Solution quality metrics

Import gd, igd, and hv from evomo.metrics. These functions evaluate sets in objective space, not decision variables. Remove nonfinite values before calling them. For constrained problems, also filter feasible solutions. Use consistent objective directions, scales, reference fronts, and reference points when comparing runs.

GD and IGD

Let A be the resulting solution set and P the reference front. Let d(a, P) denote the minimum Euclidean distance from point a to P. The current implementations use:

\[ \operatorname{GD}(A,P)=\frac{\sqrt{\sum_{a\in A}d(a,P)^2}}{|A|} \]

\[ \operatorname{IGD}_p(A,P)=\left(\frac{1}{|P|}\sum_{q\in P}d(q,A)^p\right)^{1/p} \]

gd(objs, pf) uses this fixed definition, rather than the arithmetic mean or root mean square of nearest-point distances. igd(objs, pf, p=1) defaults to the arithmetic mean of distances from reference points to the solution set. Lower values are better. Inputs must be nonempty, and IGD’s p must be positive.

from evomo.metrics import gd, igd

reference_front = problem.pf().to(device=pareto_fitness.device, dtype=pareto_fitness.dtype)
gd_score = gd(pareto_fitness, reference_front)
igd_score = igd(pareto_fitness, reference_front)

Use this after result filtering in the quickstart. torch.cdist constructs a distance matrix between the two sets, so large solution sets and reference fronts can require substantial memory. Objectives with larger scales can dominate distances. If you normalize objectives, apply the same transform to both sets and record the normalization rule.

HV: Monte Carlo estimation

hv(objs, ref, num_sample=100000) estimates hypervolume for minimization. ref has shape (M,). Choose a reference point worse than the solutions of interest and keep it fixed across comparisons. Points outside it contribute no volume. Higher HV is better for fixed scales and a fixed reference point.

import torch
from evomo.metrics import hv

objectives = torch.tensor([[0.2, 0.8], [0.5, 0.4], [0.8, 0.2]])
reference = objectives.new_tensor([1.1, 1.1])
torch.manual_seed(42)
print(hv(objectives, reference, num_sample=10000))

This is a stochastic estimate, not an exact hypervolume calculation. Increasing num_sample increases runtime and memory costs; comparisons scale with sample count, solution count, and objective count. An empty set returns scalar zero. Nonpositive sample counts raise an error.

For maximization, negate both objectives and the reference point, for example hv(-rewards, -reference_rewards). For mixed directions, transform both inputs using the same per-objective directions. Do not pass restored maximization objectives directly to HV under its minimization convention.

Reproducible comparisons

Record algorithm seeds, evaluation budgets, the extraction scope (final population or historical archive), reference front sampling, HV reference points, and sample counts. Across multiple seeds, report distributions or means with dispersion rather than treating one run as a stable result.