Optimization API
================

Members: Bergstra, Lamblin, Delalleau, Glorot, Breuleux, Bordes
Leader: Bergstra


Description
-----------

This API is for iterative optimization algorithms, such as:

 - stochastic gradient descent (incl. momentum, annealing)
 - delta bar delta
 - conjugate methods
 - L-BFGS
 - "Hessian Free"
 - SGD-QN
 - TONGA

The API includes an iterative interface based on Theano, and a one-shot
interface similar to SciPy and MATLAB that is based on Python and Numpy, that
only uses Theano for the implementation.


Theano Interface
-----------------

The theano interface to optimization algorithms is to ask for a dictionary of
updates that can be used in theano.function.  Implementations of iterative
optimization algorithms should be global functions with a signature like
'iterative_optimizer'.

def iterative_optimizer(parameters, 
        cost=None,
        gradients=None,
        stop=None, 
        updates=None,
        **kwargs):
    """
    :param parameters: list or tuple of Theano variables 
        that we want to optimize iteratively.  If we're minimizing f(x), then
        together, these variables represent 'x'.  Typically these are shared
        variables and their values are the initial values for the minimization
        algorithm.

    :param cost: scalar-valued Theano variable that computes an exact or noisy estimate of
        cost  (what are the conditions on the noise?).  Some algorithms might
        need an exact cost, some algorithms might ignore the cost if the
        gradients are given.

    :param gradients: list or tuple of Theano variables representing the gradients on
        the corresponding parameters.  These default to tensor.grad(cost,
        parameters).

    :param stop: a shared variable (scalar integer) that (if provided) will be
        updated to say when the iterative minimization algorithm has finished
        (1) or requires more iterations (0).

    :param updates: a dictionary to update with the (var, new_value) items
        associated with the iterative algorithm.  The default is a new empty
        dictionary.  A KeyError is raised in case of key collisions.

    :param kwargs: algorithm-dependent arguments

    :returns: a dictionary mapping each parameter to an expression that it
       should take in order to carry out the optimization procedure.

       If all the parameters are shared variables, then this dictionary may be
       passed as the ``updates`` argument to theano.function.

       There may be more key,value pairs in the dictionary corresponding to
       internal variables that are part of the optimization algorithm.

    """


Numpy Interface
---------------

The numpy interface to optimization algorithms is supposed to mimick
scipy's.  Its arguments are numpy arrays, and functions that manipulate numpy
arrays.

TODO: There is also room for an iterative object (that doesn't hog program
control) but which nonetheless works on numpy objects.  Actually minimize() should
use this iterative interface under the hood.

def minimize(x0, f, df, opt_algo, **kwargs):
    """
    Return a point x_new that minimizes function `f` with derivative `df`.

    This is supposed to provide an interface similar to scipy's minimize
    routines, or MATLAB's.

    :type x0: numpy ndarray
    :param x0: starting point for minimization

    :type f: python callable mapping something like x0 to a scalar
    :param f: function to minimize

    :type df: python callable mapping something like x0 to the derivative of f at that point
    :param df: derivative of `f`

    :param opt_algo: one of the functions that implements the
    `iterative_optimizer` interface.

    :param kwargs: passed through to `opt_algo`

    """

OD asks: Could it be more convenient for x0 to be a list?
 
JB replies: Yes, but that's not the interface used by other minimize()
routines (e.g. in scipy).  Maybe another list-based interface is required?

OD replies: I think most people would prefer to use a list-based interface, so
    they don't have to manually pack / unpack multiple arrrays of parameters. So I
    would vote in favor or having both (where the main reason to also provide a
    non-list interface would be to allow one to easily switch e.g. to scipy's
    minimize). 
    I would guess the reason scipy's interface is like this is because it makes
    it easier for the optimization algorithm. However, this does not really
    matter if we are just wrapping a theano-based algorithm (that already has
    to handle multiple parameters), and avoiding useless data copies on each call
    to f / df can only help speed-wise.

OD asks: Why make a difference between iterative and one-shot versions? A one-shot
    algorithm can be seen as an iterative one that stops after its first
    iteration. The difference I see between the two interfaces proposed here
    is mostly that one relies on Theano while the other one does not, but
    hopefully a non-Theano one can be created by simply wrapping around the
    Theano one.

JB replies: Right, it would make more sense to distinguish them by the fact that
one works on Theano objects, and the other on general Python callable functions.
There is room for an iterative numpy interface, but I didn't make it yet.  Would
that answer your question?

OD replies and asks: Partly. Do we really need a non-iterative interface?

Examples
--------


Simple stochastic gradient descent with extra updates:

sgd([p], gradients=[g], updates={a:b}, step_size=.1) will return {a:b, p:p-.1*g}