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Simple and Practical Python library for CMA-ES. Please refer to the paper [Nomura and Shibata 2024] for detailed information, including the design philosophy and advanced examples.
Supported Python versions are 3.7 or later.
$ pip install cmaes
Or you can install via conda-forge.
$ conda install -c conda-forge cmaes
This library provides an "ask-and-tell" style interface. We employ the standard version of CMA-ES [Hansen 2016].
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 [Akiba et al. 2019], 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)The performance of the CMA-ES can deteriorate when faced with difficult problems such as multimodal or noisy ones, if its hyperparameter values are not properly configured. The Learning Rate Adaptation CMA-ES (LRA-CMA) effectively addresses this issue by autonomously adjusting the learning rate. Consequently, LRA-CMA eliminates the need for expensive hyperparameter tuning.
LRA-CMA can be used by simply adding lr_adapt=True to the initialization of CMA().
Source code
import numpy as np
from cmaes import CMA
def rastrigin(x):
dim = len(x)
return 10 * dim + sum(x**2 - 10 * np.cos(2 * np.pi * x))
if __name__ == "__main__":
dim = 40
optimizer = CMA(mean=3*np.ones(dim), sigma=2.0, lr_adapt=True)
for generation in range(50000):
solutions = []
for _ in range(optimizer.population_size):
x = optimizer.ask()
value = rastrigin(x)
if generation % 500 == 0:
print(f"#{generation} {value}")
solutions.append((x, value))
optimizer.tell(solutions)
if optimizer.should_stop():
breakThe full source code is available here.
Warm Starting CMA-ES (WS-CMA) is a method that transfers prior knowledge from similar tasks through the initialization of the CMA-ES. This is useful especially when the evaluation budget is limited (e.g., hyperparameter optimization of machine learning algorithms).
Source code
import numpy as np
from cmaes import CMA, get_warm_start_mgd
def source_task(x1: float, x2: float) -> float:
b = 0.4
return (x1 - b) ** 2 + (x2 - b) ** 2
def target_task(x1: float, x2: float) -> float:
b = 0.6
return (x1 - b) ** 2 + (x2 - b) ** 2
if __name__ == "__main__":
# Generate solutions from a source task
source_solutions = []
for _ in range(1000):
x = np.random.random(2)
value = source_task(x[0], x[1])
source_solutions.append((x, value))
# Estimate a promising distribution of the source task,
# then generate parameters of the multivariate gaussian distribution.
ws_mean, ws_sigma, ws_cov = get_warm_start_mgd(
source_solutions, gamma=0.1, alpha=0.1
)
optimizer = CMA(mean=ws_mean, sigma=ws_sigma, cov=ws_cov)
# Run WS-CMA-ES
print(" g f(x1,x2) x1 x2 ")
print("=== ========== ====== ======")
while True:
solutions = []
for _ in range(optimizer.population_size):
x = optimizer.ask()
value = target_task(x[0], x[1])
solutions.append((x, value))
print(
f"{optimizer.generation:3d} {value:10.5f}"
f" {x[0]:6.2f} {x[1]:6.2f}"
)
optimizer.tell(solutions)
if optimizer.should_stop():
breakThe full source code is available here.
CMA-ES with Margin (CMAwM) introduces a lower bound on the marginal probability for each discrete dimension, ensuring that samples avoid being fixed to a single point. This method can be applied to mixed spaces consisting of continuous (such as float) and discrete elements (including integer and binary types).
| CMA | CMAwM |
|---|---|
 |
 |
The above figures are taken from EvoConJP/CMA-ES_with_Margin.
Source code
import numpy as np
from cmaes import CMAwM
def ellipsoid_onemax(x, n_zdim):
n = len(x)
n_rdim = n - n_zdim
r = 10
if len(x) < 2:
raise ValueError("dimension must be greater one")
ellipsoid = sum([(1000 ** (i / (n_rdim - 1)) * x[i]) ** 2 for i in range(n_rdim)])
onemax = n_zdim - (0.0 < x[(n - n_zdim) :]).sum()
return ellipsoid + r * onemax
def main():
binary_dim, continuous_dim = 10, 10
dim = binary_dim + continuous_dim
bounds = np.concatenate(
[
np.tile([-np.inf, np.inf], (continuous_dim, 1)),
np.tile([0, 1], (binary_dim, 1)),
]
)
steps = np.concatenate([np.zeros(continuous_dim), np.ones(binary_dim)])
optimizer = CMAwM(mean=np.zeros(dim), sigma=2.0, bounds=bounds, steps=steps)
print(" evals f(x)")
print("====== ==========")
evals = 0
while True:
solutions = []
for _ in range(optimizer.population_size):
x_for_eval, x_for_tell = optimizer.ask()
value = ellipsoid_onemax(x_for_eval, binary_dim)
evals += 1
solutions.append((x_for_tell, value))
if evals % 300 == 0:
print(f"{evals:5d} {value:10.5f}")
optimizer.tell(solutions)
if optimizer.should_stop():
break
if __name__ == "__main__":
main()Source code is also available here.
CatCMA is a method for mixed-category optimization problems, which is the problem of simultaneously optimizing continuous and categorical variables. CatCMA employs the joint probability distribution of multivariate Gaussian and categorical distributions as the search distribution.
Source code
import numpy as np
from cmaes import CatCMA
def sphere_com(x, c):
dim_co = len(x)
dim_ca = len(c)
if dim_co < 2:
raise ValueError("dimension must be greater one")
sphere = sum(x * x)
com = dim_ca - sum(c[:, 0])
return sphere + com
def rosenbrock_clo(x, c):
dim_co = len(x)
dim_ca = len(c)
if dim_co < 2:
raise ValueError("dimension must be greater one")
rosenbrock = sum(100 * (x[:-1] ** 2 - x[1:]) ** 2 + (x[:-1] - 1) ** 2)
clo = dim_ca - (c[:, 0].argmin() + c[:, 0].prod() * dim_ca)
return rosenbrock + clo
def mc_proximity(x, c, cat_num):
dim_co = len(x)
dim_ca = len(c)
if dim_co < 2:
raise ValueError("dimension must be greater one")
if dim_co != dim_ca:
raise ValueError(
"number of dimensions of continuous and categorical variables "
"must be equal in mc_proximity"
)
c_index = np.argmax(c, axis=1) / cat_num
return sum((x - c_index) ** 2) + sum(c_index)
if __name__ == "__main__":
cont_dim = 5
cat_dim = 5
cat_num = np.array([3, 4, 5, 5, 5])
# cat_num = 3 * np.ones(cat_dim, dtype=np.int64)
optimizer = CatCMA(mean=3.0 * np.ones(cont_dim), sigma=1.0, cat_num=cat_num)
for generation in range(200):
solutions = []
for _ in range(optimizer.population_size):
x, c = optimizer.ask()
value = mc_proximity(x, c, cat_num)
if generation % 10 == 0:
print(f"#{generation} {value}")
solutions.append(((x, c), value))
optimizer.tell(solutions)
if optimizer.should_stop():
breakThe full source code is available here.
MAP-CMA is a method that is introduced to interpret the rank-one update in the CMA-ES from the perspective of the natural gradient. The rank-one update derived from the natural gradient perspective is extensible, and an additional term, called momentum update, appears in the update of the mean vector. The performance of MAP-CMA is not significantly different from that of CMA-ES, as the primary motivation for MAP-CMA comes from the theoretical understanding of CMA-ES.
Source code
import numpy as np
from cmaes import MAPCMA
def rosenbrock(x):
dim = len(x)
if dim < 2:
raise ValueError("dimension must be greater one")
return sum(100 * (x[:-1] ** 2 - x[1:]) ** 2 + (x[:-1] - 1) ** 2)
if __name__ == "__main__":
dim = 20
optimizer = MAPCMA(mean=np.zeros(dim), sigma=0.5, momentum_r=dim)
print(" evals f(x)")
print("====== ==========")
evals = 0
while True:
solutions = []
for _ in range(optimizer.population_size):
x = optimizer.ask()
value = rosenbrock(x)
evals += 1
solutions.append((x, value))
if evals % 1000 == 0:
print(f"{evals:5d} {value:10.5f}")
optimizer.tell(solutions)
if optimizer.should_stop():
breakThe full source code is available here.
Sep-CMA-ES is an algorithm that limits the covariance matrix to a diagonal form. This reduction in the number of parameters enhances scalability, making Sep-CMA-ES well-suited for high-dimensional optimization tasks. Additionally, the learning rate for the covariance matrix is increased, leading to superior performance over the (full-covariance) CMA-ES on separable functions.
Source code
import numpy as np
from cmaes import SepCMA
def ellipsoid(x):
n = len(x)
if len(x) < 2:
raise ValueError("dimension must be greater one")
return sum([(1000 ** (i / (n - 1)) * x[i]) ** 2 for i in range(n)])
if __name__ == "__main__":
dim = 40
optimizer = SepCMA(mean=3 * np.ones(dim), sigma=2.0)
print(" evals f(x)")
print("====== ==========")
evals = 0
while True:
solutions = []
for _ in range(optimizer.population_size):
x = optimizer.ask()
value = ellipsoid(x)
evals += 1
solutions.append((x, value))
if evals % 3000 == 0:
print(f"{evals:5d} {value:10.5f}")
optimizer.tell(solutions)
if optimizer.should_stop():
breakFull source code is available here.
IPOP-CMA-ES is a method that involves restarting the CMA-ES with an incrementally increasing population size, as described below.
Source code
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}")Full source code is available here.
If you use our library in your work, please cite our paper:
Masahiro Nomura, Masashi Shibata.
cmaes : A Simple yet Practical Python Library for CMA-ES
https://arxiv.org/abs/2402.01373
Bibtex:
@article{nomura2024cmaes,
title={cmaes : A Simple yet Practical Python Library for CMA-ES},
author={Nomura, Masahiro and Shibata, Masashi},
journal={arXiv preprint arXiv:2402.01373},
year={2024}
}
For any questions, feel free to raise an issue or contact me at [email protected].
Projects using cmaes:
- Optuna : A hyperparameter optimization framework that supports CMA-ES using this library under the hood.
- Kubeflow/Katib : Kubernetes-based system for hyperparameter tuning and neural architecture search
- (If you are using
cmaesin your project and would like it to be listed here, please submit a GitHub issue.)
Other libraries:
We have great respect for all libraries involved in CMA-ES.
- pycma : Most renowned CMA-ES implementation, created and maintained by Nikolaus Hansen.
- pymoo : A library for multi-objective optimization in Python.
- evojax : evojax offers a JAX-port of this library.
- evosax : evosax provides a JAX-based implementation of CMA-ES and sep-CMA-ES, inspired by this library.
References:
- [Akiba et al. 2019] T. Akiba, S. Sano, T. Yanase, T. Ohta, M. Koyama, Optuna: A Next-generation Hyperparameter Optimization Framework, KDD, 2019.
- [Auger and Hansen 2005] A. Auger, N. Hansen, A Restart CMA Evolution Strategy with Increasing Population Size, CEC, 2005.
- [Hamano et al. 2022] R. Hamano, S. Saito, M. Nomura, S. Shirakawa, CMA-ES with Margin: Lower-Bounding Marginal Probability for Mixed-Integer Black-Box Optimization, GECCO, 2022.
- [Hamano et al. 2024a] R. Hamano, S. Saito, M. Nomura, K. Uchida, S. Shirakawa, CatCMA : Stochastic Optimization for Mixed-Category Problems, GECCO, 2024.
- [Hamano et al. 2024b] R. Hamano, S. Shirakawa, M. Nomura, Natural Gradient Interpretation of Rank-One Update in CMA-ES, PPSN, 2024.
- [Hansen 2016] N. Hansen, The CMA Evolution Strategy: A Tutorial. arXiv:1604.00772, 2016.
- [Nomura et al. 2021] M. Nomura, S. Watanabe, Y. Akimoto, Y. Ozaki, M. Onishi, Warm Starting CMA-ES for Hyperparameter Optimization, AAAI, 2021.
- [Nomura et al. 2023] M. Nomura, Y. Akimoto, I. Ono, CMA-ES with Learning Rate Adaptation: Can CMA-ES with Default Population Size Solve Multimodal and Noisy Problems?, GECCO, 2023.
- [Nomura and Shibata 2024] M. Nomura, M. Shibata, cmaes : A Simple yet Practical Python Library for CMA-ES, arXiv:2402.01373, 2024.
- [Ros and Hansen 2008] R. Ros, N. Hansen, A Simple Modification in CMA-ES Achieving Linear Time and Space Complexity, PPSN, 2008.