Using Analytic QP and Sparseness to Speed Training of Support Vector Machines

Using Analytic QP and Sparseness to Speed Training of

Support Vector Machines

John C. Platt

Presented by: Travis Desell

Overview• Introduction

– Motivation– General SVMs– General SVM training– Related Work

• Sequential Minimal Optimization (SMO)– Choosing the smallest optimization problem– Solving the smallest optimization problem

• Benchmarks• Conclusion• Remarks & Future Work• References

Motivation

• Traditional SVM Training Algorithms– Require quadratic programming (QP) package– SVM training is slow, especially for large problems

• Sequential Minimal Optimization (SMO)– Requires no QP package– Easy to implement– Often faster– Good scalability properties

General SVMs

u = i iyiK(xi,x) – b (1)• u : SVM output• : weights to blend different kernels• y in {-1, +1} : desired output• b : threshold• xi : stored training example (vector)• x : input (vector)• K : kernel function to measure similarity of xi to xi

General SVMs (2)

• For linear SVMs, K is linear, thus (1) can be expressed as the dot product of w and x minus the threshold:

u = w * x – b(2)w = i iyixi (3)

• Where w, x, and xi are vectors

General SVM Training• Training an SVM is finding i, expressed as

minimizing a dual quadratic form:min () = min ½ i j yiyjK(xi, xj)ij – ii (4)

• Subject to box constraints:0 <= i <= C, for all I (5)

• And the linear equality constraint:i yii = 0 (6)

• The i are Lagrange multipliers of a primal QP problem: there is a one-to-one correspondence between each i and each training example xi

General SVM Training (2)

• SMO solves the QP expressed in (4-6)• Terminates when all of the Karush-Kuhn-Tucker

(KKT) optimality conditions are fulfilled:i = 0 -> yiui >= 1 (7)

0 < i < C -> yiui = 1 (8)

i = C -> yiui <= 1 (9)

• Where ui is the SVM output for the ith training example

Related Work• “Chunking” [9]

– Removing training examples with i = 0 does not change solution.– Breaks down large QP problem into smaller sub-problems to

identify non-zero i.– The QP sub-problem consists of every non-zero i from previous

sub-problem combined with M worst examples that violate (7-9) for some M [1].

– Last step solves the entire QP problem as all non-zero i have been found.

– Cannot handle large-scale training problems if standard QP techniques are used. Kaufman [3] describes QP algorithm to overcome this.

Related Work (2)

• Decomposition [6]:– Breaks the large QP problem into smaller QP sub-

problems.– Osuna et al. [6] suggest using fixed size matrix for

every sub-problem – allows very large training sets.– Joachims [2] suggests adding and subtracting examples

according to heuristics for rapid convergence.– Until SMO, requires numerical QP library, which can

be costly or slow.

Sequential Minimal Optimization

• SMO decomposes the overall QP problem (4-6), into fixed size QP sub-problems.

• Chooses the smallest optimization problem (SOP) at each step.– This optimization consists of two elements of

because of the linear equality constraint.

• SMO repeatedly chooses two elements of to jointly optimize until the overall QP problem is solved.

Choosing the SOP

• Heuristic based approach• Terminates when the entire training set

obeys (7-9) within (typically <= 10-3)• Repeatedly finds 1 and 2 and optimizes

until termination

Finding 1

• “First choice heuristic”– Searches through examples most likely to violate

conditions (non-bound subset)– i at the bounds likely to stay there, non-bound i will

move as others are optimized• “Shrinking Heuristic”

– Finds examples which fulfill (7-9) more than the worst example failed

– Ignores these examples until a final pass at the end to ensure all examples fulfill (7-9)

Finding 2

• Chosen to maximize the size of the step taken during the joint optimization of 1 and 2

• Each non-bound has a cached error value E for each non-bound example

• If E1 is negative, chooses 2 with minimum E2

• If E1 is positive, chooses 2 with maximum E2

Solving the SOP• Computes minimum along the direction of the

linear equality constant:2

new = y2(E1-E2)/(K(x1,x1)+K(x2,x2)–2K(x1, x2)) (10)Ei = ui-yi (11)

• Clips 2new within [L,H]:

L = max(0,2+s1-0.5(s+1)C) (12)H = min(C,2+s1-0.5(s-1)C) (13)

s = y1y2 (14)

• Calculates 1new:

1new = 1 + s(2 – 2

new,clipped) (15)

Benchmarks

• UCI Adult: SVM is given 14 attributes of a census and is asked to predict if household income is greater than $50k. 8 categorical attributes, 6 continues = 123 binary attributes.

• Web: classify if a web page is in a category or not. 300 sparse binary keyword attributes.

• MNIST: One classifier is trained. 784-dimensional, non-binary vectors stored as sparse vectors.

Description of Benchmarks

• Web and Adult are trained with linear and Gaussian SVMs.

• Performed with and without sparse inputs, with and without kernel caching

• PCG chunking always uses caching

Benchmarking SMO

Conclusions

• PCG chunking slower than SMO, SMO ignores examples whose Lagrange multipliers are at C.

• Overhead of PCG chunking not involved with kernel (kernel optimizations do not greatly effect time).

Conclusions (2)• SVMlight solves 10 dimensional QP sub-problems.• Differences mostly due to kernel optimizations

and numerical QP overhead.• SMO faster on linear problems due to linear SVM

folding, but SVMlight can potentially use this as well.

• SVMlight benefits from complex kernel cache while SVM does have a complex kernel cache and thus does not benefit from it at large problem sizes.

Remarks & Future Work

• Heuristic based approach to finding 1 and 2 to optimize:– Possible to determine optimal choice strategy to

minimize the number of steps?

• Proof that SMO always minimizes the QP problem?

References

• [1] C. J. C. Burges. A tutorial on support vector machines for pattern recognition. Data Mining and Knowledge Discovery, 2(2), 1998.

• [2] T. Joachims. Making large-scale SVM learning practical. In B. Sch¨olkopf, C. J. C. Burges, and A. J. Smola, editors, Advances in Kernel Methods — Support Vector Learning, pages 169–184. MIT Press, 1998.

References (2)• [3] L. Kaufman. Solving the quadratic

programming problem arising in support vector classification. In B. Sch¨olkopf, C. J. C. Burges, and A. J. Smola, editors, Advances in Kernel Methods — Support Vector Learning, pages 147–168. MIT Press, 1998.

• [6] E. Osuna, R. Freund, and F. Girosi. Improved training algorithm for support vector machines. In Proc. IEEE Neural Networks in Signal Processing ’97, 1997.

References (3)

• [9] V. Vapnik. Estimation of Dependences Based on Empirical Data. Springer-Verlag, 1982.