import numpy as np # engine for numerical computing
from pypop7.optimizers.core.optimizer import Optimizer
from pypop7.optimizers.sa.sa import SA
[docs]class NSA(SA):
"""Noisy Simulated Annealing (NSA).
.. note:: This is a *slightly modified* version of discrete `NSA` for continuous optimization.
Parameters
----------
problem : dict
problem arguments with the following common settings (`keys`):
* 'fitness_function' - objective function to be **minimized** (`func`),
* 'ndim_problem' - number of dimensionality (`int`),
* 'upper_boundary' - upper boundary of search range (`array_like`),
* 'lower_boundary' - lower boundary of search range (`array_like`).
options : dict
optimizer options with the following common settings (`keys`):
* 'max_function_evaluations' - maximum of function evaluations (`int`, default: `np.Inf`),
* 'max_runtime' - maximal runtime to be allowed (`float`, default: `np.Inf`),
* 'seed_rng' - seed for random number generation needed to be *explicitly* set (`int`);
and with the following particular settings (`keys`):
* 'x' - initial (starting) point (`array_like`),
* 'sigma' - initial global step-size (`float`),
* 'is_noisy' - whether or not to minimize a **noisy** cost function (`bool`, default: `False`),
* 'schedule' - schedule for sampling intensity (`str`, default: `linear`),
* currently only two (*linear* or *quadratic*) schedules are supported for sampling intensity,
* 'n_samples' - number of samples (`int`),
* 'rt' - reducing factor of annealing temperature (`float`, default: `0.99`).
Examples
--------
Use the optimizer to minimize the well-known test function
`Rosenbrock <http://en.wikipedia.org/wiki/Rosenbrock_function>`_:
.. code-block:: python
:linenos:
>>> import numpy # engine for numerical computing
>>> from pypop7.benchmarks.base_functions import rosenbrock # function to be minimized
>>> from pypop7.optimizers.sa.nsa import NSA
>>> problem = {'fitness_function': rosenbrock, # define problem arguments
... 'ndim_problem': 2,
... 'lower_boundary': -5*numpy.ones((2,)),
... 'upper_boundary': 5*numpy.ones((2,))}
>>> options = {'max_function_evaluations': 5000, # set optimizer options
... 'seed_rng': 2022,
... 'x': 3*numpy.ones((2,)),
... 'sigma': 1.0,
... 'temperature': 100.0}
>>> nsa = NSA(problem, options) # initialize the optimizer class
>>> results = nsa.optimize() # run the optimization process
>>> # return the number of function evaluations and best-so-far fitness
>>> print(f"NSA: {results['n_function_evaluations']}, {results['best_so_far_y']}")
NSA: 5000, 0.006086567926462302
For its correctness checking of coding, the *code-based repeatability report* cannot be provided owing to
the lack of some details of its experiments in the original paper.
Attributes
----------
is_noisy : `bool`
whether or not to minimize a noisy cost function.
n_samples : `int`
number of samples for each iteration.
rt : `float`
reducing factor of annealing temperature.
schedule : `str`
schedule for sampling intensity.
sigma : `float`
global step-size (fixed during optimization).
x : `array_like`
initial (starting) point.
References
----------
Bouttier, C. and Gavra, I., 2019.
Convergence rate of a simulated annealing algorithm with noisy observations.
Journal of Machine Learning Research, 20(1), pp.127-171.
https://www.jmlr.org/papers/v20/16-588.html
"""
def __init__(self, problem, options):
SA.__init__(self, problem, options)
self.sigma = options.get('sigma')
assert self.sigma > 0.0
self.is_noisy = options.get('is_noisy', False)
assert self.is_noisy in [True, False]
self.schedule = options.get('schedule', 'linear') # schedule for sampling intensity
assert self.schedule in ['linear', 'quadratic'],\
'Currently only two (*linear* or *quadratic*) schedules are supported for sampling intensity.'
if self.is_noisy:
self.n_samples = options.get('n_samples')
else:
self.n_samples = 1 # a mandatory setting
assert self.n_samples > 0
self.rt = options.get('cr', 0.99) # reducing factor of temperature
self._tk = 0
def initialize(self, args=None):
if self.x is None: # starting point
x = self.rng_initialization.uniform(self.initial_lower_boundary, self.initial_upper_boundary)
else:
x = np.copy(self.x)
assert len(x) == self.ndim_problem
y = self._evaluate_fitness(x, args)
self.parent_x, self.parent_y = np.copy(x), np.copy(y)
return y
def iterate(self, args=None):
x = self.parent_x + self.sigma*self.rng_optimization.standard_normal((self.ndim_problem,))
self._tk += self.rng_optimization.exponential()
if self.schedule == 'linear':
n_tk = self._tk
else: # quadratic
n_tk = np.square(self._tk)
if self.n_samples is None:
n_samples = self.rng_optimization.poisson(n_tk) + 1
else:
n_samples = self.n_samples
y, parent_y = [], []
for _ in range(n_samples):
if self._check_terminations():
break
y.append(self._evaluate_fitness(x, args))
if self.is_noisy: # for noisy optimization
for _ in range(n_samples):
if self._check_terminations():
break
parent_y.append(self._evaluate_fitness(self.parent_x, args))
else: # for static optimization
parent_y = self.parent_y
diff = np.mean(parent_y) - np.mean(y)
if (diff >= 0) or (self.rng_optimization.random() < np.exp(diff/self.temperature)):
self.parent_x, self.parent_y = np.copy(x), np.copy(y)
if not self.is_noisy: # for static optimization
parent_y = []
if len(parent_y) > 0:
y.extend(parent_y)
return y
def optimize(self, fitness_function=None, args=None):
fitness = Optimizer.optimize(self, fitness_function)
y = self.initialize(args)
while not self._check_terminations():
self._print_verbose_info(fitness, y)
y = self.iterate(args)
self._n_generations += 1
self.temperature = np.maximum(self.temperature*self.rt, 1e-200)
return self._collect(fitness, y)