from pypop7.optimizers.rs.prs import PRS
[docs]class RHC(PRS):
"""Random (stochastic) Hill Climber (RHC).
.. note:: Currently `RHC` only supports normally-distributed random sampling during optimization. It often suffers
from **slow convergence** for large-scale black-box optimization (LSBBO), owing to its *relatively limited*
exploration ability originating from its **individual-based** sampling strategy. Therefore, it is **highly
recommended** to first attempt more advanced (e.g. population-based) methods for LSBBO.
`"The hill-climbing search algorithm is the most basic local search technique. They have two key advantages:
(1) they use very little memory; and (2) they can often find reasonable solutions in large or infinite state
spaces for which systematic algorithms are unsuitable."---[Russell&Norvig, 2021]
<http://aima.cs.berkeley.edu/>`_
AKA `"stochastic local search (steepest ascent or greedy search)"---[Murphy., 2022]
<https://probml.github.io/pml-book/book2.html>`_.
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`),
* '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.rhc import RHC
>>> 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}
>>> rhc = RHC(problem, options) # initialize the optimizer class
>>> results = rhc.optimize() # run the optimization process
>>> # return the number of used function evaluations and found best-so-far fitness
>>> print(f"RHC: {results['n_function_evaluations']}, {results['best_so_far_y']}")
RHC: 5000, 7.13722829962456e-05
For its correctness checking of coding, refer to `this code-based repeatability report
<https://tinyurl.com/3u864ju3>`_ for more details.
Attributes
----------
init_distribution : `int`
random sampling distribution for starting-point initialization.
sigma : `float`
global step-size (fixed during optimization).
x : `array_like`
initial (starting) point.
References
----------
The following code from PyBrain directly inspired the coding of `RHC`:
https://github.com/pybrain/pybrain/blob/master/pybrain/optimization/hillclimber.py
For the following book, Chapter 6.7 (DFO) gives an introduction of `RHC`:
https://probml.github.io/pml-book/book2.html
For the following book, Chapter 4 (SEARCH IN COMPLEX ENVIRONMENTS) gives an introduction of `RHC`:
Russell, S. and Norvig P., 2021.
`Artificial intelligence: A modern approach (Global Edition).
<http://aima.cs.berkeley.edu/>`_
Pearson Education.
Hoos, H.H. and Stützle, T., 2004.
`Stochastic local search: Foundations and applications.
<https://www.elsevier.com/books/stochastic-local-search/hoos/978-1-55860-872-6>`_
Elsevier.
Baluja, S., 1996.
`Genetic algorithms and explicit search statistics.
<https://proceedings.neurips.cc/paper/1996/hash/e6d8545daa42d5ced125a4bf747b3688-Abstract.html>`_
In Advances in Neural Information Processing Systems (pp.319-325).
Juels, A. and Wattenberg, M., 1995.
`Stochastic hillclimbing as a baseline method for evaluating genetic algorithms.
<https://proceedings.neurips.cc/paper/1995/hash/36a1694bce9815b7e38a9dad05ad42e0-Abstract.html>`_
In Advances in Neural Information Processing Systems (pp. 430-436).
"""
def __init__(self, problem, options):
# only support normally-distributed random sampling during optimization
options['_sampling_type'] = 0 # 0 -> normally distributed random sampling (a mandatory setting)
PRS.__init__(self, problem, options)
# set default: 1 -> uniformly distributed random sampling
self.init_distribution = options.get('init_distribution', 1)
if self.init_distribution not in [0, 1]: # 0 -> normally distributed random sampling
info = 'For currently {:s}, only support uniformly or normally distributed random initialization.'
raise ValueError(info.format(self.__class__.__name__))
def _sample(self, rng): # only for function `initialize(self)` inherited from the parent class `PRS`
if self.init_distribution == 0: # normally distributed
x = rng.standard_normal(size=(self.ndim_problem,))
else: # uniformly distributed
x = rng.uniform(self.initial_lower_boundary, self.initial_upper_boundary)
return x
def iterate(self): # sampling via mutating the best-so-far individual
noise = self.rng_optimization.standard_normal(size=(self.ndim_problem,))
return self.best_so_far_x + self.sigma*noise # mutation based on Gaussian-noise perturbation