# Copyright 2024 United Kingdom Research and Innovation
# Copyright 2024 The University of Manchester
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# Authors:
# - CIL Developers, listed at: https://github.com/TomographicImaging/CIL/blob/master/NOTICE.txt
from abc import ABC, abstractmethod
import numpy
from numbers import Number
[docs]
class StepSizeRule(ABC):
"""
Abstract base class for a step size rule. The abstract method, `get_step_size` takes in an algorithm and thus can access all parts of the algorithm (e.g. current iterate, current gradient, objective functions etc) and from this should return a float as a step size.
"""
def __init__(self):
'''Initialises the step size rule
'''
pass
[docs]
@abstractmethod
def get_step_size(self, algorithm):
"""
Returns
--------
the calculated step size:float
"""
pass
[docs]
class ConstantStepSize(StepSizeRule):
"""
Step-size rule that always returns a constant step-size.
Parameters
----------
step_size: float
The step-size to be returned with each call.
"""
def __init__(self, step_size):
'''Initialises the constant step size rule
Parameters:
-------------
step_size : float, the constant step size
'''
self.step_size = step_size
[docs]
def get_step_size(self, algorithm):
"""
Returns
--------
the calculated step size:float
"""
return self.step_size
[docs]
class ArmijoStepSizeRule(StepSizeRule):
r""" Applies the Armijo rule to calculate the step size (step_size).
The Armijo rule runs a while loop to find the appropriate step_size by starting from a very large number (`alpha`). The step_size is found by reducing the step size (by a factor `beta`) in an iterative way until a certain criterion is met. To avoid infinite loops, we add a maximum number of times (`max_iterations`) the while loop is run.
Parameters
----------
alpha: float, optional, default=1e6
The starting point for the step size iterations
beta: float between 0 and 1, optional, default=0.5
The amount the step_size is reduced if the criterion is not met
max_iterations: integer, optional, default is numpy.ceil (2 * numpy.log10(alpha) / numpy.log10(2))
The maximum number of iterations to find a suitable step size
Reference
------------
- Algorithm 3.1 in Nocedal, J. and Wright, S.J. eds., 1999. Numerical optimization. New York, NY: Springer New York. https://www.math.uci.edu/~qnie/Publications/NumericalOptimization.pdf)
- https://projecteuclid.org/download/pdf_1/euclid.pjm/1102995080
"""
def __init__(self, alpha=1e6, beta=0.5, max_iterations=None):
'''Initialises the step size rule
'''
self.alpha_orig = alpha
if self.alpha_orig is None: # Can be removed when alpha and beta are deprecated in GD
self.alpha_orig = 1e6
self.beta = beta
if self.beta is None: # Can be removed when alpha and beta are deprecated in GD
self.beta = 0.5
self.max_iterations = max_iterations
if self.max_iterations is None:
self.max_iterations = numpy.ceil(2 * numpy.log10(self.alpha_orig) / numpy.log10(2))
[docs]
def get_step_size(self, algorithm):
"""
Applies the Armijo rule to calculate the step size (`step_size`)
Returns
--------
the calculated step size:float
"""
k = 0
self.alpha = self.alpha_orig
f_x = algorithm.objective_function(algorithm.solution)
self.x_armijo = algorithm.solution.copy()
while k < self.max_iterations:
algorithm.gradient_update.multiply(self.alpha, out=self.x_armijo)
algorithm.solution.subtract(self.x_armijo, out=self.x_armijo)
f_x_a = algorithm.objective_function(self.x_armijo)
sqnorm = algorithm.gradient_update.squared_norm()
if f_x_a - f_x <= - (self.alpha/2.) * sqnorm:
break
k += 1.
self.alpha *= self.beta
if k == self.max_iterations:
raise ValueError(
'Could not find a proper step_size in {} loops. Consider increasing alpha or max_iterations.'.format(self.max_iterations))
return self.alpha
[docs]
class BarzilaiBorweinStepSizeRule(StepSizeRule):
r""" Applies the Barzilai- Borwein rule to calculate the step size (step_size).
Let :math:`\Delta x=x_k-x_{k-1}` and :math:`\Delta g=g_k-g_{k-1}`. Where :math:`x_k` is the :math:`k` th iterate (current solution after iteration :math:`k` ) and :math:`g_k` is the gradient calculation in the :math:`k` th iterate, found in :code:`algorithm.gradient_update`. A Barzilai-Borwein (BB) iteration is :math:`x_{k+1}=x_k-\alpha_kg_k` where the step size :math:`\alpha _k` is either
- :math:`\alpha_k^{LONG}=\frac{\Delta x\cdot\Delta x}{\Delta x\cdot\Delta g}`, or
- :math:`\alpha_k^{SHORT}=\frac{\Delta x \cdot\Delta g}{\Delta g \cdot\Delta g}`.
Where the operator :math:`\cdot` is the standard inner product between two vectors.
This is suitable for use with gradient based iterative methods where the calculated gradient is stored as `algorithm.gradient_update`.
Parameters
----------
initial: float, greater than zero
The step-size for the first iteration. We recommend something of the order :math:`1/f.L` where :math:`f` is the (differentiable part of) the objective you wish to minimise.
mode: One of 'long', 'short' or 'alternate', default is 'short'.
This calculates the step-size based on the LONG, SHORT or alternating between the two, starting with short.
stabilisation_param: 'auto', float or 'off', default is 'auto'
In order to add stability the step-size has an upper limit of :math:`\Delta/\|g_k\|` where by 'default', the `stabilisation_param`, :math:`\Delta` is determined automatically to be the minimium of :math:`\Delta x` from the first 3 iterations. The user can also pass a fixed constant or turn "off" the stabilisation, equivalently passing `np.inf`.
Reference
---------
- Barzilai, Jonathan; Borwein, Jonathan M. (1988). "Two-Point Step Size Gradient Methods". IMA Journal of Numerical Analysis. 8: 141–148, https://doi.org/10.1093/imanum/8.1.141
- Burdakov, O., Dai, Y. and Huang, N., 2019. STABILIZED BARZILAI-BORWEIN METHOD. Journal of Computational Mathematics, 37(6). https://doi.org/10.4208/jcm.1911-m2019-0171
- https://en.wikipedia.org/wiki/Barzilai-Borwein_method
"""
def __init__(self, initial, mode='short', stabilisation_param="auto"):
'''Initialises the step size rule
'''
self.mode=mode
if self.mode == 'short':
self.is_short = True
elif self.mode == 'long' or self.mode == 'alternate':
self.is_short = False
else:
raise ValueError('Mode should be chosen from "long", "short" or "alternate". ')
self.store_grad=None
self.store_x=None
self.initial=initial
if stabilisation_param == 'auto':
self.adaptive = True
stabilisation_param = numpy.inf
elif stabilisation_param == "off":
self.adaptive = False
stabilisation_param = numpy.inf
elif ( isinstance(stabilisation_param, Number) and stabilisation_param >=0):
self.adaptive = False
else:
raise TypeError(" The stabilisation_param should be 'auto', a positive number or 'off'")
self.stabilisation_param=stabilisation_param
[docs]
def get_step_size(self, algorithm):
"""
Applies the B-B rule to calculate the step size (`step_size`)
Returns
--------
the calculated step size:float
"""
#For the first iteration we use an initial step size because the BB step size requires a previous iterate.
if self.store_x is None:
self.store_x=algorithm.x.copy() # We store the last iterate in order to calculate the BB step size
self.store_grad=algorithm.gradient_update.copy()# We store the last gradient in order to calculate the BB step size
return self.initial
gradient_norm = algorithm.gradient_update.norm()
#If the gradient is zero, gradient based algorithms will not update and te step size calculation will divide by zero so we stop iterations.
if gradient_norm < 1e-8:
raise StopIteration
algorithm.x.subtract(self.store_x, out=self.store_x)
algorithm.gradient_update.subtract(self.store_grad, out=self.store_grad)
if self.is_short:
ret = (self.store_x.dot(self.store_grad))/ (self.store_grad.dot(self.store_grad))
else:
ret = (self.store_x.dot(self.store_x))/ (self.store_x.dot(self.store_grad))
#This computes the default stabilisation parameter, using the first three iterations
if (algorithm.iteration <=3 and self.adaptive):
self.stabilisation_param = min(self.stabilisation_param, self.store_x.norm() )
# Computes the step size as the minimum of the ret, above, and :math:`\Delta/\|g_k\|` ignoring any NaN values.
ret = numpy.nanmin( numpy.array([ret, self.stabilisation_param/gradient_norm]))
# We store the last iterate and gradient in order to calculate the BB step size
self.store_x.fill(algorithm.x)
self.store_grad.fill(algorithm.gradient_update)
if self.mode == "alternate":
self.is_short = not self.is_short
return ret
