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

Project description

CMA-ES

Software License PyPI - Downloads

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

visualize-six-hump-camel

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 [2], 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

Warm Starting CMA-ES [3]

Warm Starting CMA-ES is a method to transfer prior knowledge on similar HPO tasks through the initialization of CMA-ES. It estimates a promising distribution and generates the parameters of the multivariate gaussian distribution like below:

Rot Ellipsoid function Ellipsoid function
rot-ellipsoid quadratic
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

The full source code is available here.

Separable CMA-ES [4]

sep-CMA-ES is an algorithm which constrains the covariance matrix to be diagonal. Due to reduce the model complexity, the learning rate for the covariance matrix is reduced. Consequently, this algorithm outperforms CMA-ES on separable functions.

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.

IPOP-CMA-ES [5]

IPOP-CMA-ES is a method to restart CMA-ES with increasing population size like below.

visualize-ipop-cmaes-himmelblau

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.

BIPOP-CMA-ES [6]

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

visualize-bipop-cmaes-himmelblau

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))

Full source code is available here.

Benchmark results

Rosenbrock function Six-Hump Camel function
rosenbrock six-hump-camel

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:

Project details


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