Discus
The problem
The discus squares the first gene a million times more than the others:
f(x) = 10⁶ x₁² + Σᵢ₌₂ⁿ xᵢ², each xᵢ in [−100, 100]
Its minimum is 0, at the origin. Here n = 30. It's BBOB's f11 (Hansen et al. 2009), with an oscillation and a rotation there; this plain form and the bounds are the CEC 2014 (Liang, Qu and Suganthan 2013, function 3) and CEC 2017 (Awad et al. 2016, function 11) reports' basic function, which those suites shift and rotate.
What makes it hard
The level sets are discs: one direction is a thousand times more sensitive than all the others. A step that suits the 29 flat directions overshoots in the steep one, and a step that suits the steep one crawls in the others: a search has to give that one direction its own scale. As it is, the direction is a gene's axis; shifted and rotated, a random one.
Representation
A Real genome of 30 genes: the point x itself. The fitness is f(x), to minimize. The function is
genoxide's problems::Discus, 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, 10⁶ x₁² + x₂², rotated with seed 1, so that the population can be drawn on its contour. It meets the target after 684 evaluations.
Good results
The minimum is 0. As it is, sep-CMA-ES reaches 1e-8 after 6,594 evaluations; PSO takes 27,720, CMA-ES 29,778 and SHADE 33,800. The genetic algorithm ends at 0.075.
Shifted and rotated, CMA-ES takes 29,288 evaluations, as many as before. SHADE reaches 1e-4 after 268,800 evaluations and ends at 6.5·10⁻⁶; sep-CMA-ES ends at 3.7·10⁴, PSO at 2,500 and the genetic algorithm at 640.
Known optimum: 0 (at the origin)
Source: examples/discus
Interactive run: tachsin.gr/projects/genoxide/examples/discus
cargo run --release --example discus
//! Discus: minimize a sphere squashed along one axis, a thousand times more sensitive than
//! the others, in 30 dimensions, as it is and shifted and rotated.
//!
//! 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::Discus`. 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 discus
//! ```
mod trace;
use genoxide::observer::Snapshot;
use genoxide::prelude::*;
use genoxide::problems::{Discus, 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!("Discus in {DIMENSIONS} dimensions, {BUDGET} evaluations at most");
compare("Discus", &Discus::new(DIMENSIONS))?;
// the CEC 2014 and 2017 suites' form, with genoxide's own shift and rotation
let rotated = Rotated::new(Shifted::new(Discus::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/discus/main.py
"""Discus: minimize a sphere squashed along one axis, a thousand times more sensitive than the
others, in 30 dimensions, as it is and shifted and rotated.
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::Discus`. 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/discus/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"Discus in {DIMENSIONS} dimensions, {BUDGET} evaluations at most")
compare("Discus", gx.problems.Discus(DIMENSIONS))
# the CEC 2014 and 2017 suites' form, with genoxide's own shift and rotation
rotated = gx.problems.Rotated(gx.problems.Shifted(gx.problems.Discus(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:
Discus in 30 dimensions, 300000 evaluations at most
Discus: evaluations until the error is at most
algorithm 1 1e-2 1e-4 1e-6 1e-8 best
CMA-ES 22442 24990 26978 28532 29778 1.0e-8
sep-CMA-ES 3766 4424 5138 5852 6594 9.0e-9
DE 14600 19600 24100 29300 33800 9.8e-9
PSO 12400 16240 20040 23560 27720 8.3e-9
GA 119237 - - - - 7.5e-2
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 23408 25564 26824 28210 29288 9.5e-9
sep-CMA-ES - - - - - 3.7e4
DE 159900 216600 268800 - - 6.5e-6
PSO - - - - - 2.5e3
GA - - - - - 6.4e2