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Artificial Intelligence 1995

Support-Vector Networks

Corinna Cortes & Vladimir Vapnik (AT&T Bell Labs)

Separate two classes by the widest margin — and only the borderline points decide where the line falls.

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In depth · the introduction

Of all the lines that could split two groups of dots, a support-vector machine picks exactly one — the one with the most elbow room — and that single choice is why it generalizes so well.

The idea, unpacked

Suppose you have two kinds of things — spam and not-spam, say — each described by some numbers, and you want a rule that tells them apart. Geometrically, you are hunting for a line (in higher dimensions, a flat sheet) that puts one kind on each side. Usually many lines would work. Which is best?

The support-vector machine answers: the line that leaves the widest empty gap on both sides. It pushes the boundary as far as it can from the nearest examples of each class, so there is the most possible room before a new, slightly different example would be misjudged. Remarkably, only the closest few examples — the support vectors — matter; the rest could vanish and the line would not move.

Where it came from

The method grew at AT&T Bell Labs, the same storied lab behind the transistor and information theory. Vladimir Vapnik had spent decades in the Soviet Union building a mathematical theory of learning; after emigrating, he and his colleagues turned it into an algorithm. A 1992 paper handled the clean case where the two classes can be separated perfectly. Then in 1995 Corinna Cortes and Vapnik published "Support-Vector Networks," which added the crucial ability to cope with messy, overlapping real data — and the method took off.

Why it mattered

Before deep learning's resurgence, the support-vector machine was the workhorse of practical machine learning. It was accurate, it came with genuine mathematical guarantees about when it would generalize, and — through a clever shortcut called the kernel trick — it could draw curved, complicated boundaries while still solving a simple, well-behaved problem with a single best answer. For fifteen years it was the method to beat.

A precise analogy

Think of building the widest possible road between two neighbourhoods of houses. You don't care about the houses deep inside each neighbourhood; only the ones nearest the other side constrain how wide the road can be and where it must run. Those few border houses are the support vectors. Move a house in the interior and the road is unchanged; nudge a border house and the whole road shifts. Try it on the dots below.

An interactive scatter plot of two classes of points with a slider that rotates a straight dividing line. For each angle the widget shades the empty 'street' between the classes, draws its two margin edges, and circles the support vectors that touch it, while a readout reports the current margin width and the widest margin attainable; an Expert panel lists the margin, the maximum margin, the line angle, and the number of support vectors.

Where it sits

This is one of the great branches of machine learning: statistical learning, where the goal is provable generalization rather than brain-inspired wiring. It shares the Library's AI shelf with the neural-network lineage — Rosenblatt's perceptron, Rumelhart's backpropagation, and the deep networks of AlexNet and the Transformer that later overtook it. For much of the 1990s and 2000s, though, the support-vector machine quietly won.

The original document
Original source text
C. Cortes & V. Vapnik · Machine Learning 20(3): 273–297 · 1995
Abstract
The support-vector network is a new learning machine for two-group classification problems.
The machine conceptually implements the following idea: input vectors are non-linearly mapped to a very high-dimension feature space. In this feature space a linear decision surface is constructed. Special properties of the decision surface ensures high generalization ability of the learning machine.
The abstract continues that the idea had previously been implemented only for the separable case; that this paper extends it to non-separable training data; and that the resulting classifier is benchmarked against classical learning algorithms on optical character recognition.
The optimal hyperplane
The paper takes the separating boundary that maximizes the margin between the two classes as the optimal hyperplane, and shows it is determined by a small subset of the training vectors — the support vectors — which lie exactly on the margin.
Soft margin
To handle data that cannot be separated without error, the authors introduce non-negative slack variables and a penalty, allowing some points to violate the margin while keeping the boundary as wide as possible. This soft margin is the central new contribution of the 1995 paper.
Convolution of the dot-product (kernels)
Because the optimization depends on the data only through dot products, each dot product is replaced by a kernel function — equivalent to constructing the optimal hyperplane in a high-dimensional feature space without ever computing the mapping explicitly. Polynomial kernels are used in the experiments.
[ … ]
Experiments
On the U.S. Postal Service handwritten-digit benchmark the support-vector network attains an error rate of about 4%, comparable to the best competing methods of the day — evidence for the high generalization ability claimed in the abstract.
AT&T Bell Laboratories · 1995