Solvers

The pyunlocbox.solvers module implements a solving function (which will minimize your objective function) as well as common solvers.

Solving

Call solve() to solve your convex optimization problem using your instantiated solver and functions objects.

Interface

The solver base class defines a common interface to all solvers:

solver.pre(functions, x0)	Solver-specific pre-processing.
solver.algo(objective, niter)	Call the solver iterative algorithm and the provided acceleration scheme.
solver.post()	Solver-specific post-processing.

Solvers

Then, derived classes implement various common solvers.

gradient_descent(**kwargs)	Gradient descent algorithm.
forward_backward([accel])	Forward-backward proximal splitting (FISTA and ISTA) algorithm.
douglas_rachford([lambda_])	Douglas-Rachford proximal splitting algorithm.
generalized_forward_backward([lambda_])	Generalized forward-backward proximal splitting algorithm.

Primal-dual solvers (based on primal_dual)

mlfbf([L, Lt, d0])	Monotone+Lipschitz forward-backward-forward primal-dual algorithm.
projection_based([lambda_])	Projection-based primal-dual algorithm.

pyunlocbox.solvers.solve(functions, x0, solver=None, atol=None, dtol=None, rtol=0.001, xtol=None, maxit=200, verbosity='LOW', inplace=False)

Solve an optimization problem whose objective function is the sum of some convex functions.

This function minimizes the objective function \(f(x) = \sum\limits_{k=0}^{k=K} f_k(x)\), i.e. solves \(\operatorname{arg\,min}\limits_x f(x)\) for \(x \in \mathbb{R}^{n \times N}\) where \(n\) is the dimensionality of the data and \(N\) the number of independent problems. It returns a dictionary with the found solution and some informations about the algorithm execution.

Parameters

functionslist of objects

A list of convex functions to minimize. These are objects who must implement the pyunlocbox.functions.func.eval() method. The pyunlocbox.functions.func.grad() and / or 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.

x0array_like

Starting point of the algorithm, \(x_0 \in \mathbb{R}^{n \times N}\).

solversolver class instance, optional

The solver algorithm. It is an object who must inherit from pyunlocbox.solvers.solver and implement the _pre(), _algo() and _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.

atolfloat, optional

The absolute tolerance stopping criterion. The algorithm stops when \(f(x^t) < atol\) where \(f(x^t)\) is the objective function at iteration \(t\). Default is None.

dtolfloat, optional

Stop when the objective function is stable enough, i.e. when \(\left|f(x^t) - f(x^{t-1})\right| < dtol\). Default is None.

rtolfloat, optional

The relative tolerance stopping criterion. The algorithm stops when \(\left|\frac{ f(x^t) - f(x^{t-1}) }{ f(x^t) }\right| < rtol\). Default is \(10^{-3}\).

xtolfloat, optional

Stop when the variable is stable enough, i.e. when \(\frac{\|x^t - x^{t-1}\|_2}{\sqrt{n N}} < xtol\). Note that additional memory will be used to store \(x^{t-1}\). Default is None.

maxitint, 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'.

inplacebool, optional

If True and x0 is a numpy array, then x0 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.

Returns

solndarray

The problem solution.

solverstr

The used solver.

crit{‘ATOL’, ‘DTOL’, ‘RTOL’, ‘XTOL’, ‘MAXIT’}

The used stopping criterion. See above for definitions.

niterint

The number of iterations.

timefloat

The execution time in seconds.

objectivendarray

The successive evaluations of the objective function at each iteration.

Examples

>>> import numpy as np
>>> from pyunlocbox import functions, solvers

Define a problem:

>>> y = [4, 5, 6, 7]
>>> f = functions.norm_l2(y=y)

Solve it:

>>> x0 = np.zeros(len(y))
>>> ret = solvers.solve([f], x0, atol=1e-2, verbosity='ALL')
INFO: Dummy objective function added.
INFO: Selected solver: forward_backward
INFO: Forward-backward method
    dummy evaluation: 0.000000e+00
    norm_l2 evaluation: 1.260000e+02
Iteration 1 of forward_backward:
    dummy evaluation: 0.000000e+00
    norm_l2 evaluation: 1.400000e+01
    objective = 1.40e+01
Iteration 2 of forward_backward:
    dummy evaluation: 0.000000e+00
    norm_l2 evaluation: 2.963739e-01
    objective = 2.96e-01
Iteration 3 of forward_backward:
    dummy evaluation: 0.000000e+00
    norm_l2 evaluation: 7.902529e-02
    objective = 7.90e-02
Iteration 4 of forward_backward:
    dummy evaluation: 0.000000e+00
    norm_l2 evaluation: 5.752265e-02
    objective = 5.75e-02
Iteration 5 of forward_backward:
    dummy evaluation: 0.000000e+00
    norm_l2 evaluation: 5.142032e-03
    objective = 5.14e-03
Solution found after 5 iterations:
    objective function f(sol) = 5.142032e-03
    stopping criterion: ATOL

Verify the stopping criterion (should be smaller than atol=1e-2):

>>> np.linalg.norm(ret['sol'] - y)**2  
0.00514203...

Show the solution (should be close to y w.r.t. the L2-norm measure):

>>> ret['sol']
array([4.02555301, 5.03194126, 6.03832952, 7.04471777])

Show the used solver:

>>> ret['solver']
'forward_backward'

Show some information about the convergence:

>>> ret['crit']
'ATOL'
>>> ret['niter']
5
>>> ret['time']  
0.0012578964233398438
>>> ret['objective']  
[[126.0, 0], [13.99999999..., 0], [0.29637392..., 0], [0.07902528..., 0],
[0.05752265..., 0], [0.00514203..., 0]]

class pyunlocbox.solvers.solver(step=1.0, accel=None)

Bases: object

Defines the solver object interface.

This class defines the interface of a solver object intended to be passed to the 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

stepfloat

The gradient-descent step-size. This parameter is bounded by 0 and \(\frac{2}{\beta}\) where \(\beta\) is the Lipschitz constant of the gradient of the smooth function (or a sum of smooth functions). Default is 1.

accelpyunlocbox.acceleration.accel

User-defined object used to adaptively change the current step size and solution while the algorithm is running. Default is a dummy object that returns unchanged values.

pre(functions, x0)

Solver-specific pre-processing. See parameters documentation in pyunlocbox.solvers.solve() documentation.

Notes

When preprocessing the functions, the solver should split them into two lists: * self.smooth_funs, for functions involved in gradient steps. * self.non_smooth_funs, for functions involved proximal steps. This way, any method that takes in the solver as argument, such as the methods in pyunlocbox.acceleration.accel, can have some context as to how the solver is using the functions.

algo(objective, niter)

Call the solver iterative algorithm and the provided acceleration scheme. See parameters documentation in pyunlocbox.solvers.solve()

Notes

The method self.accel.update_sol() is called before self._algo() because the acceleration schemes usually involves some sort of averaging of previous solutions, which can add some unwanted artifacts on the output solution. With this ordering, we guarantee that the output of solver.algo is not corrupted by the acceleration scheme.

Similarly, the method self.accel.update_step() is called after self._algo() to allow the step update procedure to act directly on the solution output by the underlying algorithm, and not on the intermediate solution output by the acceleration scheme in self.accel.update_sol().

post()

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 pyunlocbox.solvers.solve().

objective(x)

Return the objective function at x.

Necessitate solver._pre(…) to be run first.

class pyunlocbox.solvers.gradient_descent(**kwargs)

Bases: pyunlocbox.solvers.solver

Gradient descent algorithm.

This algorithm solves optimization problems composed of the sum of any number of smooth functions.

See generic attributes descriptions of the pyunlocbox.solvers.solver base class.

Notes

This algorithm requires each function implement the pyunlocbox.functions.func.grad() method.

See pyunlocbox.acceleration.regularized_nonlinear for a very efficient acceleration scheme for this method.

Examples

>>> import numpy as np
>>> from pyunlocbox import functions, solvers
>>> dim = 25
>>> np.random.seed(0)
>>> xstar = np.random.rand(dim)  # True solution
>>> x0 = np.random.rand(dim)
>>> x0 = xstar + 5*(x0 - xstar) / np.linalg.norm(x0 - xstar)
>>> A = np.random.rand(dim, dim)
>>> step = 1 / np.linalg.norm(np.dot(A.T, A))
>>> f = functions.norm_l2(lambda_=0.5, A=A, y=np.dot(A, xstar))
>>> fd = functions.dummy()
>>> solver = solvers.gradient_descent(step=step)
>>> params = {'rtol':0, 'maxit':14000, 'verbosity':'NONE'}
>>> ret = solvers.solve([f, fd], x0, solver, **params)
>>> pctdiff = 100 * np.sum((xstar - ret['sol'])**2) / np.sum(xstar**2)
>>> print('Difference: {0:.1f}%'.format(pctdiff))
Difference: 1.3%

class pyunlocbox.solvers.forward_backward(accel=<pyunlocbox.acceleration.fista object>, **kwargs)

Bases: pyunlocbox.solvers.solver

Forward-backward proximal splitting (FISTA and ISTA) 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 pyunlocbox.solvers.solver base class.

Parameters

accelpyunlocbox.acceleration.accel

Acceleration scheme to use. Default is pyunlocbox.acceleration.fista(), which corresponds to the ‘FISTA’ solver. Passing pyunlocbox.acceleration.dummy() instead results in the ISTA solver. Note that while FISTA is much more time-efficient, it is less memory-efficient.

Notes

This algorithm requires one function to implement the pyunlocbox.functions.func.prox() method and the other one to implement the pyunlocbox.functions.func.grad() method.

See [BT09a] for details about the algorithm.

Examples

>>> import numpy as np
>>> from pyunlocbox import functions, solvers
>>> y = [4, 5, 6, 7]
>>> x0 = np.zeros(len(y))
>>> f1 = functions.norm_l2(y=y)
>>> f2 = functions.dummy()
>>> solver = solvers.forward_backward(step=0.5)
>>> ret = solvers.solve([f1, f2], x0, solver, atol=1e-5)
Solution found after 15 iterations:
    objective function f(sol) = 4.957288e-07
    stopping criterion: ATOL
>>> ret['sol']
array([4.0002509 , 5.00031362, 6.00037635, 7.00043907])

class pyunlocbox.solvers.generalized_forward_backward(lambda_=1, *args, **kwargs)

Bases: pyunlocbox.solvers.solver

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 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 pyunlocbox.functions.func.prox() method or the pyunlocbox.functions.func.grad() method.

See [RFP13] for details about the algorithm.

Examples

>>> import numpy as np
>>> from pyunlocbox import functions, solvers
>>> 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])

class pyunlocbox.solvers.douglas_rachford(lambda_=1, *args, **kwargs)

Bases: pyunlocbox.solvers.solver

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 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 pyunlocbox.functions.func.prox() method.

See [CP07] for details about the algorithm.

Examples

>>> import numpy as np
>>> from pyunlocbox import functions, solvers
>>> 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])

class pyunlocbox.solvers.primal_dual(L=None, Lt=None, d0=None, *args, **kwargs)

Bases: pyunlocbox.solvers.solver

Parent class of all primal-dual algorithms.

See generic attributes descriptions of the pyunlocbox.solvers.solver base class.

Parameters

Lfunction or ndarray, optional

The transformation L that maps from the primal variable space to the dual variable space. Default is the identity, \(L(x)=x\). If L is an ndarray, it will be converted to the operator form.

Ltfunction or ndarray, optional

The adjoint operator. If Lt is an ndarray, it will be converted to the operator form. If L is an ndarray, default is the transpose of L. If L is a function, default is L, \(Lt(x)=L(x)\).

d0: ndarray, optional

Initialization of the dual variable.

class pyunlocbox.solvers.mlfbf(L=None, Lt=None, d0=None, *args, **kwargs)

Bases: pyunlocbox.solvers.primal_dual

Monotone+Lipschitz forward-backward-forward primal-dual algorithm.

This algorithm solves convex optimization problems with objective of the form \(f(x) + g(Lx) + h(x)\), where \(f\) and \(g\) are proper, convex, lower-semicontinuous functions with easy-to-compute proximity operators, and \(h\) has Lipschitz-continuous gradient with constant \(\beta\).

See generic attributes descriptions of the pyunlocbox.solvers.primal_dual base class.

Notes

The order of the functions matters: set \(f\) first on the list, \(g\) second, and \(h\) third.

This algorithm requires the first two functions to implement the pyunlocbox.functions.func.prox() method, and the third function to implement the pyunlocbox.functions.func.grad() method.

The step-size should be in the interval \(\left] 0, \frac{1}{\beta + \|L\|_{2}}\right[\).

See [KP15], Algorithm 6, for details.

Examples

>>> import numpy as np
>>> from pyunlocbox import functions, solvers
>>> y = np.array([294, 390, 361])
>>> L = np.array([[5, 9, 3], [7, 8, 5], [4, 4, 9], [0, 1, 7]])
>>> x0 = np.zeros(len(y))
>>> f = functions.dummy()
>>> f._prox = lambda x, T: np.maximum(np.zeros(len(x)), x)
>>> g = functions.norm_l2(lambda_=0.5)
>>> h = functions.norm_l2(y=y, lambda_=0.5)
>>> max_step = 1/(1 + np.linalg.norm(L, 2))
>>> solver = solvers.mlfbf(L=L, step=max_step/2.)
>>> ret = solvers.solve([f, g, h], x0, solver, maxit=1000, rtol=0)
Solution found after 1000 iterations:
    objective function f(sol) = 1.839060e+05
    stopping criterion: MAXIT
>>> ret['sol']
array([1., 1., 1.])

class pyunlocbox.solvers.projection_based(lambda_=1.0, *args, **kwargs)

Bases: pyunlocbox.solvers.primal_dual

Projection-based primal-dual algorithm.

This algorithm solves convex optimization problems with objective of the form \(f(x) + g(Lx)\), where \(f\) and \(g\) are proper, convex, lower-semicontinuous functions with easy-to-compute proximity operators.

See generic attributes descriptions of the pyunlocbox.solvers.primal_dual base class.

Parameters

lambda_float, optional

The update term weight. It should be between 0 and 2. Default is 1.

Notes

The order of the functions matters: set \(f\) first on the list, and \(g\) second.

This algorithm requires the two functions to implement the pyunlocbox.functions.func.prox() method.

The step-size should be in the interval \(\left] 0, \infty \right[\).

See [KP15], Algorithm 7, for details.

Examples

>>> import numpy as np
>>> from pyunlocbox import functions, solvers
>>> y = np.array([294, 390, 361])
>>> L = np.array([[5, 9, 3], [7, 8, 5], [4, 4, 9], [0, 1, 7]])
>>> x0 = np.array([500, 1000, -400])
>>> f = functions.norm_l1(y=y)
>>> g = functions.norm_l1()
>>> solver = solvers.projection_based(L=L, step=1.)
>>> ret = solvers.solve([f, g], x0, solver, maxit=1000, rtol=None, xtol=.1)
Solution found after 996 iterations:
    objective function f(sol) = 1.045000e+03
    stopping criterion: XTOL
>>> ret['sol']
array([0, 0, 0])