# -*- coding: utf-8 -*-
r"""
This module implements solver objects who minimize an objective function. Call
:func:`solve` to solve your convex optimization problem using your instantiated
solver and functions objects. The :class:`solver` base class defines the
interface of all solver objects. The specialized solver objects inherit from
it and implement the class methods. The following solvers are included :
* :class:`forward_backward`: Forward-backward proximal splitting algorithm.
* :class:`douglas_rachford`: Douglas-Rachford proximal splitting algorithm.
* :class:`generalized_forward_backward`: Generalized Forward-Backward.
"""
import numpy as np
import time
from pyunlocbox.functions import dummy
[docs]def solve(functions, x0, solver=None, atol=None, dtol=None, rtol=1e-3,
xtol=None, maxit=200, verbosity='LOW'):
r"""
Solve an optimization problem whose objective function is the sum of some
convex functions.
This function minimizes the objective function :math:`f(x) =
\sum\limits_{k=0}^{k=K} f_k(x)`, i.e. solves
:math:`\operatorname{arg\,min}\limits_x f(x)` for :math:`x \in
\mathbb{R}^{n \times N}` where :math:`n` is the dimensionality of the data
and :math:`N` the number of independent problems. It returns a dictionary
with the found solution and some informations about the algorithm
execution.
Parameters
----------
functions : list of objects
A list of convex functions to minimize. These are objects who must
implement the :meth:`pyunlocbox.functions.func.eval` method. The
:meth:`pyunlocbox.functions.func.grad` and / or
:meth:`pyunlocbox.functions.func.prox` methods are required by some
solvers. Note also that some solvers can only handle two convex
functions while others may handle more. Please refer to the
documentation of the considered solver.
x0 : array_like
Starting point of the algorithm, :math:`x_0 \in \mathbb{R}^{n \times
N}`. Note that if you pass a numpy array it will be modified in place
during execution to save memory. It will then contain the solution. Be
careful to pass data of the type (int, float32, float64) you want your
computations to use.
solver : solver class instance, optional
The solver algorithm. It is an object who must inherit from
:class:`pyunlocbox.solvers.solver` and implement the :meth:`_pre`,
:meth:`_algo` and :meth:`_post` methods. If no solver object are
provided, a standard one will be chosen given the number of convex
function objects and their implemented methods.
atol : float, optional
The absolute tolerance stopping criterion. The algorithm stops when
:math:`f(x^t) < atol` where :math:`f(x^t)` is the objective function at
iteration :math:`t`. Default is None.
dtol : float, optional
Stop when the objective function is stable enough, i.e. when
:math:`\left|f(x^t) - f(x^{t-1})\right| < dtol`. Default is None.
rtol : float, optional
The relative tolerance stopping criterion. The algorithm stops when
:math:`\left|\frac{ f(x^t) - f(x^{t-1}) }{ f(x^t) }\right| < rtol`.
Default is :math:`10^{-3}`.
xtol : float, optional
Stop when the variable is stable enough, i.e. when :math:`\frac{\|x^t -
x^{t-1}\|_2}{\sqrt{n N}} < xtol`. Note that additional memory will be
used to store :math:`x^{t-1}`. Default is None.
maxit : int, optional
The maximum number of iterations. Default is 200.
verbosity : {'NONE', 'LOW', 'HIGH', 'ALL'}, optional
The log level : ``'NONE'`` for no log, ``'LOW'`` for resume at
convergence, ``'HIGH'`` for info at all solving steps, ``'ALL'`` for
all possible outputs, including at each steps of the proximal operators
computation. Default is ``'LOW'``.
Returns
-------
sol : ndarray
The problem solution.
solver : str
The used solver.
crit : {'ATOL', 'DTOL', 'RTOL', 'XTOL', 'MAXIT'}
The used stopping criterion. See above for definitions.
niter : int
The number of iterations.
time : float
The execution time in seconds.
objective : ndarray
The successive evaluations of the objective function at each iteration.
Examples
--------
>>> import pyunlocbox
>>> import numpy as np
Define a problem:
>>> y = [4, 5, 6, 7]
>>> f = pyunlocbox.functions.norm_l2(y=y)
Solve it:
>>> x0 = np.zeros(len(y))
>>> ret = pyunlocbox.solvers.solve([f], x0, atol=1e-2, verbosity='ALL')
INFO: Dummy objective function added.
INFO: Selected solver : forward_backward
norm_l2 evaluation : 1.260000e+02
dummy evaluation : 0.000000e+00
INFO: Forward-backward method : FISTA
Iteration 1 of forward_backward :
norm_l2 evaluation : 1.400000e+01
dummy evaluation : 0.000000e+00
objective = 1.40e+01
Iteration 2 of forward_backward :
norm_l2 evaluation : 1.555556e+00
dummy evaluation : 0.000000e+00
objective = 1.56e+00
Iteration 3 of forward_backward :
norm_l2 evaluation : 3.293044e-02
dummy evaluation : 0.000000e+00
objective = 3.29e-02
Iteration 4 of forward_backward :
norm_l2 evaluation : 8.780588e-03
dummy evaluation : 0.000000e+00
objective = 8.78e-03
Solution found after 4 iterations :
objective function f(sol) = 8.780588e-03
stopping criterion : ATOL
Verify the stopping criterion (should be smaller than atol=1e-2):
>>> np.linalg.norm(ret['sol'] - y)**2
0.008780587752251795
Show the solution (should be close to y w.r.t. the L2-norm measure):
>>> ret['sol']
array([ 4.03339154, 5.04173943, 6.05008732, 7.0584352 ])
Show the used solver:
>>> ret['solver']
'forward_backward'
Show some information about the convergence:
>>> ret['crit']
'ATOL'
>>> ret['niter']
4
>>> ret['time'] # doctest:+SKIP
0.0012578964233398438
>>> ret['objective'] # doctest:+NORMALIZE_WHITESPACE
[[126.0, 0], [13.999999999999998, 0], [1.5555555555555558, 0],
[0.032930436204105726, 0], [0.0087805877522517933, 0]]
"""
if verbosity not in ['NONE', 'LOW', 'HIGH', 'ALL']:
raise ValueError('Verbosity should be either NONE, LOW, HIGH or ALL.')
# Add a second dummy convex function if only one function is provided.
if len(functions) < 1:
raise ValueError('At least 1 convex function should be provided.')
elif len(functions) == 1:
functions.append(dummy())
if verbosity in ['LOW', 'HIGH', 'ALL']:
print('INFO: Dummy objective function added.')
# Choose a solver if none provided.
if not solver:
if len(functions) == 2:
fb0 = 'GRAD' in functions[0].cap(x0) and 'PROX' in functions[1].cap(x0)
fb1 = 'GRAD' in functions[1].cap(x0) and 'PROX' in functions[0].cap(x0)
dg0 = 'PROX' in functions[0].cap(x0) and 'PROX' in functions[1].cap(x0)
if fb0 or fb1:
solver = forward_backward() # Need one prox and 1 grad.
elif dg0:
solver = douglas_rachford() # Need two prox.
else:
raise ValueError('No suitable solver for the given functions.')
elif len(functions) > 2:
solver = generalized_forward_backward()
if verbosity in ['LOW', 'HIGH', 'ALL']:
print('INFO: Selected solver : %s' % (solver.__class__.__name__,))
# Set solver and functions verbosity.
translation = {'ALL': 'HIGH', 'HIGH': 'HIGH', 'LOW': 'LOW', 'NONE': 'NONE'}
solver.verbosity = translation[verbosity]
translation = {'ALL': 'HIGH', 'HIGH': 'LOW', 'LOW': 'NONE', 'NONE': 'NONE'}
functions_verbosity = []
for f in functions:
functions_verbosity.append(f.verbosity)
f.verbosity = translation[verbosity]
tstart = time.time()
crit = None
niter = 0
objective = [[f.eval(x0) for f in functions]]
rtol_only_zeros = True
# Solver specific initialization.
solver.pre(functions, x0)
while not crit:
niter += 1
if xtol != None:
last_sol = np.array(solver.sol, copy=True)
if verbosity in ['HIGH', 'ALL']:
print('Iteration %d of %s :' % (niter, solver.__class__.__name__))
# Solver iterative algorithm.
solver.algo(objective, niter)
objective.append([f.eval(solver.sol) for f in functions])
current = np.sum(objective[-1])
last = np.sum(objective[-2])
# Verify stopping criteria.
if atol != None and current < atol:
crit = 'ATOL'
if dtol != None and np.abs(current - last) < dtol:
crit = 'DTOL'
if rtol != None:
div = current # Prevent division by 0.
if div == 0:
if verbosity in ['LOW', 'HIGH', 'ALL']:
print('WARNING: objective function is equal to 0 !')
if last != 0:
div = last
else:
div = 1.0 # Result will be zero anyway.
else:
rtol_only_zeros = False
relative = np.abs((current - last) / div)
if relative < rtol and not rtol_only_zeros:
crit = 'RTOL'
if xtol != None:
err = np.linalg.norm(solver.sol - last_sol)
err /= np.sqrt(last_sol.size)
if err < xtol:
crit = 'XTOL'
if maxit != None and niter >= maxit:
crit = 'MAXIT'
if verbosity in ['HIGH', 'ALL']:
print(' objective = %.2e' % current)
# Restore verbosity for functions. In case they are called outside solve().
for k, f in enumerate(functions):
f.verbosity = functions_verbosity[k]
if verbosity in ['LOW', 'HIGH', 'ALL']:
print('Solution found after %d iterations :' % niter)
print(' objective function f(sol) = %e' % current)
print(' stopping criterion : %s' % crit)
# Returned dictionary.
result = {'sol': solver.sol,
'solver': solver.__class__.__name__, # algo for consistency ?
'crit': crit,
'niter': niter,
'time': time.time() - tstart,
'objective': objective}
# Solver specific post-processing (e.g. delete references).
solver.post()
return result
[docs]class solver(object):
r"""
Defines the solver object interface.
This class defines the interface of a solver object intended to be passed
to the :func:`pyunlocbox.solvers.solve` solving function. It is intended to
be a base class for standard solvers which will implement the required
methods. It can also be instantiated by user code and dynamically modified
for rapid testing. This class also defines the generic attributes of all
solver objects.
Parameters
----------
step : float
The gradient-descent step-size. This parameter is bounded by 0 and
:math:`\frac{2}{\beta}` where :math:`\beta` is the Lipschitz constant
of the gradient of the smooth function (or a sum of smooth functions).
Default is 1.
post_step : function
User defined function to post-process the step size. This function is
called every iteration and permits the user to alter the solver
algorithm. The user may start with a high step size and progressively
lower it while the algorithm runs to accelerate the convergence. The
function parameters are the following : `step` (current step size),
`sol` (current problem solution), `objective` (list of successive
evaluations of the objective function), `niter` (current iteration
number). The function should return a new value for `step`. Default is
to return an unchanged value.
post_sol : function
User defined function to post-process the problem solution. This
function is called every iteration and permits the user to alter the
solver algorithm. Same parameter as :func:`post_step`. Default is to
return an unchanged value.
"""
def __init__(self, step=1, post_step=None, post_sol=None):
if step < 0:
raise ValueError('Gamma should be a positive number.')
self.step = step
if post_step:
self.post_step = post_step
else:
self.post_step = lambda step, sol, objective, niter: step
if post_sol:
self.post_sol = post_sol
else:
self.post_sol = lambda step, sol, objective, niter: sol
[docs] def pre(self, functions, x0):
"""
Solver specific initialization. See parameters documentation in
:func:`pyunlocbox.solvers.solve` documentation.
"""
self.sol = np.asarray(x0)
self._pre(functions, np.asarray(x0))
def _pre(self, functions, x0):
raise NotImplementedError("Class user should define this method.")
[docs] def algo(self, objective, niter):
"""
Call the solver iterative algorithm while allowing the user to alter
it. This makes it possible to dynamically change the `step` step size
while the algorithm is running. See parameters documentation in
:func:`pyunlocbox.solvers.solve` documentation.
"""
self._algo()
self.step = self.post_step(self.step, self.sol, objective, niter)
self.sol = self.post_sol(self.step, self.sol, objective, niter)
def _algo(self):
raise NotImplementedError("Class user should define this method.")
[docs] def post(self):
"""
Solver specific post-processing. Mainly used to delete references added
during initialization so that the garbage collector can free the
memory. See parameters documentation in
:func:`pyunlocbox.solvers.solve` documentation.
"""
self._post()
del self.sol
def _post(self):
raise NotImplementedError("Class user should define this method.")
[docs]class forward_backward(solver):
r"""
Forward-backward proximal splitting algorithm.
This algorithm solves convex optimization problems composed of the sum of
a smooth and a non-smooth function.
See generic attributes descriptions of the
:class:`pyunlocbox.solvers.solver` base class.
Parameters
----------
method : {'FISTA', 'ISTA'}, optional
The method used to solve the problem. It can be 'FISTA' or 'ISTA'. Note
that while FISTA is much more time efficient, it is less memory
efficient. Default is 'FISTA'.
lambda_ : float, optional
The update term weight for ISTA. It should be between 0 and 1. Default
is 1.
Notes
-----
This algorithm requires one function to implement the
:meth:`pyunlocbox.functions.func.prox` method and the other one to
implement the :meth:`pyunlocbox.functions.func.grad` method.
See :cite:`beck2009FISTA` for details about the algorithm.
Examples
--------
>>> from pyunlocbox import functions, solvers
>>> import numpy as np
>>> y = [4, 5, 6, 7]
>>> x0 = np.zeros(len(y))
>>> f1 = functions.norm_l2(y=y)
>>> f2 = functions.dummy()
>>> solver = solvers.forward_backward(method='FISTA', lambda_=1, step=0.5)
>>> ret = solvers.solve([f1, f2], x0, solver, atol=1e-5)
Solution found after 12 iterations :
objective function f(sol) = 4.135992e-06
stopping criterion : ATOL
>>> ret['sol']
array([ 3.99927529, 4.99909411, 5.99891293, 6.99873176])
"""
def __init__(self, method='FISTA', lambda_=1, *args, **kwargs):
super(forward_backward, self).__init__(*args, **kwargs)
self.method = method
self.lambda_ = lambda_
def _pre(self, functions, x0):
if self.verbosity is 'HIGH':
print('INFO: Forward-backward method : %s' % (self.method,))
if self.lambda_ <= 0 or self.lambda_ > 1:
raise ValueError('Lambda is bounded by 0 and 1.')
if self.method is 'ISTA':
self._algo = self._ista
elif self.method is 'FISTA':
self._algo = self._fista
self.z = np.array(x0, copy=True)
self.t = 1.
else:
raise ValueError('The method should be either FISTA or ISTA.')
if len(functions) != 2:
raise ValueError('Forward-backward requires two convex functions.')
if 'PROX' in functions[0].cap(x0) and 'GRAD' in functions[1].cap(x0):
self.f1 = functions[0]
self.f2 = functions[1]
elif 'PROX' in functions[1].cap(x0) and 'GRAD' in functions[0].cap(x0):
self.f1 = functions[1]
self.f2 = functions[0]
else:
raise ValueError('Forward-backward requires a function to '
'implement prox() and the other grad().')
def _ista(self):
x = self.sol - self.step * self.f2.grad(self.sol)
self.sol[:] += self.lambda_ * (self.f1.prox(x, self.step) - self.sol)
def _fista(self):
x = self.z - self.step * self.f2.grad(self.z)
x[:] = self.f1.prox(x, self.step)
tn = (1. + np.sqrt(1.+4.*self.t**2.)) / 2.
self.z[:] = x + (self.t-1.) / tn * (x-self.sol)
self.t = tn
self.sol[:] = x
def _post(self):
del self._algo, self.f1, self.f2
if self.method is 'FISTA':
del self.z, self.t
[docs]class generalized_forward_backward(solver):
r"""
Generalized forward-backward proximal splitting algorithm.
This algorithm solves convex optimization problems composed of the sum of
any number of non-smooth (or smooth) functions.
See generic attributes descriptions of the
:class:`pyunlocbox.solvers.solver` base class.
Parameters
----------
lambda_ : float, optional
A relaxation parameter bounded by 0 and 1. Default is 1.
Notes
-----
This algorithm requires each function to either implement the
:meth:`pyunlocbox.functions.func.prox` method or the
:meth:`pyunlocbox.functions.func.grad` method.
See :cite:`raguet2013generalizedFB` for details about the algorithm.
Examples
--------
>>> from pyunlocbox import functions, solvers
>>> import numpy as np
>>> y = [0.01, 0.2, 8, 0.3, 0 , 0.03, 7]
>>> x0 = np.zeros(len(y))
>>> f1 = functions.norm_l2(y=y)
>>> f2 = functions.norm_l1()
>>> solver = solvers.generalized_forward_backward(lambda_=1, step=0.5)
>>> ret = solvers.solve([f1, f2], x0, solver)
Solution found after 2 iterations :
objective function f(sol) = 1.463100e+01
stopping criterion : RTOL
>>> ret['sol']
array([ 0. , 0. , 7.5, 0. , 0. , 0. , 6.5])
"""
def __init__(self, lambda_=1, *args, **kwargs):
super(generalized_forward_backward, self).__init__(*args, **kwargs)
self.lambda_ = lambda_
def _pre(self, functions, x0):
if self.lambda_ <= 0 or self.lambda_ > 1:
raise ValueError('Lambda is bounded by 0 and 1.')
self.f = [] # Smooth functions.
self.g = [] # Non-smooth functions.
self.z = []
for f in functions:
if 'GRAD' in f.cap(x0):
self.f.append(f)
elif 'PROX' in f.cap(x0):
self.g.append(f)
self.z.append(np.array(x0, copy=True))
else:
raise ValueError('Generalized forward-backward requires each '
'function to implement prox() or grad().')
if self.verbosity is 'HIGH':
print('INFO: Generalized forward-backward minimizing %i smooth '
'functions and %i non-smooth functions.'
% (len(self.f), len(self.g)))
def _algo(self):
# Smooth functions.
grad = np.zeros(self.sol.shape)
for f in self.f:
grad += f.grad(self.sol)
# Non-smooth functions.
if not self.g:
self.sol[:] -= self.step * grad # Reduces to gradient descent.
else:
sol = np.zeros(self.sol.shape)
for i, g in enumerate(self.g):
tmp = 2 * self.sol - self.z[i] - self.step * grad
tmp[:] = g.prox(tmp, self.step * len(self.g))
self.z[i] += self.lambda_ * (tmp - self.sol)
sol += 1. * self.z[i] / len(self.g)
self.sol[:] = sol
def _post(self):
del self.f, self.g, self.z
[docs]class douglas_rachford(solver):
r"""
Douglas-Rachford proximal splitting algorithm.
This algorithm solves convex optimization problems composed of the sum of
two non-smooth (or smooth) functions.
See generic attributes descriptions of the
:class:`pyunlocbox.solvers.solver` base class.
Parameters
----------
lambda_ : float, optional
The update term weight. It should be between 0 and 1. Default is 1.
Notes
-----
This algorithm requires the two functions to implement the
:meth:`pyunlocbox.functions.func.prox` method.
See :cite:`combettes2007DR` for details about the algorithm.
Examples
--------
>>> from pyunlocbox import functions, solvers
>>> import numpy as np
>>> y = [4, 5, 6, 7]
>>> x0 = np.zeros(len(y))
>>> f1 = functions.norm_l2(y=y)
>>> f2 = functions.dummy()
>>> solver = solvers.douglas_rachford(lambda_=1, step=1)
>>> ret = solvers.solve([f1, f2], x0, solver, atol=1e-5)
Solution found after 8 iterations :
objective function f(sol) = 2.927052e-06
stopping criterion : ATOL
>>> ret['sol']
array([ 3.99939034, 4.99923792, 5.99908551, 6.99893309])
"""
def __init__(self, lambda_=1, *args, **kwargs):
super(douglas_rachford, self).__init__(*args, **kwargs)
self.lambda_ = lambda_
def _pre(self, functions, x0):
if self.lambda_ <= 0 or self.lambda_ > 1:
raise ValueError('Lambda is bounded by 0 and 1.')
if len(functions) != 2:
raise ValueError('Douglas-Rachford requires two convex functions.')
self.f1 = functions[0]
self.f2 = functions[1]
self.z = np.array(x0, copy=True)
def _algo(self):
tmp = self.f1.prox(2 * self.sol - self.z, self.step)
self.z[:] = self.z + self.lambda_ * (tmp - self.sol)
self.sol[:] = self.f2.prox(self.z, self.step)
def _post(self):
del self.f1, self.f2, self.z