SVM Tutorial
Zoya Gavrilov
Just the basics with a little bit of spoon-feeding…
1 Simplest case: linearly-separable data, binary classification
Goal: we want to find the hyperplane (i.e. decision boundary) linearly separating our classes. Our boundary will have equation: wT x + b = 0.
Anything above the decision boundary should have label 1.
i.e., xi s.t. wT xi + b > 0 will have corresponding yi = 1.
Similarly, anything below the decision boundary should have label −1. i.e., xi s.t. wT xi + b < 0 will have corresponding yi = −1.
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The reason for this labelling scheme is that it lets us condense the formula- tion for the decision function to f (x) = sign(wT x + b) since f (x) = +1 for all x above the boundary, and f(x) = −1 for all x below the boundary.
Thus, we can figure out if an instance has been classified properly by checking that y(wT x+b) ≥ 1 (which will be the case as long as either both y, wT x+b > 0 or else y, wT x + b < 0).
You’ll notice that we will now have some space between our decision bound- ary and the nearest data points of either class. Thus, let’s rescale the data such that anything on or above the boundary wT x + b = 1 is of one class (with label 1), and anything on or below the boundary wT x + b = −1 is of the other class (with label −1).
What is the distance between these newly added boundaries?
First note that the two lines are parallel, and thus share their parameters w, b.
Pick an arbirary point x1 to lie on line wT x + b = −1. Then, the closest point
on line wTx+b = 1 to x1 is the point x2 = x1 +λw (since the closest point
will always lie on the perpendicular; recall that the vector w is perpendicular
to both lines). Using this formulation, λw will be the line segment connecting
x1 and x2, and thus, λ∥w∥, the distance between x1 and x2, is the shortest
distance between the two lines/boundaries. Solving for λ:
⇒ wT x2 + b = 1 where x2 = x1 + λw
⇒wT(x1 +λw)+b=1
⇒ wT x1 + b + λwT w = 1 where wT x1 + b = −1
⇒ −1 + λwT w = 1
⇒ λwT w = 2
⇒λ=2=2 wT w ∥w∥2
Andso,thedistanceλ∥w∥is 2 ∥w∥= 2 =√ 2 ∥w∥2 ∥w∥ wTw
It’s intuitive that we would want to maximize the distance between the two
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boundaries demarcating the classes (Why? We want to be as sure as possible that we are not making classification mistakes, and thus we want our data points from the two classes to lie as far away from each other as possible). This distance is called the margin, so what we want to do is to obtain the maximal margin.
√
Thus, we want to maximize √ 2 , which is equivalent to minimizing wTw
wTw, 2
which is in turn equivalent to minimizing wTw (since square root is a monotonic 2
function).
This quadratic programming problem is expressed as: minw,b wTw
2
subject to: yi(wT xi + b) ≥ 1 (∀ data points xi). 2 Soft-margin extention
Consider the case that your data isn’t perfectly linearly separable. For instance, maybe you aren’t guaranteed that all your data points are correctly labelled, so you want to allow some data points of one class to appear on the other side of the boundary.
We can introduce slack variables - an εi ≥ 0 for each xi. Our quadratic program-
ming problem becomes:
minw,b,ε wTw + C εi 2i
subjectto:yi(wTxi+b)≥1−εi andεi ≥0(∀datapointsxi). 3 Nonlinear decision boundary
Mapping your data vectors, xi, into a higher-dimension (even infinite) feature space may make them linearly separable in that space (whereas they may not be linearly separable in the original space). The formulation of the quadratic programming problem is as above, but with all xi replaced with φ(xi), where φ provides the higher-dimensional mapping. So we have the standard SVM formu- lation:
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minw,b,ε wTw + C εi 2i
subjectto:yi(wTφ(xi)+b)≥1−εi andεi ≥0(∀datapointsxi).
4 Reformulating as a Lagrangian
We can introduce Lagrange multipliers to represent the condition:
yi(wT φ(xi) + b) must be as close to 1 as possible.
This condition is captured by: maxαi≥0αi[1 − yi(wT φ(xi) + b)]
This ensures that when yi(wT φ(xi) + b) ≥ 1, the expression above is maximal when αi = 0 (since [1 − yi(wT φ(xi) + b)] ends up being negative). Otherwise, yi(wT φ(xi) + b) < 1, so [1 − yi(wT φ(xi) + b)] is a positive value, and the expres- sion is maximal when αi → ∞. This has the effect of penalizing any misclassified data points, while assigning 0 penalty to properly classified instances.
We thus have the following formulation: minw,b[wTw + maxαi≥0αi[1−yi(wTφ(xi)+b)]]
2i
To allow for slack (soft-margin), preventing the α variables from going to ∞, we
can impose constraints on the Lagrange multipliers to lie within: 0 ≤ αi ≤ C.
We can define the dual problem by interchanging the max and min as follows
(i.e. we minimize after fixing alpha):
maxα≥0[minw,bJ(w, b; α)] where J(w, b; α) = wTw + αi[1 − yi(wT φ(xi) + b)] 2i
Since we’re solving an optimization problem, we set ∂J = 0 and discover that ∂w
the optimal setting of w is αi yi φ(xi ), while setting ∂ J = 0 yields the con- i ∂b
straint i αiyi = 0.
Thus, after substituting and simplifying, we get:
minw,bJ(w,b;α)=αi−1 αiαjyiyjφ(xi)Tφ(xj) i 2 i,j
And thus our dual is:
maxα≥0[ αi − 1 αiαj yiyj φ(xi)T φ(xj)]
i 2 i,j Subjectto:iαiyi =0and0≤αi ≤C
5 Kernel Trick
Because we’re working in a higher-dimension space (and potentially even an infinite-dimensional space), calculating φ(xi)T φ(xj) may be intractable. How- ever, it turns out that there are special kernel functions that operate on the lower dimension vectors xi and xj to produce a value equivalent to the dot- product of the higher-dimensional vectors.
For instance, consider the function φ : R3 → R10, where:
φ(x) = (1, √2x(1), √2x(2), √2x(3), [x(1)]2, [x(2)]2, [x(3)]2, √2x(1)x(2), √2x(1)x(3), √2x(2)x(3))
Instead, we have the kernel trick, which tells us that K(xi, xj) = (1 + xiT xj)2 = φ(xi)T φ(xj) for the given φ. Thus, we can simplify our calculations.
Re-writing the dual in terms of the kernel, yields: maxα≥0[ αi − 1 αiαjyiyjK(xi,xj)]
6 Decision function
To classify a novel instance x once you’ve learned the optimal αi parameters, all you have to do is calculate f(x) = sign(wT x + b) = i αiyiK(xi, x) + b
(by setting w = i αiyiφ(xi) and using the kernel trick).
Note that αi is only non-zero for instances φ(xi) on or near the boundary - those are called the support vectors since they alone specify the decision boundary. We can toss out the other data points once training is complete. Thus, we only sum over the xi which constitute the support vectors.
7 Learn more
A great resource is the MLSS 2006 (Machine Learning Summer School) talk by Chih-Jen Lin, available at: videolectures.net/mlss06tw lin svm/. Make sure to download his presentation slides. Starting with the basics, the talk continues on
i 2 i,j
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to discussing practical issues, parameter/kernel selection, as well as more ad- vanced topics.
The typical reference for such matters: Pattern Recognition and Machine Learn- ing by Christopher M. Bishop.