Different powers
The problem
BBOB's different powers raises each gene's absolute value to a power from 2 to 6, and takes the square root of the sum:
f(x) = √(Σ |xᵢ|^(2 + 4 (i−1)/(n−1))), i from 1 to n, each xᵢ in [−5, 5]
Its minimum is 0, at the origin. Here n = 30. It's BBOB's f14 (Hansen et al. 2009), which rotates it, with its search domain. It isn't the sum of different powers of Molga and Smutnicki, with powers 2 to n + 1 and no square root.
What makes it hard
Near the minimum, the genes' sensitivities drift apart: an error of 1e-8, under the square root, needs the first gene within 10⁻⁸ of 0, but allows the last one, with its sixth power, to be 10⁻²·⁷ ≈ 0.002 away. The closer the search gets, the more different the scales it needs, so a search must keep adapting them. Shifted and rotated, every direction mixes the powers.
Representation
A Real genome of 30 genes: the point x itself. The fitness is f(x), to minimize. The function is
genoxide's problems::DifferentPowers, which brings its bounds and its minimum, and the shifted and
rotated instance problems::Rotated::new(problems::Shifted::new(function, 1), 1), which keeps them.
Algorithm
Five algorithms, each with a budget of 10,000 evaluations per dimension, 300,000 in all, and a target of 1e-8, from seed 1:
- CMA-ES (Hansen and Ostermeier, 2001, Evolutionary Computation 9(2): 159-195), which samples a population of 14 from a normal distribution and adapts its mean, its step size and its covariance matrix, from a step size of 0.3 of each gene's range and a random start;
- sep-CMA-ES (Ros and Hansen, 2008, PPSN X: 296-305), the same with a diagonal covariance matrix: a scale per gene but no correlations;
- differential evolution with genoxide's defaults, SHADE (Tanabe and Fukunaga, CEC 2013), with a population of 100;
- particle swarm optimization (Kennedy and Eberhart, 1995), 40 particles with Clerc and Kennedy's constriction coefficients and a global topology;
- a real-coded genetic algorithm: a population of 100, tournaments of 3, simulated binary crossover (Deb and Agrawal, 1995) with η = 15 and polynomial mutation with η = 20 at a rate of 1/30 per gene.
The second table runs the same algorithms on the function shifted and rotated, with genoxide's
problems::Shifted and problems::Rotated and seed 1: the minimum moves to a random point in the
middle 80% of the box, and an orthogonal matrix, drawn from normal numbers made orthonormal by
Gram-Schmidt as BBOB draws its rotations, turns the function about it. That's how the CEC and BBOB
suites use the function, with their own data; genoxide generates its instances instead.
Output
The first line gives the dimension and the budget. Then two tables, the function as it is and
shifted and rotated: a row per algorithm, the evaluations it had used when its best error first
reached each value of the heading, and the best error it found, to two significant digits. A dash is
an error not reached. The function is evaluated with genoxide's portable math, so the runs are the
same on every platform, and in Python, run evaluates it in Rust, so both versions print the same.
The project page plays back another run: CMA-ES on the function in 2 dimensions, √(x₁² + x₂⁶), rotated with seed 1, so that the population can be drawn on its contour. It meets the target after 714 evaluations.
Good results
The minimum is 0. As it is, sep-CMA-ES reaches 1e-8 first, after 7,420 evaluations, then PSO (23,080), SHADE (30,000) and CMA-ES (47,208); the genetic algorithm ends at 1.3e-6.
Shifted and rotated, only CMA-ES reaches 1e-8, after 48,734 evaluations, about as many as before. sep-CMA-ES ends at 3.9e-5, SHADE at 1.1e-5, PSO at 1.6e-4 and the genetic algorithm at 3.1e-3: their scales per gene, or steps along the axes, no longer fit.
Known optimum: 0 (at the origin)
Source: examples/different_powers
Interactive run: tachsin.gr/projects/genoxide/examples/different-powers
cargo run --release --example different_powers
//! Different powers: minimize BBOB's different powers in 30 dimensions, exponents from 2 to 6 under
//! a square root, as it is and shifted and rotated, as BBOB does.
//!
//! Compares how fast CMA-ES, with a full and with a diagonal covariance matrix (sep-CMA-ES),
//! differential evolution, particle swarm optimization and a real-coded genetic algorithm close in
//! on the minimum, 0 at the origin: the evaluations each takes until its error is at most 1, 1e-2, 1e-4, 1e-6 and 1e-8.
//! The function is genoxide's `problems::DifferentPowers`. Then the same on the function shifted and rotated, with genoxide's `problems::Shifted`
//! and `problems::Rotated`, as the CEC and BBOB suites transform it.
//!
//! With `GENOXIDE_TRACE=<file>`, it also writes a trace of a run for the plot on the example's
//! page, with `trace.rs`.
//!
//! ```text
//! cargo run --release --example different_powers
//! ```
mod trace;
use genoxide::observer::Snapshot;
use genoxide::prelude::*;
use genoxide::problems::{DifferentPowers, Problem, Rotated, Shifted};
const DIMENSIONS: usize = 30;
const BUDGET: u64 = 10_000 * DIMENSIONS as u64;
// the errors at which the table gives each run's evaluations
const ERRORS: [f64; 5] = [1e0, 1e-2, 1e-4, 1e-6, 1e-8];
const COLUMNS: [&str; 5] = ["1", "1e-2", "1e-4", "1e-6", "1e-8"];
fn main() -> Result<()> {
println!("Different powers in {DIMENSIONS} dimensions, {BUDGET} evaluations at most");
compare("Different powers", &DifferentPowers::new(DIMENSIONS))?;
// BBOB's f14, with genoxide's own shift and rotation
let rotated = Rotated::new(Shifted::new(DifferentPowers::new(DIMENSIONS), 1), 1);
compare("Shifted and rotated (seed 1)", &rotated)?;
// with GENOXIDE_TRACE=<file>, a trace for the plot on the example's page, of a separate
// run in 2 dimensions: the plot is the function's contour
trace::record_small()?;
Ok(())
}
// the table of the five algorithms on `problem`, after a line that names it
fn compare<P>(name: &str, problem: &P) -> Result<()>
where
P: Problem<Representation = Real> + FitnessFunction<Reals, Output = f64> + Clone,
{
let minimum = problem.optimum().expect("known").value();
let stop = || Stop::target(minimum + 1e-8).or(Stop::evaluations(BUDGET));
println!("{name}: evaluations until the error is at most");
print!("{:<10}", "algorithm");
COLUMNS.iter().for_each(|column| print!("{column:>9}"));
println!("{:>9}", "best");
for (name, covariance) in [
("CMA-ES", cmaes::Covariance::Full),
("sep-CMA-ES", cmaes::Covariance::Diagonal),
] {
let cmaes = Cmaes::builder(problem.representation())
.covariance(covariance)
.minimize()
.seed(1)
.build()?;
let mut reached = Reached::new(minimum);
let outcome = Engine::new(cmaes, problem.clone())
.stop_when(stop())
.on_generation(|snapshot| reached.record(snapshot))
.run()?;
reached.print(name, &outcome);
}
let de = De::builder(problem.representation())
.minimize()
.seed(1)
.build()?;
let mut reached = Reached::new(minimum);
let outcome = Engine::new(de, problem.clone())
.stop_when(stop())
.on_generation(|snapshot| reached.record(snapshot))
.run()?;
reached.print("DE", &outcome);
let pso = Pso::builder(problem.representation())
.population_size(40)
.minimize()
.seed(1)
.build()?;
let mut reached = Reached::new(minimum);
let outcome = Engine::new(pso, problem.clone())
.stop_when(stop())
.on_generation(|snapshot| reached.record(snapshot))
.run()?;
reached.print("PSO", &outcome);
let ga = Ga::builder(problem.representation())
.population_size(100)
.select(Tournament::new(3)?)
.crossover(SimulatedBinaryCrossover::new(15.0)?)
.mutate(PolynomialMutation::per_gene(1.0 / DIMENSIONS as f64, 20.0)?)
.minimize()
.seed(1)
.build()?;
let mut reached = Reached::new(minimum);
let outcome = Engine::new(ga, problem.clone())
.stop_when(stop())
.on_generation(|snapshot| reached.record(snapshot))
.run()?;
reached.print("GA", &outcome);
Ok(())
}
// the evaluations after the first generation whose best error was at most each of ERRORS, for a
// function whose minimum is `minimum`
struct Reached {
minimum: f64,
evaluations: [Option<u64>; 5],
}
impl Reached {
fn new(minimum: f64) -> Self {
let evaluations = [None; 5];
Self {
minimum,
evaluations,
}
}
fn record(&mut self, snapshot: &Snapshot<'_, Reals>) {
let progress = snapshot.progress();
let Some(best) = progress.best().and_then(Fitness::score) else {
return;
};
let error = best - self.minimum;
for (reached, bound) in self.evaluations.iter_mut().zip(ERRORS) {
if reached.is_none() && error <= bound {
*reached = Some(progress.evaluations());
}
}
}
// a row of the table: the evaluations, "-" for an error not reached, and the best error
fn print(&self, name: &str, outcome: &Outcome<Reals>) {
print!("{name:<10}");
for reached in self.evaluations {
let reached = reached.map_or("-".to_string(), |evaluations| evaluations.to_string());
print!("{reached:>9}");
}
// rounding can put a solution a few ulps below the minimum
let best = outcome.best_fitness().score().expect("valid");
let error = (best - self.minimum).max(0.0);
println!("{:>9}", format!("{error:.1e}"));
}
}
python examples/different_powers/main.py
"""Different powers: minimize BBOB's different powers in 30 dimensions, exponents from 2 to 6 under
a square root, as it is and shifted and rotated, as BBOB does.
Compares how fast CMA-ES, with a full and with a diagonal covariance matrix (sep-CMA-ES),
differential evolution, particle swarm optimization and a real-coded genetic algorithm close in on
the minimum, 0 at the origin: the evaluations each takes until its error is at most 1, 1e-2, 1e-4,
1e-6 and 1e-8. The function is genoxide's `problems::DifferentPowers`. Then the same on the function
shifted and rotated, with genoxide's `problems::Shifted` and `problems::Rotated`, as the CEC and
BBOB suites transform it.
With ``GENOXIDE_TRACE=<file>``, it also writes a trace of a run for the plot on the example's page,
with trace.py.
python examples/different_powers/main.py
"""
import genoxide as gx
from trace import record_small
DIMENSIONS = 30
BUDGET = 10_000 * DIMENSIONS
# the errors at which the table gives each run's evaluations
ERRORS = [1e0, 1e-2, 1e-4, 1e-6, 1e-8]
COLUMNS = ["1", "1e-2", "1e-4", "1e-6", "1e-8"]
def error_text(error):
"""An error to two significant digits, as Rust writes it: 9.9e-9."""
mantissa, exponent = f"{error:.1e}".split("e")
return f"{mantissa}e{int(exponent)}"
class Reached:
"""The evaluations after the first generation whose best error was at most each of ERRORS,
for a function whose minimum is ``minimum``."""
def __init__(self, minimum):
self.minimum = minimum
self.evaluations = [None] * len(ERRORS)
def record(self, progress):
if progress.best_fitness is None:
return
error = progress.best_fitness - self.minimum
for i, bound in enumerate(ERRORS):
if self.evaluations[i] is None and error <= bound:
self.evaluations[i] = progress.evaluations
def print(self, name, result):
"""A row of the table: the evaluations, "-" for an error not reached, and the best
error."""
cells = ["-" if reached is None else str(reached) for reached in self.evaluations]
# rounding can put a solution a few ulps below the minimum
cells.append(error_text(max(result.best_fitness - self.minimum, 0.0)))
print(f"{name:<10}" + "".join(f"{cell:>9}" for cell in cells))
def compare(name, problem):
"""The table of the five algorithms on ``problem``, after a line that names it."""
minimum = problem.optimum.value
print(f"{name}: evaluations until the error is at most")
print(f"{'algorithm':<10}" + "".join(f"{column:>9}" for column in COLUMNS) + f"{'best':>9}")
genome = problem.genome
for label, algorithm in (
("CMA-ES", gx.Cmaes(genome, objective="minimize", seed=1)),
("sep-CMA-ES", gx.Cmaes(genome, covariance="diagonal", objective="minimize", seed=1)),
("DE", gx.De(genome, objective="minimize", seed=1)),
("PSO", gx.Pso(genome, population_size=40, objective="minimize", seed=1)),
(
"GA",
gx.Ga(
genome,
population_size=100,
select=gx.Tournament(3),
crossover=gx.SimulatedBinaryCrossover(15.0),
mutation=gx.PolynomialMutation(20.0, rate=1 / DIMENSIONS),
objective="minimize",
seed=1,
),
),
):
reached = Reached(minimum)
result = algorithm.run(
problem, target=minimum + 1e-8, evaluations=BUDGET, on_generation=reached.record
)
reached.print(label, result)
print(f"Different powers in {DIMENSIONS} dimensions, {BUDGET} evaluations at most")
compare("Different powers", gx.problems.DifferentPowers(DIMENSIONS))
# BBOB's f14, with genoxide's own shift and rotation
rotated = gx.problems.Rotated(
gx.problems.Shifted(gx.problems.DifferentPowers(DIMENSIONS), seed=1), seed=1
)
compare("Shifted and rotated (seed 1)", rotated)
# with GENOXIDE_TRACE=<file>, a trace for the plot on the example's page, of a separate run in
# 2 dimensions: the plot is the function's contour
record_small()
What it prints, from a seeded run:
Different powers in 30 dimensions, 300000 evaluations at most
Different powers: evaluations until the error is at most
algorithm 1 1e-2 1e-4 1e-6 1e-8 best
CMA-ES 798 2954 12026 27342 47208 9.9e-9
sep-CMA-ES 812 2240 3864 5558 7420 9.9e-9
DE 4800 11100 16500 23400 30000 7.1e-9
PSO 2720 8000 13000 17680 23080 9.6e-9
GA 4280 18230 82461 - - 1.3e-6
Shifted and rotated (seed 1): evaluations until the error is at most
algorithm 1 1e-2 1e-4 1e-6 1e-8 best
CMA-ES 1078 3010 12488 28714 48734 9.6e-9
sep-CMA-ES 1064 2772 99750 - - 3.9e-5
DE 5500 15500 63200 - - 1.1e-5
PSO 6760 23760 - - - 1.6e-4
GA 6702 65310 - - - 3.1e-3