Kernel Learning with a Million Kernels

SVN VishwanathanPurdue University

Ashesh JainIIT Delhi

Manik VarmaMicrosoft Research India

To appear SIGKDD 2012

• The objective in kernel learning is to jointly learn both SVM and kernel parameters from training data.

• Kernel parameterizations• Linear : • Non-linear :

• Regularizers• Sparse l1 • Sparse and non-sparse lp>1

• Log determinant

Kernel Learning

P = Minw,b,d ½s. t.

• • and exist and are continuous

The GMKL Primal Formulation

• The GMKL primal formulation for binary classification.

P = Minw,b,d, 𝜉 ½s. t.

&

• Intermediate Dual

D = Mind Max 1t – ½tYKdY + r(d)s. t. 1tY = 0

0 C &

The GMKL Primal Formulation

Outer loop: Mind W(d)s. t.

Inner loop: Max 1t – ½tYKdY + r(d) W(d) = s. t. 1tY = 0

0 C

• [Chapelle et al. MLJ 2002].• can be obtained using a standard SVM solver.• Optimized using Projected Gradient Descent (PGD).

Wrapper Method for Optimizing GKML

Projected Gradient Descent

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

d1

d 2

x0

x2

z1

x1

x∗

x3

• PGD requires many function and gradient evaluations as• No step size information is available.• The Armijo rule might reject many step size proposals.• Inaccurate gradient values can lead to many tiny steps.

• Noisy function and gradient values can cause PGD to converge to points far away from the optimum.

• Solving SVMs to high precision to obtain accurate function and gradient values is very expensive.

• Repeated projection onto the feasible set might also be expensive.

PGD Limitations

SPG Solution – Spectral Step Length• Quadratic approximation : ½• Spectral step length :

x0

x1

x0

x1

x*

Original Function Approximation

SPG Solution – Spectral Step Length

-1 -0.5 0 0.5 1-1

-0.5

0

0.5

1

x0x1

• Spectral step length :

• Handling function and gradient noise.• Non-monotone rule:

SPG Solution – Non-Monotone Rule

0 10 20 30 40 50 601426

1428

1430

1432

1434

1436

1438

time (s)

f(x)

Global MinimumSPG-GMKL

00.1

0.20.3

0.40.5

0.60.7

0.80.9

1

00.1

0.20.3

0.40.5

0.60.7

0.80.9

13.5

4

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5

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6

6.5

SPG Solution – SVM Precision Tuning

3 hr

2 hr

1 hr

0.5 hr

0.2 hr

0.3 hr

0.1 hr

0.1 hr

SPGPGD

• SPG requires fewer function and gradient evaluations due to• The 2nd order spectral step length estimation.• The non-monotone line search criterion.

• SPG is more robust to noisy function and gradient values due to the non-monotone line search criterion.

• SPG never needs to solve an SVM with high precision due to our precision tuning strategy.

• SPG needs to perform only a single projection per step.

SPG Advantages

1: 2: 3: 4:5:6:7:8:9:10:

SPG Algorithm

• Sum of kernels subject to regularization

• Product of kernels subject to regularization

Results on Large Scale Data Sets

Data Sets # Train # KernelsPGD (hrs) SPG (hrs) PGD (hrs) SPG (hrs)

Adult - 9 32,561 50 35.84 4.55 31.77 4.42

Cod - RNA 59,535 50 – 25.17 66.48 19.10

KDDCup04 50,000 50 – 40.10 – 42.20

Sonar 208 1 Million – 105.62

Covertype 581,012 5 – 64.46

Data Sets # Train # KernelsPGD (hrs) SPG (hrs) PGD (hrs) SPG (hrs)

Letter 20,000 16 18.66 0.67 18.69 0.66

Poker 25,010 10 5.57 0.49 2.29 0.96

Adult - 8 22,696 42 – 1.73 – 3.42

Web - 7 24,692 43 – 0.88 – 1.33

RCV1 20,242 50 – 18.17 – 15.93

Cod - RNA 59,535 8 – 3.45 – 8.99

• Sum of kernels subject to regularization

Results on Small Scale Data Sets

Data Sets SimpleMKL (s) Shogun (s) PGD (s) SPG (s)Wpbc 400356.4 157.7 3817.6 64.2Breast - Cancer 676356.4 121.2 5785.1 50.6 Australian 38333.5 1094621.6 297.1 100.8Ionosphere 1247680.0 10718.8 1392824.2 396.8Sonar 14681252.7 93565.0 – 27364.0

• Scaling with the number of training points

SPG Scaling Properties

3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.81.5

2

2.5

3

3.5

4

4.5

5

5.5

log(#Training Points)

log(

Tim

e)

Adult

SPG p=1.0SPG p=1.33SPG p=1.66PGD p=1.0PGD p=1.33PGD p=1.66

• Scaling with the number of kernels

SPG Scaling Properties

3.6 3.8 4 4.2 4.4 4.6 4.8 5 5.22

2.5

3

3.5

4

4.5

5

5.5

log(#Kernel)

log(

Tim

e)

Sonar

SPG p=1.33SPG p=1.66PGD p=1.33PGD p=1.66

Acknowledgements• Kamal Gupta (IITD)

• Subhashis Banerjee (IITD)

• The Computer Services Center at IIT Delhi