import numpy as np
from pypop7.optimizers.rs.rhc import RHC
[docs]class ARHC(RHC):
"""Annealed Random Hill Climber (ARHC).
.. note:: The search performance of `ARHC` depends **heavily** on the *temperature* setting of the annealing
process. However, its proper setting is a **non-trivial** task, since it may vary among different problems
and even between different optimization stages for the problem at hand. Therefore, it is **highly recommended**
to first attempt more advanced (e.g. population-based) methods for large-scale black-box optimization.
Here we include it mainly for *benchmarking* purpose.
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`):
* 'sigma' - initial global step-size (`float`),
* 'temperature' - annealing temperature (`float`),
* 'x' - initial (starting) point (`array_like`),
* if not given, it will draw a random sample from the uniform distribution whose search range is
bounded by `problem['lower_boundary']` and `problem['upper_boundary']`, when `init_distribution`
is `1`. Otherwise, *standard normal* distributed random sampling is used.
* 'init_distribution' - random sampling distribution for starting-point initialization (`int`,
default: `1`). Only when `x` is not set *explicitly*, it will be used.
* `1`: *uniform* distributed random sampling only for starting-point initialization,
* `0`: *standard normal* distributed random sampling only for starting-point initialization.
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
>>> from pypop7.benchmarks.base_functions import rosenbrock # function to be minimized
>>> from pypop7.optimizers.rs.arhc import ARHC
>>> 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': 0.1,
... 'temperature': 1.5}
>>> arhc = ARHC(problem, options) # initialize the optimizer class
>>> results = arhc.optimize() # run the optimization process
>>> # return the number of used function evaluations and found best-so-far fitness
>>> print(f"ARHC: {results['n_function_evaluations']}, {results['best_so_far_y']}")
ARHC: 5000, 0.0002641143073543329
For its correctness checking of coding, refer to `this code-based repeatability report
<https://tinyurl.com/2s3v8z7h>`_ for more details.
Attributes
----------
init_distribution : `int`
random sampling distribution for starting-point initialization.
sigma : `float`
global step-size (fixed during optimization).
temperature : `float`
annealing temperature.
x : `array_like`
initial (starting) point.
References
----------
https://probml.github.io/pml-book/book2.html (See CHAPTER 6.7 Derivative-free optimization)
Russell, S. and Norvig P., 2021.
Artificial intelligence: A modern approach (Global Edition).
Pearson Education.
http://aima.cs.berkeley.edu/ (See CHAPTER 4: SEARCH IN COMPLEX ENVIRONMENTS)
Hoos, H.H. and Stützle, T., 2004.
Stochastic local search: Foundations and applications.
Elsevier.
https://www.elsevier.com/books/stochastic-local-search/hoos/978-1-55860-872-6
https://github.com/pybrain/pybrain/blob/master/pybrain/optimization/hillclimber.py
"""
def __init__(self, problem, options):
RHC.__init__(self, problem, options)
self.temperature = options.get('temperature') # annealing temperature
assert self.temperature is not None
self._parent_x, self._parent_y = np.copy(self.best_so_far_x), np.copy(self.best_so_far_y)
def iterate(self): # sampling via mutating the parent individual
noise = self.rng_optimization.standard_normal(size=(self.ndim_problem,))
return self._parent_x + self.sigma*noise # mutation based on Gaussian-noise perturbation
def _evaluate_fitness(self, x, args=None):
y = RHC._evaluate_fitness(self, x, args)
# update parent solution and fitness
diff = y - self._parent_y
if (diff < 0) or (self.rng_optimization.random() < np.exp(-diff/self.temperature)): # annealing
self._parent_x, self._parent_y = np.copy(x), y
return y