Lévy Flight Sampler

Class or Function Names

• LevyFlightSampler

Overview

This sampler proposes new hyperparameter candidates by taking a Lévy-flight step from the current best trial. A Lévy flight is a random walk whose step lengths follow a heavy-tailed (Lévy stable) distribution: most steps are small, enabling fine local search, but occasional large steps jump far away, helping to escape shallow local optima.

The step is computed via the Mantegna algorithm, an efficient closed-form approximation of the Lévy stable distribution that requires only standard Gaussian random variates.

When to use

Key parameters

Installation

pip install optunahub numpy

Or install dependencies directly from the registry:

pip install -r https://hub.optuna.org/samplers/levy_flight_sampler/requirements.txt

Example

import optuna
import optunahub


def objective(trial):
    x = trial.suggest_float("x", -5, 5)
    y = trial.suggest_float("y", -5, 5)
    return (x - 1.5) ** 2 + (y + 2.0) ** 2


mod = optunahub.load_module("samplers/levy_flight_sampler")
sampler = mod.LevyFlightSampler(beta=1.5, step_scale=0.1, seed=42)
study = optuna.create_study(sampler=sampler)
study.optimize(objective, n_trials=100)

print(study.best_params)
print(study.best_value)

See example.py for a full comparison against TPESampler on the Rosenbrock benchmark.

Algorithm Details

1. Initialization — first two trials use random sampling to establish an initial best.
2. Lévy step — for each parameter p with range [low, high]:
   • Draw u ~ N(0, σ²) and v ~ N(0, 1) using the Mantegna sigma for beta.
   • Step length = step_scale × (high − low) × u / |v|^(1/beta).
   • Clip new value to [low, high].
3. Fallback — CategoricalDistribution and unknown distributions use RandomSampler.

The Mantegna sigma is:

σ = ( Γ(1+β) · sin(πβ/2) / (Γ((1+β)/2) · β · 2^((β−1)/2)) )^(1/β)

References

• Yang, X.-S. & Deb, S. (2010). Engineering Optimisation by Cuckoo Search. International Journal of Mathematical Modelling and Numerical Optimisation, 1(4), 330–343. https://doi.org/10.1504/IJMMNO.2010.035430

• Mantegna, R. N. (1994). Fast, accurate algorithm for numerical simulation of Lévy stable stochastic processes. Physical Review E, 49(5), 4677. https://doi.org/10.1103/PhysRevE.49.4677