Example for evolutionary regression on a piecewise target function¶

Example demonstrating the use of Cartesian genetic programming for regression on a piecewise target function by using the conditional (if/else) operator.

# The docopt str is added explicitly to ensure compatibility with
# sphinx-gallery.
docopt_str = """
   Usage:
     example_piecewise_target_function.py [--max-generations=<N>]

   Options:
     -h --help
     --max-generations=<N>  Maximum number of generations [default: 2000]
"""

import functools

import matplotlib.pyplot as plt
import numpy as np
from docopt import docopt

import cgp

args = docopt(docopt_str)

We define a piecewise target function. The function applies different transformations to the input depending whether the input is less or greater than zero. Thus to achieve high fitness, an individual must make use of the if/else operator.

def f_target(x):
    return np.select([x >= 0, x < 0], [x ** 2 + 1.0, -x])

Then we define the objective function for the evolution. It uses the mean-squared error between the output of the expression represented by a given individual and the target function evaluated on a set of pseudo-random points.

def objective(individual, rng):
    """Objective function of the regression task.

    Parameters
    ----------
    individual : Individual
        Individual of the Cartesian Genetic Programming Framework.
    rng: numpy.random.RandomState

    Returns
    -------
    Individual
        Modified individual with updated fitness value.
    """
    if not individual.fitness_is_None():
        return individual

    n_function_evaluations = 1000

    f = individual.to_numpy()
    x = rng.uniform(-5, 5, size=n_function_evaluations)
    y = f(x)

    loss = np.mean((f_target(x) - y) ** 2)
    individual.fitness = -loss

    return individual

Next, we set up the evolutionary search. First, we define the parameters for the population, the genomes of individuals, and the evolutionary algorithm.

population_params = {
    "n_parents": 1,
    "seed": 8188211,
}

genome_params = {
    "n_inputs": 1,
    "n_outputs": 1,
    "n_columns": 20,
    "n_rows": 1,
    "levels_back": None,
    "primitives": (cgp.IfElse, cgp.Mul, cgp.Add, cgp.Sub, cgp.ConstantFloat,),
}

ea_params = {"n_offsprings": 4, "mutation_rate": 0.03, "n_processes": 2}

evolve_params = {"max_generations": int(args["--max-generations"]), "termination_fitness": 0.0}

# create population that will be evolved
pop = cgp.Population(**population_params, genome_params=genome_params)

# create instance of evolutionary algorithm
ea = cgp.ea.MuPlusLambda(**ea_params)

# define callback for recording of fitness over generations
history = {}
history["fitness_champion"] = []
history["expr_champion"] = []


def recording_callback(pop):
    history["fitness_champion"].append(pop.champion.fitness)
    try:
        sympy_expression = pop.champion.to_sympy()
    except TypeError:
        sympy_expression = pop.champion.to_sympy(simplify=False)

    history["expr_champion"].append(sympy_expression)


# the objective passed to evolve should only accept one argument,
# the individual
rng = np.random.RandomState(seed=population_params["seed"])
obj = functools.partial(objective, rng=rng)

# Perform the evolution
cgp.evolve(pop, obj, ea, **evolve_params, print_progress=True, callback=recording_callback)

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After the evolutionary search has ended, we print the expression with the highest fitness and plot the search progression and target and evolved functions.

print(f"Final expression {pop.champion.to_sympy()} with fitness {pop.champion.fitness}")

fig = plt.figure(1)
plt.plot(history["fitness_champion"])
plt.ylim(1.1 * min(history["fitness_champion"]), 5)
plt.xlabel("Generation")
plt.ylabel("Loss (Fitness)")
plt.legend(["Champion loss per generation"])
plt.title({pop.champion.to_sympy()})
fig.savefig("example_piecewise_fitness_history.pdf")

x = np.arange(-5, 5, 0.01)
champion_numpy = pop.champion.to_numpy()

fig = plt.figure(2)
plt.subplot(121)
plt.plot(x, f_target(x), "b")
plt.xlabel("x")
plt.ylabel("y")
plt.title("Target function")
plt.legend(["target"])
plt.subplot(122)
plt.plot(x, champion_numpy(x), "r")
plt.xlabel("x")
plt.ylabel("y")
plt.title("Evolved function")
plt.legend(["champion"])
fig.savefig("example_piecewise_target_function.pdf", dpi=300)

${Piecewise((x_0**2 + 1.0, x_0 >= 0), (-x_0, True))}$

Out:

Final expression Piecewise((x_0**2 + 1.0, x_0 >= 0), (-x_0, True)) with fitness 0.0

Total running time of the script: ( 2 minutes 44.778 seconds)

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