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# Using Analytic QP and Sparseness to Speed Training of Support Vector Machines - PowerPoint PPT Presentation

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

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

John C. Platt

Presented by: Travis Desell

Overview Support Vector Machines

• 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 Support Vector Machines

• 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 Support Vector Machines

u = SiaiyiK(xi,x) – b (1)

• u : SVM output

• a : 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) Support Vector Machines

• 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 = Siaiyixi (3)

• Where w, x, and xi are vectors

General SVM Training Support Vector Machines

• Training an SVM is finding ai, expressed as minimizing a dual quadratic form:

minaY(a) = mina ½ Si SjyiyjK(xi, xj)aiaj – Siai (4)

• Subject to box constraints:

0 <= ai <= C, for all I (5)

• And the linear equality constraint:

Siyiai = 0 (6)

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

General SVM Training (2) Support Vector Machines

• SMO solves the QP expressed in (4-6)

• Terminates when all of the Karush-Kuhn-Tucker (KKT) optimality conditions are fulfilled:

ai = 0 -> yiui >= 1 (7)

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

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

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

Related Work Support Vector Machines

• “Chunking” [9]

• Removing training examples with ai = 0 does not change solution.

• Breaks down large QP problem into smaller sub-problems to identify non-zero ai.

• The QP sub-problem consists of every non-zero ai 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 ai 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) Support Vector Machines

• 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 Support Vector Machines

• 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 a, because of the linear equality constraint.

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

Choosing the SOP Support Vector Machines

• Heuristic based approach

• Terminates when the entire training set obeys (7-9) within e (typically <= 10-3)

• Repeatedly finds a1 and a2 and optimizes until termination

Finding Support Vector Machinesa1

• “First choice heuristic”

• Searches through examples most likely to violate conditions (non-bound subset)

• ai at the bounds likely to stay there, non-bound ai 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 Support Vector Machinesa2

• Chosen to maximize the size of the step taken during the joint optimization of a1 and a2

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

• If E1 is negative, chooses a2 with minimum E2

• If E1 is positive, chooses a2 with maximum E2

Solving the SOP Support Vector Machines

• Computes minimum along the direction of the linear equality constant:

a2new = y2(E1-E2)/(K(x1,x1)+K(x2,x2)–2K(x1, x2)) (10)

Ei = ui-yi (11)

• Clips a2new within [L,H]:

L = max(0,a2+sa1-0.5(s+1)C) (12)

H = min(C,a2+sa1-0.5(s-1)C) (13)

s = y1y2 (14)

• Calculates a1new:

a1new = a1 + s(a2 – a2new,clipped) (15)

Benchmarks Support Vector Machines

• 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 Support Vector Machines

• 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 Support Vector Machines

Conclusions Support Vector Machines

• 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) Support Vector Machines

• 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 Support Vector Machines

• Heuristic based approach to finding a1 and a2 to optimize:

• Possible to determine optimal choice strategy to minimize the number of steps?

• Proof that SMO always minimizes the QP problem?

References Support Vector Machines

• [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) Support Vector Machines

• [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) Support Vector Machines

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