Solver class hierarchy

Solver object interface

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:

step : float

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.

accel : pyunlocbox.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.

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().

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.

Gradient descent algorithm

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

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

Forward-backward proximal splitting algorithm

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

Bases: pyunlocbox.solvers.solver

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 pyunlocbox.solvers.solver base class.

Parameters:

accel : pyunlocbox.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

>>> 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(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])

Douglas-Rachford proximal splitting algorithm

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

>>> 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])

Generalized forward-backward proximal splitting algorithm

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

>>> 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])

Primal-dual algorithms

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:

L : function 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.

Lt : function 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.

Monotone+Lipschitz forward-backward-forward algorithm

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

>>> from pyunlocbox import functions, solvers
>>> import numpy as np
>>> 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) = 1.833865e+05
    stopping criterion: MAXIT
>>> ret['sol']
array([ 1.,  1.,  1.])

Projection-based primal-dual algorithm

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

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