Support vector machines In a nutshell Linear classifiers selecting hyperplane maximizing separation margin between classes (large margin classifiers) Solution only depends on a small subset of training examples (support vectors) Sound generalization theory (bounds or error based on margin) Can be easily extended to nonlinear separation (kernel machines) Maximum margin classifier
Maximum margin classifier Classifier margin Given a training set D, a classifier confidence margin is: ρ = min (x,y) Dyf(x) It is the minimal confidence margin (for predicting the true label) among training examples 2
A classifier geometric margin is: ρ w = min yf(x) (x,y) D w Maximum margin classifier Canonical hyperplane There is an infinite number of equivalent formulation for the same hyperplane: w T x + w 0 = 0 α(w T x + w 0 ) = 0 α 0 The canonical hyperplane is the hyperplane having confidence margin equal to : ρ = min (x,y) Dyf(x) = Its geometric margin is: ρ w = w Maximum margin classifier 3
Theorem (Margin Error Bound). Consider the set of decision functions f(x) = signw T x with w Λ and x R, for some R,Λ > 0. Moreover, let ρ > 0 and ν denote the fraction of training examples with margin smaller than ρ/ w, referred to as the margin error. For all distributions P generating the data, with probability at least δ over the drawing of the m training patterns, and for any ρ > 0 and δ (0, ), the probability that a test pattern drawn from P will be misclassified is 4
bound from above by Here, c is a universal constant. ( ) c R2 Λ ν + 2 m ρ 2 ln 2 m + ln(/δ). Margin Error Bound: interpretation ( ) c R2 Λ ν + 2 m ρ 2 ln 2 m + ln(/δ). The probability of test error depends on (among other components): number of margin errors ν (examples with margin smaller than ρ/ w ) ln number of training examples (error depends on 2 m m ) size of the margin (error depends on /ρ 2 ) Note If ρ is fixed to (canonical hyperplane), maximizing margin corresponds to minimizing w Learning problem minw,w 0 2 w 2 subject to: y i (w T x i + w 0 ) (x i, y i ) D Note constraints guarantee that all points are correctly classified (plus canonical form) minimization corresponds to maximizing the (squared) margin quadratic optimization problem (objective is quadratic, points satisfying constraints form a convex set) Learning problem minw,w 0 2 w 2 subject to: y i (w T x i + w 0 ) (x i, y i ) D Note 5
constraints guarantee that all points are correctly classified (plus canonical form) minimization corresponds to maximizing the (squared) margin quadratic optimization problem (objective is quadratic, points satisfying constraints form a convex set) Digression: constrained optimization Karush-Kuhn-Tucker (KKT) approach A constrained optimization problem can be addressed by converting it into an unconstrained problem with the same solution Let s have a constrained optimization problem as: min z subject to: f(z) g i (z) 0 i Let s introduce a non-negative variable α i 0 (called Lagrange multiplier) for each constraint and rewrite the optimization problem as (Lagrangian): min z maxα 0 f(z) i α i g i (z) Digression: constrained optimization Karush-Kuhn-Tucker (KKT) approach min z maxα 0 f(z) i α i g i (z) The optimal solutions z for this problem are the same as the optimal solutions for the original (constrained) problem: If for a given z at least one constraint is not satisfied, i.e. g i (z ) < 0 for some i, maximizing over α i leads to an infinite value (not a minimum, unless there is no non-infinite minimum) If all constraints are satisfied (i.e. g i (z ) 0 for all i), maximization over the α will set all elements of the summation to zero, so that z is a solution of min z f(z). Karush-Kuhn-Tucker (KKT) approach minw,w 0 2 w 2 subject to: y i (w T x i + w 0 ) (x i, y i ) D The constraints can be included in the minimization using Lagrange multipliers α i 0 (m = D ): L(w, w 0, α) = 2 w 2 α i (y i (w T x i + w 0 ) ) The Lagrangian is minimized wrt w, w 0 and maximized wrt α i (solution is a saddle point) 6
Dual formulation L(w, w 0, α) = 2 w 2 α i (y i (w T x i + w 0 ) ) Vanishing derivatives wrt primal variables we get: w 0 L(w, w 0, α) = 0 α i y i = 0 w L(w, w 0, α) = 0 w = α i y i x i Dual formulation Substituting in the Lagrangian we get: 2 w 2 2 α i (y i (w T x i + w 0 ) ) = α i α j y i y j x T i x j i,j= α i y i w 0 + α i = }{{} =0 α i 2 α i α j y i y j x T i x j i,j= α i α j y i y j x T i x j = L(α) i,j= which is to be maximized wrt the dual variables α Dual formulation max α IR m α i 2 α i α j y i y j x T i x j i,j= subject to α i 0 i =,..., m α i y i = 0 The resulting maximization problem including the constraints Still a quadratic optimization problem 7
Note The dual formulation has simpler contraints (box), easier to solve The primal formulation has d + variables (number of features +): minw,w 0 2 w 2 The dual formulation has m variables (number of training examples): max α IR m α i 2 α i α j y i y j x T i x j i,j= One can choose the primal formulation if it has much less variables (problem dependent) Decision function Substituting w = m α iy i x i in the decision function we get: f(x) = w T x + w 0 = α i y i x T i x + w 0 The decision function is linear combination of dot products between training points and the test point dot product is kind of similarity between points Weights of the combination are α i y i : large α i implies large contribution towards class y i (times the similarity) Karush-Khun-Tucker conditions (KKT) L(w, w 0, α) = 2 w 2 α i (y i (w T x i + w 0 ) ) At the saddle point it holds that for all i: α i (y i (w T x i + w 0 ) ) = 0 Thus, either the example does not contribute to the final f(x): α i = 0 or the example stays on the minimal confidence hyperplane from the decision one: y i (w T x i + w 0 ) = 8
Support vectors points staying on the minimal confidence hyperplanes are called support vectors All other points do not contribute to the final decision function (i.e. they could be removed from the training set) SVM are sparse i.e. they typically have few support vectors Decision function bias The bias w 0 can be computed from the KKT conditions Given an arbitrary support vector x i (with α i > 0) the KKT conditions imply: y i (w T x i + w 0 ) = y i w T x i + y i w 0 = w 0 = y iw T x i y i For robustness, the bias is usually averaged over all support vectors 9
Soft margin SVM Soft margin SVM Slack variables 0
min w X,w 0 IR, ξ IR m 2 w 2 + C subject to ξ i y i (w T x i + w 0 ) ξ i ξ i 0 i =,..., m i =,..., m A slack variable ξ i represents the penalty for example x i not satisfying the margin constraint The sum of the slacks is minimized together to the inverse margin The regularization parameter C 0 trades-off data fitting and size of the margin Soft margin SVM Regularization theory Regularized loss minimization problem min w X,w 0 IR, ξ IR m 2 w 2 + C The loss term accounts for error minimization l(y i, f(x i )) The margin maximization term accounts for regularization i.e. solutions with larger margin are preferred Note Regularization is a standard approach to prevent overfitting It corresponds to a prior for simpler (more regular, smoother) solutions Soft margin SVM
Hinge loss l(y i, f(x i )) = y i f(x i ) + = y i (w T x i + w 0 ) + z + = z if z > 0 and 0 otherwise (positive part) it corresponds to the slack variable ξ i (violation of margin costraint) all examples not violating margin costraint have zero loss (sparse set of support vectors) Soft margin SVM Lagrangian L = C ξ i + 2 w 2 α i (y i (w T x i + w 0 ) + ξ i ) β i ξ i where α i 0 and β i 0 Vanishing derivatives wrt primal variables we get: w 0 L = 0 α i y i = 0 m w L = 0 w = α i y i x i ξ i L = 0 C α i β i = 0 2
Soft margin SVM Dual formulation Substituting in the Lagrangian we get C ξ i + 2 w 2 α i (y i (w T x i + w 0 ) + ξ i ) β i ξ i = ξ i (C α i β i ) + }{{} 2 }{{} =0 =0 α i y i w 0 + α i = α i 2 α i α j y i y j x T i x j α i α j y i y j x T i x j i,j= α i α j y i y j x T i x j = L(α) i,j= i,j= Soft margin SVM Dual formulation max α IR m α i 2 α i α j y i y j x T i x j i,j= subject to 0 α i C i =,..., m α i y i = 0 The box constraint for α i comes from C α i β i = 0 (and the fact that both α i 0 and β i 0) Soft margin SVM Karush-Khun-Tucker conditions (KKT) L = C ξ i + 2 w 2 α i (y i (w T x i + w 0 ) + ξ i ) β i ξ i At the saddle point it holds that for all i: α i (y i (w T x i + w 0 ) + ξ i ) = 0 β i ξ i = 0 Thus, support vectors (α i > 0) are examples for which (y i (w T x i + w 0 ) 3
Soft margin SVM Support Vectors α i (y i (w T x i + w 0 ) + ξ i ) = 0 β i ξ i = 0 If α i < C, C α i β i = 0 and β i ξ i = 0 imply that ξ i = 0 These are called unbound SV ((y i (w T x i + w 0 ) =, they stay on the confidence one hyperplane If α i = C (bound SV) then ξ i can be greater the zero, in which case the SV are margin errors Support vectors 4
Large-scale SVM learning Stochastic gradient descent Objective for a single example (x i, y i ): λ min w X 2 w 2 + m y i w, x i + E(w; (x i, y i )) = λ 2 w 2 + y i w, x i + 5
Subgradient: w E(w; (x i, y i )) = λw [y i w, x i < ]y i x i Large-scale SVM learning Note Indicator function [y i w, x i ] = { if yi w, x i < 0 otherwise The subgradient of a function f at a point x 0 is any vector v such that for any x: f(x) f(x 0 ) v T (x x 0 ) Large-scale SVM learning Pseudocode (pegasus). Initialize w = 0 2. for t = to T: (a) Randomly choose (x it, y it ) from D (b) Set η t = λt (c) Update w: w t+ = w t η t w E(w; (x it, y it )) 3. Return w T + Note The choice of the learning rate allows to bound the runtime for an ɛ-accurate solution to O(d/λɛ) with d maximum number of non-zero features in an example. Large-scale SVM learning Dual version It is easy to show that: t w t+ = λt [y it w t, x it < ]y it x it We can represent w t+ implicitly by storing in vector α t+ the number of times each example was selected and had a non-zero loss, i.e.: α t+ [j] = {t t : i t = j y j w t, x j < } 6
Large-scale SVM learning Pseudocode (pegasus dual). Initialize α = 0 2. for t = to T: (a) Randomly choose (x it, y it ) from D (b) Set α t+ = α t (c) If y it λt t j= α t[j]y j x j, x it < i. α t+ [i t ] = α t+ [i t ] + 3. Return α T + Note This will be useful when combined with kernels. References Biblio C. Burges, A tutorial on support vector machines for pattern recognition, Data Mining and Knowledge Discovery, 2(2), 2-67, 998. S. Shalev-Shwartz et al., Pegasos: primal estimated sub-gradient solver for SVM, Mathematical Programming, 27(), 3-30, 20. Software svm module in scikit-learn http://scikit-learn.org/stable/index.html libsvm http://www.csie.ntu.edu.tw/ cjlin/libsvm/ svmlight http://svmlight.joachims.org/ 7