A Trust-Region Newton Method in Python

.. DANGER:: This is alpha quality software and still quite rough on the edges. Specifically the error management is still lacking (which means that if something goes wrong in the optimization you won't see an error message but just get garbage). These things are being worked out but we're not quite there yet.

.. image:: http://fa.bianp.net/blog/static/images/2013/comparison_logistic_corr_10.png

The main function is pytron.minimize::

def minimize(func, grad_hess, x0, args=(), max_iter=1000, tol=1e-6):

    Parameters
    ----------
    func : callable
        func(w, *args) is the evaluation of the function at w, It
        should return a float.
    grad_hess: callable
        returns the gradient and a callable with the hessian times
        an arbitrary vector.
    tol: float
        stopping criterion. XXX TODO. what is the stopping criterion ?

    Returns
    -------
    w : array

Stopping criterion

It stops whenever ||grad(x)|| < eps or the maximum number of iterations is attained.

TODO: add tol

Examples

Code

This software uses the C++ implementation of TRON optimization software <http://www.mcs.anl.gov/~more/tron/>_ (files src/tron.{h,cpp}) distributed from the LIBLINEAR sources (v1.93), which is BSD licensed. Note that the original Fortran TRON implementation (available here <http://fa.bianp.net/projects/pytron/tron-1.2.tar.gz>_) is not open source and is not used in this project.

The modifications with respect to the orginal code are:

* Do not initialize values to zero, allow arbitrary initializations

* Modify stopping criterion to comply with scipy.optimize API. Stop
  whenever gradient is smaller than a given quantity, specified in the
  gtol argument

* Return the gradient from TRON::tron (pass by reference)

* Add `tol` option to TRON

* Rename `eps` to `gtol`.

* Use infinity norm as stopping criterion for gradient instead of L2.

TODO

* return status from TRON::TRON
* callback argument

References

If you use the software please consider citing some of the references below.

The method is described in the paper "Newton's Method for Large Bound-Constrained Optimization Problems", Chih-Jen Lin and Jorge J. Moré (http://epubs.siam.org/doi/abs/10.1137/S1052623498345075)

It is also discussed in the contex of Logistic Regression in the paper "Trust Region Newton Method for Logistic Regression", Chih-Jen Lin, Ruby C. Weng, S. Sathiya Keerthi (http://dl.acm.org/citation.cfm?id=1390703)

The website http://www.mcs.anl.gov/~more/tron/ contains reference to this implementation, although the links to the software seem to be currently broken (May 2013).

License

This code is licensed under the terms of the BSD license. See file COPYING for more details.

Acknowledgement

The source code for the