Multi-objective CMA-ES (MO-CMA-ES) Sampler

Abstract

MoCmaSampler provides the implementation of the s-MO-CMA-ES algorithm. This algorithm extends (1+1)-CMA-ES to multi-objective optimization by introducing a selection strategy based on non-domination sorting and contributing hypervolume (S-metric). It inherits important properties of CMA-ES, invariance against order-preserving transformations of the fitness function value and rotation and translation of the search space.

Class or Function Names

• MoCmaSampler(*, search_space: dict[str, BaseDistribution] | None = None, popsize: int | None = None, seed: int | None = None)
  • search_space: A dictionary containing the search space that defines the parameter space. The keys are the parameter names and the values are the parameter’s distribution. If the search space is not provided, the sampler will infer the search space dynamically. Example:

    search_space = {
        "x": optuna.distributions.FloatDistribution(-5, 5),
        "y": optuna.distributions.FloatDistribution(-5, 5),
    }
    MoCmaSampler(search_space=search_space)
    

  • popsize: Population size of the CMA-ES algorithm. If not provided, the population size will be set based on the search space dimensionality. If you have a sufficient evaluation budget, it is recommended to increase the popsize.
  • seed: Seed for random number generator.

Note that because of the limitation of the algorithm, only non-conditional numerical parameters are sampled by the MO-CMA-ES algorithm, and categorical parameters and conditional parameters are handled by random sampling.

Example

import optuna
import optunahub


def objective(trial: optuna.Trial) -> tuple[float, float]:
    x = trial.suggest_float("x", 0, 5)
    y = trial.suggest_float("y", 0, 3)
    v0 = 4 * x**2 + 4 * y**2
    v1 = (x - 5) ** 2 + (y - 5) ** 2
    return v0, v1

samplers = [
    optunahub.load_module("samplers/mocma").MoCmaSampler(popsize=100, seed=42),
    optuna.samplers.NSGAIISampler(population_size=100, seed=42),
]
studies = []
for sampler in samplers:
    study = optuna.create_study(
        directions=["minimize", "minimize"],
        sampler=sampler,
        study_name=f"{sampler.__class__.__name__}",
    )
    study.optimize(objective, n_trials=1000)
    studies.append(study)

optunahub.load_module("visualization/plot_pareto_front_multi").plot_pareto_front(
    studies
).show()
optunahub.load_module("visualization/plot_hypervolume_history_multi").plot_hypervolume_history(
    studies, reference_point=[200.0, 100.0]
).show()

Others

Test

To execute the tests for MoCmaSamler, please run the following commands. The test file is provided in the package.

pip install pytest

pytest -s tests/test_sampler.py

Reference

Christian Igel, Nikolaus Hansen, Stefan Roth. Covariance Matrix Adaptation for Multi-objective Optimization, Evolutionary Computatio. (2007) 15 (1): 1–28. https://doi.org/10.1162/evco.2007.15.1.1.

BibTeX

@article{igel2007covariance,
  title={Covariance matrix adaptation for multi-objective optimization},
  author={Igel, Christian and Hansen, Nikolaus and Roth, Stefan},
  journal={Evolutionary computation},
  volume={15},
  number={1},
  pages={1--28},
  year={2007},
  publisher={MIT Press One Rogers Street, Cambridge, MA 02142-1209, USA journals-info~…}
}