Lightweight Covariance Matrix Adaptation Evolution Strategy (CMA-ES) [1] implementation.
- 2021/03/10 "Introduction to CMA-ES sampler" is published at Optuna Medium Blog. This article explains when and how to make the best use out of CMA-ES sampler. Please check it out!
- 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 🎉
- 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.
Supported Python versions are 3.6 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.
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)Warm Starting CMA-ES is a method to transfer prior knowledge on similar HPO tasks through the initialization of CMA-ES. Here is the result of an experiment that tuning LightGBM for Kaggle's Toxic Comment Classification Challenge data, a multilabel classification dataset. In this benchmark, we use 10% of a full dataset as the source task, and a full dataset as the target task. Please refer the paper and/or https://github.com/c-bata/benchmark-warm-starting-cmaes for more details of experiment settings.
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.
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.
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 to restart CMA-ES with increasing population size like 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.
BIPOP-CMA-ES applies two interlaced restart strategies, one with an increasing population size and one with varying small population sizes.
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)
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.
| Rosenbrock function | Six-Hump Camel function |
|---|---|
 |
 |
This implementation (green) stands comparison with pycma (blue). See benchmark for details.
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] T. Akiba, S. Sano, T. Yanase, T. Ohta, M. Koyama, Optuna: A Next-generation Hyperparameter Optimization Framework, KDD, 2019.
- [3] M. Nomura, S. Watanabe, Y. Akimoto, Y. Ozaki, M. Onishi, Warm Starting CMA-ES for Hyperparameter Optimization, AAAI, 2021.
- [4] R. Ros, N. Hansen, A Simple Modification in CMA-ES Achieving Linear Time and Space Complexity, PPSN, 2008.
- [5] A. Auger, N. Hansen, A restart CMA evolution strategy with increasing population size, CEC, 2005.
- [6] N. Hansen, Benchmarking a BI-Population CMA-ES on the BBOB-2009 Function Testbed, GECCO Workshop, 2009.