to:
Computing
* Kernel (operating system), the central component of most operating systems
* Kernel (image processing), a matrix used for image convolution
* Compute kernel, in GPGPU programming
* Kernel method, in machine learn ...
function. This allows the algorithm to fit the maximum-margin hyperplane in a transformed
feature space
In machine learning and pattern recognition, a feature is an individual measurable property or characteristic of a phenomenon. Choosing informative, discriminating and independent features is a crucial element of effective algorithms in pattern r ...
. The transformation may be nonlinear and the transformed space high-dimensional; although the classifier is a hyperplane in the transformed feature space, it may be nonlinear in the original input space.
It is noteworthy that working in a higher-dimensional feature space increases the
generalization error of support vector machines, although given enough samples the algorithm still performs well.
Some common kernels include:
*
Polynomial (homogeneous):
. Particularly, when
, this becomes the linear kernel.
*
Polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example ...
(inhomogeneous):
.
* Gaussian
radial basis function A radial basis function (RBF) is a real-valued function \varphi whose value depends only on the distance between the input and some fixed point, either the origin, so that \varphi(\mathbf) = \hat\varphi(\left\, \mathbf\right\, ), or some other fixed ...
:
for
. Sometimes parametrized using
.
*
Sigmoid function
A sigmoid function is a mathematical function having a characteristic "S"-shaped curve or sigmoid curve.
A common example of a sigmoid function is the logistic function shown in the first figure and defined by the formula:
:S(x) = \frac = \ ...
(
Hyperbolic tangent
In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points form a circle with a unit radius, the points form the right half of the u ...
):
for some (not every)
and
.
The kernel is related to the transform
by the equation
. The value is also in the transformed space, with
. Dot products with for classification can again be computed by the kernel trick, i.e.
.
Computing the SVM classifier
Computing the (soft-margin) SVM classifier amounts to minimizing an expression of the form
We focus on the soft-margin classifier since, as noted above, choosing a sufficiently small value for
yields the hard-margin classifier for linearly classifiable input data. The classical approach, which involves reducing to a
quadratic programming
Quadratic programming (QP) is the process of solving certain mathematical optimization problems involving quadratic functions. Specifically, one seeks to optimize (minimize or maximize) a multivariate quadratic function subject to linear constr ...
problem, is detailed below. Then, more recent approaches such as sub-gradient descent and coordinate descent will be discussed.
Primal
Minimizing can be rewritten as a constrained optimization problem with a differentiable objective function in the following way.
For each
we introduce a variable
. Note that
is the smallest nonnegative number satisfying
Thus we can rewrite the optimization problem as follows
This is called the ''primal'' problem.
Dual
By solving for the
Lagrangian dual of the above problem, one obtains the simplified problem
This is called the ''dual'' problem. Since the dual maximization problem is a quadratic function of the
subject to linear constraints, it is efficiently solvable by
quadratic programming
Quadratic programming (QP) is the process of solving certain mathematical optimization problems involving quadratic functions. Specifically, one seeks to optimize (minimize or maximize) a multivariate quadratic function subject to linear constr ...
algorithms.
Here, the variables
are defined such that
Moreover,
exactly when
lies on the correct side of the margin, and
when
lies on the margin's boundary. It follows that
can be written as a linear combination of the support vectors.
The offset,
, can be recovered by finding an
on the margin's boundary and solving
(Note that
since
.)
Kernel trick
Suppose now that we would like to learn a nonlinear classification rule which corresponds to a linear classification rule for the transformed data points
Moreover, we are given a kernel function
which satisfies
.
We know the classification vector
in the transformed space satisfies
where, the
are obtained by solving the optimization problem
The coefficients
can be solved for using quadratic programming, as before. Again, we can find some index
such that
, so that
lies on the boundary of the margin in the transformed space, and then solve
Finally,
Modern methods
Recent algorithms for finding the SVM classifier include sub-gradient descent and coordinate descent. Both techniques have proven to offer significant advantages over the traditional approach when dealing with large, sparse datasets—sub-gradient methods are especially efficient when there are many training examples, and coordinate descent when the dimension of the feature space is high.
Sub-gradient descent
Sub-gradient descent algorithms for the SVM work directly with the expression
Note that
is a
convex function of
and
. As such, traditional
gradient descent
In mathematics, gradient descent (also often called steepest descent) is a first-order iterative optimization algorithm for finding a local minimum of a differentiable function. The idea is to take repeated steps in the opposite direction of the ...
(or
SGD) methods can be adapted, where instead of taking a step in the direction of the function's gradient, a step is taken in the direction of a vector selected from the function's
sub-gradient. This approach has the advantage that, for certain implementations, the number of iterations does not scale with
, the number of data points.
Coordinate descent
Coordinate descent Coordinate descent is an optimization algorithm that successively minimizes along coordinate directions to find the minimum of a function. At each iteration, the algorithm determines a coordinate or coordinate block via a coordinate selection rule, ...
algorithms for the SVM work from the dual problem
For each
, iteratively, the coefficient
is adjusted in the direction of
. Then, the resulting vector of coefficients
is projected onto the nearest vector of coefficients that satisfies the given constraints. (Typically Euclidean distances are used.) The process is then repeated until a near-optimal vector of coefficients is obtained. The resulting algorithm is extremely fast in practice, although few performance guarantees have been proven.
Empirical risk minimization
The soft-margin support vector machine described above is an example of an
empirical risk minimization
Empirical risk minimization (ERM) is a principle in statistical learning theory which defines a family of learning algorithms and is used to give theoretical bounds on their performance. The core idea is that we cannot know exactly how well an al ...
(ERM) algorithm for the ''
hinge loss
In machine learning, the hinge loss is a loss function used for training classifiers. The hinge loss is used for "maximum-margin" classification, most notably for support vector machines (SVMs).
For an intended output and a classifier score , th ...
''. Seen this way, support vector machines belong to a natural class of algorithms for statistical inference, and many of its unique features are due to the behavior of the hinge loss. This perspective can provide further insight into how and why SVMs work, and allow us to better analyze their statistical properties.
Risk minimization
In supervised learning, one is given a set of training examples
with labels
, and wishes to predict
given
. To do so one forms a
hypothesis
A hypothesis (plural hypotheses) is a proposed explanation for a phenomenon. For a hypothesis to be a scientific hypothesis, the scientific method requires that one can test it. Scientists generally base scientific hypotheses on previous obse ...
,
, such that
is a "good" approximation of
. A "good" approximation is usually defined with the help of a ''
loss function,''
, which characterizes how bad
is as a prediction of
. We would then like to choose a hypothesis that minimizes the ''
expected risk
Expected may refer to:
*Expectation (epistemic)
*Expected value
*Expected shortfall
*Expected utility hypothesis
*Expected return
*Expected loss
;See also
*Unexpected (disambiguation)
Unexpected may refer to:
Film and television
* ''Unexpecte ...
:''
In most cases, we don't know the joint distribution of
outright. In these cases, a common strategy is to choose the hypothesis that minimizes the ''empirical risk:''
Under certain assumptions about the sequence of random variables
(for example, that they are generated by a finite Markov process), if the set of hypotheses being considered is small enough, the minimizer of the empirical risk will closely approximate the minimizer of the expected risk as
grows large. This approach is called ''empirical risk minimization,'' or ERM.
Regularization and stability
In order for the minimization problem to have a well-defined solution, we have to place constraints on the set
of hypotheses being considered. If
is a
normed space (as is the case for SVM), a particularly effective technique is to consider only those hypotheses
for which
. This is equivalent to imposing a ''regularization penalty''
, and solving the new optimization problem
This approach is called ''
Tikhonov regularization
Ridge regression is a method of estimating the coefficients of multiple-regression models in scenarios where the independent variables are highly correlated. It has been used in many fields including econometrics, chemistry, and engineering. Also ...
.''
More generally,
can be some measure of the complexity of the hypothesis
, so that simpler hypotheses are preferred.
SVM and the hinge loss
Recall that the (soft-margin) SVM classifier
is chosen to minimize the following expression:
In light of the above discussion, we see that the SVM technique is equivalent to empirical risk minimization with Tikhonov regularization, where in this case the loss function is the
hinge loss
In machine learning, the hinge loss is a loss function used for training classifiers. The hinge loss is used for "maximum-margin" classification, most notably for support vector machines (SVMs).
For an intended output and a classifier score , th ...
From this perspective, SVM is closely related to other fundamental
classification algorithms such as
regularized least-squares
Regularized least squares (RLS) is a family of methods for solving the least-squares problem while using regularization to further constrain the resulting solution.
RLS is used for two main reasons. The first comes up when the number of variables ...
and
logistic regression
In statistics, the logistic model (or logit model) is a statistical model that models the probability of an event taking place by having the log-odds for the event be a linear combination of one or more independent variables. In regression a ...
. The difference between the three lies in the choice of loss function: regularized least-squares amounts to empirical risk minimization with the
square-loss,
; logistic regression employs the
log-loss,
Target functions
The difference between the hinge loss and these other loss functions is best stated in terms of ''target functions -'' the function that minimizes expected risk for a given pair of random variables
.
In particular, let
denote
conditional on the event that
. In the classification setting, we have:
The optimal classifier is therefore:
For the square-loss, the target function is the conditional expectation function,