SVM by Sequential Minimal Optimization SMO Algorithm by

SVM by Sequential Minimal Optimization (SMO) Algorithm by John Platt Lecture by David Page CS 760: Machine Learning

Quick Review • Inner product (specifically here the dot product) is defined as • Changing x to w and z to x, we may also write: or or w. Tx or w x • In our usage, x is feature vector and w is weight (or coefficient) vector • A line (or more generally a hyperplane) is written as w. Tx = b, or w. Tx - b = 0 • A linear separator is written as sign(w. Tx - b)

Quick Review (Continued) • Recall in SVMs we want to maximize margin subject to correctly separating the data… that is, we want to: • Here the y’s are the class values (+1 for positive and -1 for negative), so says xi correctly labeled with room to spare • Recall is just w. Tw

Recall Full Formulation • As last lecture showed us, we can – Solve the dual more efficiently (fewer unknowns) – Add parameter C to allow some misclassifications – Replace xi. Txj by more general kernel term

Intuitive Introduction to SMO • Perceptron learning algorithm is essentially doing same thing – find a linear separator by adjusting weights on misclassified examples • Unlike perceptron, SMO has to maintain sum over examples of example weight times example label • Therefore, when SMO adjusts weight of one example, it must also adjust weight of another

An Old Slide: Perceptron as Classifier • Output for example x is sign(w. Tx) • Candidate Hypotheses: real-valued weight vectors w • Training: Update w for each misclassified example x (target class t, predicted o) by: – wi wi + h(t-o)xi – Here h is learning rate parameter • Let E be the error o-t. The above can be rewritten as w w – h. Ex • So final weight vector is a sum of weighted contributions from each example. Predictive model can be rewritten as weighted combination of examples rather than features, where the weight on an example is sum (over iterations) of terms –h. E.

Corresponding Use in Prediction • To use perceptron, prediction is w. Tx - b. May treat -b as one more coefficient in w (and omit it here), may take sign of this value • In alternative view of last slide, prediction can be written as or, revising the weight appropriately, • Prediction in SVM is:

From Perceptron Rule to SMO Rule • Recall that SVM optimization problem has the added requirement that: • Therefore if we increase one by an amount h, in either direction, then we have to change another by an equal amount in the opposite direction (relative to class value). We can accomplish this by changing: w w – h. Ex also viewed as: – yh. E to: 1 1 – y 1 h(E 1 -E 2) or equivalently: 1 1 + y 1 h(E 2 -E 1) We then also adjust 2 so that y 1 1 + y 2 2 stays same

First of Two Other Changes • Instead of h = or some similar constant, we set h = or more generally h = • For a given difference in error between two examples, the step size is larger when the two examples are more similar. When x 1 and x 2 are similar, K(x 1, x 2) is larger (for typical kernels, including dot product). Hence the denominator is smaller and step size is larger.

Second of Two Other Changes • Recall our formulation had an additional constraint: • Therefore, we “clip” any change we make to any to respect this constraint • This yields the following SMO algorithm. (So far we have motivated it intuitively. Afterward we will show it really minimizes our objective. )

Updating Two ’s: One SMO Step • Given examples e 1 and e 2, set where: • Clip this value in the natural way: if y 1 = y 2 then: • otherwise: • Set where s = y 1 y 2

Karush-Kuhn-Tucker (KKT) Conditions • It is necessary and sufficient for a solution to our objective that all ’s satisfy the following: • An is 0 iff that example is correctly labeled with room to spare • An is C iff that example is incorrectly labeled or in the margin • An is properly between 0 and C (is “unbound”) iff that example is “barely” correctly labeled (is a support vector) • We just check KKT to within some small epsilon, typically 10 -3

SMO Algorithm • Input: C, kernel parameters, epsilon • Initialize b and all ’s to 0 • Repeat until KKT satisfied (to within epsilon): – Find an example e 1 that violates KKT (prefer unbound examples here, choose randomly among those) – Choose a second example e 2. Prefer one to maximize step size (in practice, faster to just maximize |E 1 – E 2|). If that fails to result in change, randomly choose unbound example. If that fails, randomly choose example. If that fails, re-choose e 1. – Update α 1 and α 2 in one step – Compute new threshold b

At End of Each Step, Need to Update b • Compute b 1 and b 2 as follows • Choose b = (b 1 + b 2)/2 • When α 1 and α 2 are not at bound, it is in fact guaranteed that b = b 1 = b 2

Derivation of SMO Update Rule • Recall objective function we want to optimize: α 2 with everybody else • Can write objective as: α 1 with everybody else all the other α’s where: Starred values are from end of previous round

Expressing Objective in Terms of One Lagrange Multiplier Only • Because of the constraint update has to obey: , each where w is a constant and s is y 1 y 2 • This lets us express the objective function in terms of α 2 alone:

Find Minimum of this Objective • Find extremum by first derivative wrt α 2: • If second derivative positive (usual case), then the minimum is where (by simple Algebra, remembering ss = 1):

Finishing Up • Recall that w = and • So we can rewrite as: = s(K 11 -K 12)(sα 2*+α 1*) + y 2(u 1 -u 2+y 1α 1*(K 12 -K 11) + y 2α 2*(K 22 -K 21)) +y 2 y 2 –y 1 y 2 • We can rewrite this as:

Finishing Up • Based on values of w and v, we rewrote as: u 1 – y 1 • From this, we get our update rule: where u 2 – y 2 K 11 + K 22 – 2 K 12