python协方差阵转为相关阵_cmaes-python3的轻量级协方差矩阵自适应进化策略（CMA-ES）实现。-Masashi Shibata Installation Usage Benchmar...

作者:Masashi Shibata

作者邮箱:[email protected]

首页:https://github.com/CyberAgent/cmaes

文档:None

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CMA-ES

Lightweight Covariance Matrix Adaptation Evolution Strategy (CMA-ES) [1] implementation.

Rosenbrock function.

IPOP-CMA-ES [2] on Himmelblau function.

BIPOP-CMA-ES [3] on Himmelblau function.

These GIF animations are generated by visualizer.py.

Installation

Supported Python versions are 3.6 or later.

$ pip install cmaes

Or you can install via conda-forge.

$ conda install -c conda-forge cmaes

Usage

This library provides an "ask-and-tell" style interface.

import numpy as np

from cmaes import CMA

def quadratic(x1, x2):

return (x1 - 3) ** 2 + (10 * (x2 + 2)) ** 2

if __name__ == "__main__":

optimizer = CMA(mean=np.zeros(2), sigma=1.3)

for generation in range(50):

solutions = []

for _ in range(optimizer.population_size):

x = optimizer.ask()

value = quadratic(x[0], x[1])

solutions.append((x, value))

print(f"#{generation} {value} (x1={x[0]}, x2 = {x[1]})")

optimizer.tell(solutions)

And you can use this library via Optuna [4], an automatic hyperparameter optimization framework. Optuna's built-in CMA-ES sampler which uses this library under the hood is available from v1.3.0 and stabled at v2.0.0. See the documentation or v2.0 release blog for more details.

import optuna

def objective(trial: optuna.Trial):

x1 = trial.suggest_uniform("x1", -4, 4)

x2 = trial.suggest_uniform("x2", -4, 4)

return (x1 - 3) ** 2 + (10 * (x2 + 2)) ** 2

if __name__ == "__main__":

sampler = optuna.samplers.CmaEsSampler()

study = optuna.create_study(sampler=sampler)

study.optimize(objective, n_trials=250)

Example of IPOP-CMA-ES

You can easily implement IPOP-CMA-ES which restarts CMA-ES with increasing population size.

import math

import numpy as np

from cmaes import CMA

def ackley(x1, x2):

# https://www.sfu.ca/~ssurjano/ackley.html

return (

-20 * math.exp(-0.2 * math.sqrt(0.5 * (x1 ** 2 + x2 ** 2)))

- math.exp(0.5 * (math.cos(2 * math.pi * x1) + math.cos(2 * math.pi * x2)))

+ math.e + 20

)

if __name__ == "__main__":

bounds = np.array([[-32.768, 32.768], [-32.768, 32.768]])

lower_bounds, upper_bounds = bounds[:, 0], bounds[:, 1]

mean = lower_bounds + (np.random.rand(2) * (upper_bounds - lower_bounds))

sigma = 32.768 * 2 / 5 # 1/5 of the domain width

optimizer = CMA(mean=mean, sigma=sigma, bounds=bounds, seed=0)

for generation in range(200):

solutions = []

for _ in range(optimizer.population_size):

x = optimizer.ask()

value = ackley(x[0], x[1])

solutions.append((x, value))

print(f"#{generation} {value} (x1={x[0]}, x2 = {x[1]})")

optimizer.tell(solutions)

if optimizer.should_stop():

# popsize multiplied by 2 (or 3) before each restart.

popsize = optimizer.population_size * 2

mean = lower_bounds + (np.random.rand(2) * (upper_bounds - lower_bounds))

optimizer = CMA(mean=mean, sigma=sigma, population_size=popsize)

print(f"Restart CMA-ES with popsize={popsize}")

Example of BIPOP-CMA-ES

Here is an example of BIPOP-CMA-ES which applies two interlaced restart strategies, one with an increasing population size and one with varying small population sizes.

import math

import numpy as np

from cmaes import CMA

def ackley(x1, x2):

# https://www.sfu.ca/~ssurjano/ackley.html

return (

-20 * math.exp(-0.2 * math.sqrt(0.5 * (x1 ** 2 + x2 ** 2)))

- math.exp(0.5 * (math.cos(2 * math.pi * x1) + math.cos(2 * math.pi * x2)))

+ math.e + 20

)

if __name__ == "__main__":

bounds = np.array([[-32.768, 32.768], [-32.768, 32.768]])

lower_bounds, upper_bounds = bounds[:, 0], bounds[:, 1]

mean = lower_bounds + (np.random.rand(2) * (upper_bounds - lower_bounds))

sigma = 32.768 * 2 / 5 # 1/5 of the domain width

optimizer = CMA(mean=mean, sigma=sigma, bounds=bounds, seed=0)

n_restarts = 0 # A small restart doesn't count in the n_restarts

small_n_eval, large_n_eval = 0, 0

popsize0 = optimizer.population_size

inc_popsize = 2

# Initial run is with "normal" population size; it is

# the large population before first doubling, but its

# budget accounting is the same as in case of small

# population.

poptype = "small"

for generation in range(200):

solutions = []

for _ in range(optimizer.population_size):

x = optimizer.ask()

value = ackley(x[0], x[1])

solutions.append((x, value))

print(f"#{generation} {value} (x1={x[0]}, x2 = {x[1]})")

optimizer.tell(solutions)

if optimizer.should_stop():

n_eval = optimizer.population_size * optimizer.generation

if poptype == "small":

small_n_eval += n_eval

else: # poptype == "large"

large_n_eval += n_eval

if small_n_eval < large_n_eval:

poptype = "small"

popsize_multiplier = inc_popsize ** n_restarts

popsize = math.floor(

popsize0 * popsize_multiplier ** (np.random.uniform() ** 2)

)

else:

poptype = "large"

n_restarts += 1

popsize = popsize0 * (inc_popsize ** n_restarts)

mean = lower_bounds + (np.random.rand(2) * (upper_bounds - lower_bounds))

optimizer = CMA(

mean=mean,

sigma=sigma,

bounds=bounds,

population_size=popsize,

)

print("Restart CMA-ES with popsize={} ({})".format(popsize, poptype))

Benchmark results

| Rosenbrock function | Six-Hump Camel function | | ------------------- | ----------------------- | |

|

|

This implementation (green) stands comparison with pycma (blue). See benchmark for details.

Links

Other libraries:

I respect all libraries involved in CMA-ES.

pycma : Most famous CMA-ES implementation by Nikolaus Hansen.

pymoo : Multi-objective optimization in Python.

References:

[1] [N. Hansen, The CMA Evolution Strategy: A Tutorial. arXiv:1604.00772, 2016.](https://arxiv.org/abs/1604.00772)

[2] [Auger, A., Hansen, N.: A restart CMA evolution strategy with increasing population size. In: Proceedings of the 2005 IEEE Congress on Evolutionary Computation (CEC’2005), pp. 1769–1776 (2005a)](https://sci2s.ugr.es/sites/default/files/files/TematicWebSites/EAMHCO/contributionsCEC05/auger05ARCMA.pdf)

[3] [Hansen N. Benchmarking a BI-Population CMA-ES on the BBOB-2009 Function Testbed. In the workshop Proceedings of the Genetic and Evolutionary Computation Conference, GECCO, pages 2389–2395. ACM, 2009.](https://hal.inria.fr/inria-00382093/document)

[4] [Takuya Akiba, Shotaro Sano, Toshihiko Yanase, Takeru Ohta, Masanori Koyama. 2019. Optuna: A Next-generation Hyperparameter Optimization Framework. In The 25th ACM SIGKDD Conference on Knowledge Discovery and Data Mining (KDD ’19), August 4–8, 2019.](https://dl.acm.org/citation.cfm?id=3330701)

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