Minimal example for evolutionary regression

Example demonstrating the use of Cartesian genetic programming for a simple regression task.

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

   Options:
     -h --help
     --max-generations=<N>  Maximum number of generations [default: 300]
"""

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

import cgp

args = docopt(docopt_str)

We first define a target function.

def f_target(x):
    return x ** 2 + 1.0

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 random points.

def objective(individual):

    if not individual.fitness_is_None():
        return individual

    n_function_evaluations = 1000

    np.random.seed(1234)

    f = individual.to_func()
    loss = 0
    for x in np.random.uniform(-4, 4, n_function_evaluations):
        # the callable returned from `to_func` accepts and returns
        # lists; accordingly we need to pack the argument and unpack
        # the return value
        y = f(x)
        loss += (f_target(x) - y) ** 2

    individual.fitness = -loss / n_function_evaluations

    return individual

Next, we set up the evolutionary search. We define a callback for recording of fitness over generations

history = {}
history["fitness_champion"] = []


def recording_callback(pop):
    history["fitness_champion"].append(pop.champion.fitness)

and finally perform the evolution relying on the libraries default hyperparameters except that we terminate the evolution as soon as one individual has reached fitness zero.

pop = cgp.evolve(
    objective, termination_fitness=0.0, print_progress=True, callback=recording_callback
)

Out:

[2/1000] max fitness: -51.24963399666095
[3/1000] max fitness: -51.24963399666095
[4/1000] max fitness: -51.24963399666095
[5/1000] max fitness: -44.844855976219186
[6/1000] max fitness: -44.844855976219186
[7/1000] max fitness: -44.844855976219186
[8/1000] max fitness: -44.844855976219186
[9/1000] max fitness: -44.844855976219186
[10/1000] max fitness: -44.844855976219186
[11/1000] max fitness: -44.844855976219186
[12/1000] max fitness: -44.844855976219186
[13/1000] max fitness: -44.844855976219186
[14/1000] max fitness: -44.844855976219186
[15/1000] max fitness: -44.844855976219186
[16/1000] max fitness: -44.844855976219186
[17/1000] max fitness: -44.844855976219186
[18/1000] max fitness: -44.844855976219186
[19/1000] max fitness: -44.844855976219186
[20/1000] max fitness: -44.844855976219186
[21/1000] max fitness: -44.844855976219186
[22/1000] max fitness: -44.844855976219186
[23/1000] max fitness: -44.844855976219186
[24/1000] max fitness: -36.71898739812355
[25/1000] max fitness: -36.71898739812355
[26/1000] max fitness: -36.71898739812355
[27/1000] max fitness: -36.71898739812355
[28/1000] max fitness: -36.71898739812355
[29/1000] max fitness: -36.71898739812355
[30/1000] max fitness: -36.71898739812355
[31/1000] max fitness: -36.71898739812355
[32/1000] max fitness: -36.71898739812355
[33/1000] max fitness: -36.71898739812355
[34/1000] max fitness: -36.71898739812355
[35/1000] max fitness: -36.71898739812355
[36/1000] max fitness: -36.71898739812355
[37/1000] max fitness: -36.71898739812355
[38/1000] max fitness: -36.71898739812355
[39/1000] max fitness: -36.71898739812355
[40/1000] max fitness: -36.71898739812355
[41/1000] max fitness: -36.71898739812355
[42/1000] max fitness: -36.71898739812355
[43/1000] max fitness: -36.71898739812355
[44/1000] max fitness: -36.71898739812355
[45/1000] max fitness: -36.71898739812355
[46/1000] max fitness: -1.0
[47/1000] max fitness: -1.0
[48/1000] max fitness: -1.0
[49/1000] max fitness: 0.0

After finishing the evolution, we plot the result and log the final evolved expression.

width = 9.0
fig, axes = plt.subplots(1, 2, figsize=(width, width / scipy.constants.golden))

ax_fitness, ax_function = axes[0], axes[1]
ax_fitness.set_xlabel("Generation")
ax_fitness.set_ylabel("Fitness")

ax_fitness.plot(history["fitness_champion"], label="Champion")

ax_fitness.set_yscale("symlog")
ax_fitness.set_ylim(-1.0e2, 0.1)
ax_fitness.axhline(0.0, color="0.7")

f = pop.champion.to_func()
x = np.linspace(-5.0, 5.0, 20)
y = [f(x_i) for x_i in x]
y_target = [f_target(x_i) for x_i in x]

ax_function.plot(x, y_target, lw=2, alpha=0.5, label="Target")
ax_function.plot(x, y, "x", label="Champion")
ax_function.legend()
ax_function.set_ylabel(r"$f(x)$")
ax_function.set_xlabel(r"$x$")

fig.savefig("example_minimal.pdf", dpi=300)