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Kernel Learning with a Million Kernels

Kernel Learning with a Million Kernels. Ashesh Jain IIT Delhi. S VN Vishwanathan Purdue University. Manik Varma Microsoft Research India. To appear SIGKDD 2012. The objective in kernel learning is to jointly learn both SVM and kernel parameters from training data.

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Kernel Learning with a Million Kernels

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  1. Kernel Learning with a Million Kernels Ashesh Jain IIT Delhi SVN Vishwanathan Purdue University Manik Varma Microsoft Research India To appear SIGKDD 2012

  2. 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

  3. P = Minw,b,d ½ • s. t. • and exist and are continuous The GMKL Primal Formulation

  4. The GMKL primal formulation for binary classification. • P= Minw,b,d, 𝜉 ½ • s. t. • & • Intermediate Dual • D= MindMax 1t – ½tYKdY + r(d) • s. t. 1tY = 0 • 0    C & The GMKL Primal Formulation

  5. Outer loop: MindW(d) • s. t. • Inner loop: Max 1t – ½tYKdY+ r(d) • W(d)= s. t. 1tY = 0 • 0   C • [Chapelleet al. MLJ 2002]. • can be obtained using a standard SVM solver. • Optimized using Projected Gradient Descent (PGD). Wrapper Method for Optimizing GKML

  6. Projected Gradient Descent

  7. 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

  8. Quadratic approximation : ½ • Spectral step length : SPG Solution – Spectral Step Length Original Function Approximation

  9. Spectral step length : SPG Solution – Spectral Step Length

  10. Handling function and gradient noise. • Non-monotone rule: SPG Solution – Non-Monotone Rule

  11. SPG PGD SPG Solution – SVM Precision Tuning 0.5 hr 3 hr 0.2 hr 0.3 hr 2hr 0.1 hr 1hr 0.1 hr

  12. 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

  13. 1: • 2: • 3: • 4: • 5: • 6: • 7: • 8: • 9: • 10: SPG Algorithm

  14. Sum of kernels subject to regularization • Product of kernels subject to regularization Results on Large Scale Data Sets

  15. Sum of kernels subject to regularization Results on Small Scale Data Sets

  16. Scaling with the number of training points SPG Scaling Properties

  17. Scaling with the number of kernels SPG Scaling Properties

  18. Kamal Gupta (IITD) • Subhashis Banerjee (IITD) • The Computer Services Center at IIT Delhi Acknowledgements

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