HGBat
The problem
HGBat is HappyCat's relative, with the difference of two squares in the first term:
f(x) = |(Σ xᵢ²)² − (Σ xᵢ)²|^(1/2) + (½ Σ xᵢ² + Σ xᵢ) / n + ½, each xᵢ in [−5, 5]
Its minimum is 0, at (−1, …, −1), the only one: the second part is Σ (xᵢ + 1)² / (2n), 0 only there, where the first is 0 too. Here n = 10. genoxide takes it from the CEC 2014 report (Liang, Qu and Suganthan 2013, function 12), which scales its search space [−100, 100] by 5/100, to [−5, 5], and gives no other source; it's usually credited to Beyer and Finck too, whose paper couldn't be read.
What makes it hard
The first term is 0 where ‖x‖² = |Σ xᵢ|, on two spheres through the origin, one of them through (−1, …, −1), and rises as a square root away from them: a groove whose floor curves around to the minimum, with a gentle slope along it. As on HappyCat, a search falls into the groove at once, then has to follow a curving direction with small steps across it and large ones along it.
Representation
A Real genome of 10 genes, each in [−5, 5]: the point x itself. The fitness is f(x), to minimize.
The function is genoxide's problems::HgBat, which brings its bounds and its minimum.
Algorithm
Five algorithms, each from seeds 1 to 10, with a budget of 10,000 evaluations per dimension, 100,000 per run, and a target of 1e-8:
- CMA-ES (Hansen and Ostermeier, 2001, Evolutionary Computation 9(2): 159-195), which samples a population of 10 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;
- the same with IPOP restarts (Auger and Hansen, 2005, IEEE CEC 2005: 1769-1776): a run that has converged starts again from a random point with twice the population;
- differential evolution with genoxide's defaults, SHADE (Tanabe and Fukunaga, CEC 2013), with a population of 100 and its restarts on stagnation;
- 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 1/10 per gene.
Output
The first line gives the dimension, the seeds and the budget. Then a row per algorithm: how many of
its 10 runs reached the minimum, to within 1e-8, the median of their evaluations (a dash if none
did), and the median of every run's best error, to two significant digits. 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 with IPOP restarts on the function in 2 dimensions, so that the population can be drawn on its contour. Within 20,000 evaluations it ends at an error of 0.010, in the groove: it doesn't reach the target in 2 dimensions either.
Good results
No algorithm reaches the minimum to within 1e-8. SHADE comes closest, with a median error of 0.13; PSO ends at 0.20, the genetic algorithm at 0.30, CMA-ES with IPOP restarts at 0.34 and without restarts at 0.45. They reach the groove and stall in it, as on HappyCat.
A larger budget doesn't change that. CMA-ES with IPOP restarts from seeds 1 to 5, with 1,000,000 evaluations each, ten times the page's budget, ends at errors of 0.11 to 0.33 (0.29, 0.11, 0.33, 0.25 and 0.29). Nelder-Mead started from each of those points, with an initial step of 0.01 of each range, converges after 1,347 to 2,021 evaluations at 0.10 to 0.25 (0.25, 0.10, 0.20, 0.15 and 0.15): better, but nowhere near the minimum.
That matches the function's record. The CEC 2014 competition's winner, L-SHADE, didn't reach the minimum either: on the competition's shifted and rotated HGBat (its F14) in 10 dimensions, with 100,000 evaluations, its best of 51 runs ended at an error of 4.5·10⁻² and its median at 7.6·10⁻² (Tanabe, R. and Fukunaga, A. S. (2014). Improving the search performance of SHADE using linear population size reduction. 2014 IEEE Congress on Evolutionary Computation: 1658-1665, table I, doi:10.1109/CEC.2014.6900380).
Known optimum: 0 (at (−1, …, −1))
Source: examples/hg_bat
Interactive run: tachsin.gr/projects/genoxide/examples/hg-bat
cargo run --release --example hg_bat
//! HGBat: minimize HGBat, HappyCat's relative with a groove along two spheres, in 10
//! dimensions.
//!
//! Runs CMA-ES without and with IPOP restarts (a population that doubles at each restart),
//! differential evolution (SHADE), particle swarm optimization and a real-coded genetic algorithm
//! from 10 seeds each, and counts the runs that reach the minimum, 0 at (−1, …, −1), to within
//! 1e-8. The function is genoxide's `problems::HgBat`.
//!
//! 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 hg_bat
//! ```
mod trace;
use genoxide::prelude::*;
use genoxide::problems::{HgBat, Problem};
const DIMENSIONS: usize = 10;
const SEEDS: u64 = 10;
const BUDGET: u64 = 10_000 * DIMENSIONS as u64;
// a run stops once its error to the minimum is at most this
const ERROR: f64 = 1e-8;
const ALGORITHMS: [&str; 5] = ["CMA-ES", "CMA-ES with IPOP", "DE", "PSO", "GA"];
fn main() -> Result<()> {
let problem = HgBat::new(DIMENSIONS);
let minimum = problem.optimum().expect("known").value();
println!(
"HGBat in {DIMENSIONS} dimensions, {SEEDS} seeds, {BUDGET} evaluations at most per run"
);
println!("algorithm at min evaluations median error");
for algorithm in ALGORITHMS {
// the evaluations of the runs that reach the minimum, and every run's best error
let mut evaluations = Vec::new();
let mut errors = Vec::new();
for seed in 1..=SEEDS {
let outcome = run(algorithm, problem, seed, minimum + ERROR)?;
if outcome.stop_reason() == StopReason::Target {
evaluations.push(outcome.evaluations() as f64);
}
// rounding can put a solution a few ulps below the minimum
let best = outcome.best_fitness().score().expect("valid");
errors.push((best - minimum).max(0.0));
}
let reached = format!("{}/{SEEDS}", evaluations.len());
let evaluations = median(evaluations).map_or("-".to_string(), |e| format!("{e:.0}"));
let error = median(errors).expect("a run");
println!(
"{algorithm:<16} {reached:>6} {evaluations:>11} {:>12}",
format!("{error:.1e}")
);
}
println!("evaluations: the median of the runs that reach the minimum");
// 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(())
}
// a run of `algorithm` from `seed`, until its best is at most `target` or it has used BUDGET
// evaluations
fn run(algorithm: &str, problem: HgBat, seed: u64, target: f64) -> Result<Outcome<Reals>> {
let real = problem.representation();
let stop = Stop::target(target).or(Stop::evaluations(BUDGET));
match algorithm {
"CMA-ES" | "CMA-ES with IPOP" => {
let restarts = if algorithm == "CMA-ES" {
cmaes::Restarts::Never
} else {
cmaes::Restarts::Ipop
};
let cmaes = Cmaes::builder(real)
.restarts(restarts)
.minimize()
.seed(seed)
.build()?;
Engine::new(cmaes, problem).stop_when(stop).run()
}
"DE" => {
let de = De::builder(real).minimize().seed(seed).build()?;
Engine::new(de, problem).stop_when(stop).run()
}
"PSO" => {
let pso = Pso::builder(real)
.population_size(40)
.minimize()
.seed(seed)
.build()?;
Engine::new(pso, problem).stop_when(stop).run()
}
_ => {
let ga = Ga::builder(real)
.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(seed)
.build()?;
Engine::new(ga, problem).stop_when(stop).run()
}
}
}
// the median of `values`, None without any
fn median(mut values: Vec<f64>) -> Option<f64> {
values.sort_by(f64::total_cmp);
let middle = values.len() / 2;
match values.len() {
0 => None,
n if n % 2 == 1 => Some(values[middle]),
_ => Some((values[middle - 1] + values[middle]) / 2.0),
}
}
python examples/hg_bat/main.py
"""HGBat: minimize HGBat, HappyCat's relative with a groove along two spheres, in 10 dimensions.
Runs CMA-ES without and with IPOP restarts (a population that doubles at each restart), differential
evolution (SHADE), particle swarm optimization and a real-coded genetic algorithm from 10 seeds
each, and counts the runs that reach the minimum, 0 at (−1, …, −1), to within 1e-8. The function is
genoxide's `problems::HgBat`, which run evaluates in Rust.
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/hg_bat/main.py
"""
import genoxide as gx
from trace import record_small
DIMENSIONS = 10
SEEDS = 10
BUDGET = 10_000 * DIMENSIONS
# a run stops once its error to the minimum is at most this
ERROR = 1e-8
ALGORITHMS = ["CMA-ES", "CMA-ES with IPOP", "DE", "PSO", "GA"]
def build(name, genome, seed):
"""The algorithm called ``name``, on ``genome``, from ``seed``."""
if name == "CMA-ES":
return gx.Cmaes(genome, objective="minimize", seed=seed)
if name == "CMA-ES with IPOP":
return gx.Cmaes(genome, restarts="ipop", objective="minimize", seed=seed)
if name == "DE":
return gx.De(genome, objective="minimize", seed=seed)
if name == "PSO":
return gx.Pso(genome, population_size=40, objective="minimize", seed=seed)
return 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=seed,
)
def median(values):
"""The median of ``values``, None without any."""
values = sorted(values)
middle = len(values) // 2
if not values:
return None
return values[middle] if len(values) % 2 else (values[middle - 1] + values[middle]) / 2
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)}"
problem = gx.problems.HgBat(DIMENSIONS)
minimum = problem.optimum.value
print(f"HGBat in {DIMENSIONS} dimensions, {SEEDS} seeds, {BUDGET} evaluations at most per run")
print("algorithm at min evaluations median error")
for name in ALGORITHMS:
# the evaluations of the runs that reach the minimum, and every run's best error
evaluations, errors = [], []
for seed in range(1, SEEDS + 1):
result = build(name, problem.genome, seed).run(
problem, target=minimum + ERROR, evaluations=BUDGET
)
if result.stop_reason == "target":
evaluations.append(float(result.evaluations))
# rounding can put a solution a few ulps below the minimum
errors.append(max(result.best_fitness - minimum, 0.0))
reached = f"{len(evaluations)}/{SEEDS}"
middle = median(evaluations)
evaluations_text = "-" if middle is None else f"{middle:.0f}"
error = error_text(median(errors))
print(f"{name:<16} {reached:>6} {evaluations_text:>11} {error:>12}")
print("evaluations: the median of the runs that reach the minimum")
# 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:
HGBat in 10 dimensions, 10 seeds, 100000 evaluations at most per run
algorithm at min evaluations median error
CMA-ES 0/10 - 4.5e-1
CMA-ES with IPOP 0/10 - 3.4e-1
DE 0/10 - 1.3e-1
PSO 0/10 - 2.0e-1
GA 0/10 - 3.0e-1
evaluations: the median of the runs that reach the minimum