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Topics in Convex Optimisation

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Ryota Tomioka, University of Tokyo
Thursday 22 March 2007, 13:00-15:00
5th floor meeting room, Engineering Department.

If you have a question about this talk, please contact Frederik Eaton .

This week, Ryota Tomioka will present some topics in convex optimisation. The primary reference will be subsections of Boyd and Vandenberghe, Convex Optimization

There is also a paper:

Performance Guarantees for Regularized Maximum Entropy Density Estimation M Dudik, SJ Phillips, RE Schapire – 17th Annual Conference on Learning Theory, 2004

The topics are:

1. Convex function (3.1; p67)

2. Legendre-Fenchel transformation (conjugate function) (3.3; p90)

3. norm and dual norm (appendix A.1)

4. Convex optimization problem (4.2; p136)

5. Lagrangian function (5.1.2; p216)

6. Lagrangian dual problem (5.2; p223)

7. Complementary slackness (5.5.2; p242)

8. Karush-Kuhn-Tucker (KKT) conditions (5.5.3; p243)

9. Maximum likelihood and maximum entropy (see Dudik et al. 2004)

10. Duality in information geometry

1-4 are basic definitions from sections 2,3,4

5-8 are from section 5 “duality”

9-10 are examples of duality in ML (not in the book)

Here are some interesting blogs talking about the connection between Fourier transformation and Legendre transformation:

http://sigfpe.blogspot.com/2005/10/quantum-mechanics-and-fourier-legendre.html http://math.ucr.edu/home/baez/qg-spring2004/discussion.html#idempotent

This talk is part of the Machine Learning Reading Group @ CUED series.

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Abstract

This week, Ryota Tomioka will present some topics in convex optimisation. The primary reference will be subsections of Boyd and Vandenberghe, Convex Optimization

There is also a paper:

Performance Guarantees for Regularized Maximum Entropy Density Estimation M Dudik, SJ Phillips, RE Schapire – 17th Annual Conference on Learning Theory, 2004

The topics are:

1. Convex function (3.1; p67)

2. Legendre-Fenchel transformation (conjugate function) (3.3; p90)

3. norm and dual norm (appendix A.1)

4. Convex optimization problem (4.2; p136)

5. Lagrangian function (5.1.2; p216)

6. Lagrangian dual problem (5.2; p223)

7. Complementary slackness (5.5.2; p242)

8. Karush-Kuhn-Tucker (KKT) conditions (5.5.3; p243)

9. Maximum likelihood and maximum entropy (see Dudik et al. 2004)

10. Duality in information geometry

1-4 are basic definitions from sections 2,3,4

5-8 are from section 5 “duality”

9-10 are examples of duality in ML (not in the book)

Here are some interesting blogs talking about the connection between Fourier transformation and Legendre transformation:

http://sigfpe.blogspot.com/2005/10/quantum-mechanics-and-fourier-legendre.html http://math.ucr.edu/home/baez/qg-spring2004/discussion.html#idempotent

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