cmaes 0.8.0

Lightweight Covariance Matrix Adaptation Evolution Strategy (CMA-ES) implementation for Python 3.

CMA-ES

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

Rosenbrock function.

Warm starting CMA-ES [2].

Rot Ellipsoid function	Quadratic function

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

BIPOP-CMA-ES [4] on Himmelblau function.

These GIF animations are generated by cmaes_visualizer.py and ws_cmaes_visualizer.py.

News

• 2021/02/02 The paper "Warm Starting CMA-ES for Hyperparameter Optimization" written by @nmasahiro, the maintainer of this library, is accepted at AAAI 2021 :tada:
• 2020/07/29 Optuna's built-in CMA-ES sampler which uses this library under the hood is stabled at Optuna v2.0. Please check out the v2.0 release blog.

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 [5], 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 Warm-starting CMA-ES

Warm-starting CMA-ES (WS-CMA-ES)

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
    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

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.
• [2] Masahiro Nomura, Shuhei Watanabe, Youhei Akimoto, Yoshihiko Ozaki, Masaki Onishi. “Warm Starting CMA-ES for Hyperparameter Optimization”, AAAI. 2021.
• [3] 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)
• [4] 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.
• [5] 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.