Optimization scikit: a gradient-based optimization

Last time, I’ve made a simple example of a gradient-free optimization. Now, I’d like to use the gradient of my function (analytical gradient I’ve computed) to be able to get the global minimum in less iterations.

Setting up the optimizer and the cost function

I’ve refactored the cost function and I’ve added a gradient method:

class Function(object):
    def __call__(self, x):
        t = numpy.sqrt(numpy.sum(x**2, axis=0))
        return -numpy.sinc(t)
    def gradient(self, x):
        t = numpy.sqrt(numpy.sum(x**2, axis=0))
        return x / t**2 * (- numpy.cos(numpy.pi*t) + numpy.sinc(t))

This is how the scikit works. All cost functions are instance of this kind of class. The __call__ method returns the actual cost, and additional methods provide gradient or hessian. Here, only the gradient is required.

Now, I can build the optimizer:

from scikits.optimization import *
 
mystep = step.GradientStep()
mylinesearch = line_search.GoldenSectionSearch(min_alpha_step = 0.0001, alpha_step = .5)
mycriterion = criterion.criterion(ftol = 0.0000001, iterations_max = 10)

As I’ve said before, the scikit is built around the separation principle, so what I’ve created does not contain any state. Now, I can create the glue that will steer the optimization:

myoptimizer = optimizer.StandardOptimizer(function = fun,
                                          step = mystep,
                                          line_search = mylinesearch,
                                          criterion = mycriterion,
                                          x0 = numpy.array((.8, 1.)))

As for the polytope optimizer, I can begin the optimization by calling the optimize() method.

Conclusion

This is the evolution of the cost function during the optimization:

Cost function values for each iteration

This was a simple usage of a gradient optimizer. The function is of course easy to optimize, as it has a circular symmetry. Another problem is that you need to have access to an analytical gradient. In further tutorials, I’ll introduce numerical approximation classes that will help with this task.

As usual, the code of this tutorial is available on Launchpad.

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