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 first define the parameters for the population, the genome of individuals, and the evolutionary algorithm.

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

genome_params = {
    "n_inputs": 1,
    "n_outputs": 1,
    "n_columns": 12,
    "n_rows": 1,
    "levels_back": 5,
    "primitives": (cgp.Add, cgp.Sub, cgp.Mul, 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}

We create a population that will be evolved

pop = cgp.Population(**population_params, genome_params=genome_params)

and an instance of the (mu + lambda) evolutionary algorithm

ea = cgp.ea.MuPlusLambda(**ea_params)

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

cgp.evolve(pop, objective, ea, **evolve_params, print_progress=True, callback=recording_callback)

Out:

[2/300] max fitness: -91.5955167981362
[3/300] max fitness: -67.0965931324104
[4/300] max fitness: -67.0965931324104
[5/300] max fitness: -67.0965931324104
[6/300] max fitness: -54.97072455431481
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[15/300] max fitness: -54.97072455431481
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[87/300] max fitness: -44.844855976219186
[88/300] max fitness: -1.0
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[234/300] 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)