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

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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
• 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
• 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 = 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)
• 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
• 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)
• 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
• “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)
• 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 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
• 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 a1
• “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 a2
• 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
• 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
• 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
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 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
• [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.