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:whale: Paper is now available on arXiv!

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.

visualize-six-hump-camel

Installation

Supported Python versions are 3.7 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. 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)

CMA-ES variants

Learning Rate Adaptation CMA-ES [Nomura et al. 2023]

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().

<details> <summary>Source code</summary>
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():
            break

The full source code is available here.

</details>

Warm Starting CMA-ES [Nomura et al. 2021]

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).

benchmark-lightgbm-toxic

<details> <summary>Source code</summary>
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():
            break

The full source code is available here.

</details>

CMA-ES with Margin [Hamano et al. 2022]

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).

CMACMAwM
CMA-ESCMA-ESwM

The above figures are taken from EvoConJP/CMA-ES_with_Margin.

<details> <summary>Source code</summary>
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.

</details>

CatCMA [Hamano et al. 2024a]

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.

CatCMA

<details> <summary>Source code</summary>
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():
            break

The full source code is available here.

</details>

Maximum a Posteriori CMA-ES [Hamano et al. 2024b]

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.

<details> <summary>Source code</summary>
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():
            break

The full source code is available here.

</details>

Separable CMA-ES [Ros and Hansen 2008]

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.

<details> <summary>Source code</summary>
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():
            break

Full source code is available here.

</details>

IPOP-CMA-ES [Auger and Hansen 2005]

IPOP-CMA-ES is a method that involves restarting the CMA-ES with an incrementally increasing population size, as described below.

visualize-ipop-cmaes-himmelblau

<details> <summary>Source code</summary>
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.

</details>

Citation

If you use our library in your work, please cite our paper:

Masahiro Nomura, Masashi Shibata.<br> cmaes : A Simple yet Practical Python Library for CMA-ES<br> 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}
}

Contact

For any questions, feel free to raise an issue or contact me at nomura_masahiro@cyberagent.co.jp.

Links

Projects using cmaes:

Other libraries:

We have great respect for all libraries involved in CMA-ES.

References: