The optimizer boundary

fcmaes-core deliberately remains a small, pure-Rust library for bounded, gradient-free global optimization. It does not try to contain every optimizer that may be useful before or after a global search.

The practical boundary is:

This is not a claim that the excluded methods are inferior. It is a statement about a coherent API, dependency cost, and the problems for which this project has evidence. External solvers can run inside the existing retry closure; a permanent core dependency is not needed.

Evidence behind the decision

The dependency-isolated optimizer-boundary experiment tested two tempting additions: bounded Nelder–Mead (NM) and Bayesian optimization (BO). Twenty paired optimizer seeds were used. The raw per-seed and per-evaluation files, exact commands, compiler, and host are checked in with the rendered comparison.

The corrected protocol addresses weaknesses in an earlier exploratory run:

• budgets are measured in wall-resource rounds, not just objective calls;
• DE and DE→NM use the same seed and bit-identical DE prefix;
• every simplex has enough calls to initialize and descend;
• 16-worker populations and multistarts actually execute concurrently;
• fixed-common-seed and freshly resampled ReBop objectives are separate;
• stochastic candidates are scored on disjoint held-out paths; and
• BO and DE are reconstructed at equal wall deadlines, charging measured surrogate overhead.

Nelder–Mead result

The broad promotion case failed. Standalone serial NM had the worst median on all eight problem/protocol blocks. A 16-way NM multistart also lost to full DE on all four parallel blocks.

The DE→NM tail was not reliably better than spending the same resource rounds on DE. At 16 workers it lost to full DE on 18 of 20 CFD seeds and all 20 optical-lens seeds. On fixed-seed ReBop it recorded 3 wins, 10 losses, and 7 ties; on freshly resampled ReBop it split 10/10. None of the one-worker paired comparisons established a general advantage. A simplex often improved the shorter DE prefix, especially on optical-lens, but could not repay the global evaluations sacrificed to its sequential tail.

That distinction matters: “local refinement improves an incumbent” does not imply “the hybrid beats continued global search under the same resources.” A problem-specific NM adapter can still be reasonable after a campaign, but the experiment does not justify a native general-purpose core algorithm.

Bayesian result

BO showed a real but narrow use case. On the nine-dimensional CFD problem and a nominal 25-call deadline, EGO beat DE on 17 of 20 seeds once an evaluation was assumed to cost at least 10 ms. At a nominal 60 calls it retained an observed advantage from 10 ms upward. By 150 calls DE had the better median at every tested latency.

The hard-boundary optical problem behaved differently. BO was better at the 25-call deadline for assumed costs of 10 ms or more, but DE was already better at 60 calls and dominated 18 of 20 seeds at 150 calls for 100 ms through 1 s latencies. Measured end-of-trace optimizer overhead was about 2.96 seconds for CFD and 4.02 seconds for optical, versus about 0.1 milliseconds for DE.

The conclusion is therefore conditional: BO deserves consideration for very small, sequential, expensive-evaluation campaigns, but performance depends on dimension, landscape, invalid regions, surrogate, and acquisition policy. A specialist crate can evolve those choices without making them permanent dependencies of fcmaes-core.

Gradient-aware quality diversity

fcmaes-core supports CMA-ES emitters for MAP-Elites and diversify, which places it in the CMA-ME family. It stops short of the differentiable extensions — CMA-MEGA and the wider differentiable-quality-diversity family — for a specific reason rather than a general one.

Differentiable quality diversity requires first-order differentiable measures, not only a differentiable objective. Across the applications in tutorials/, every behaviour descriptor is derived from simulator output: measured oscillation period and amplitude, half-power beamwidth, hydraulic coverage, structural redundancy. None of them is differentiable. The precondition fails in every case, so an implementation would have no problem here to validate against.

That is a statement about this project’s problem set, not about the method. Where measures are differentiable, pyribs is the reference implementation of CMA-ME, CMA-MEGA and CMA-MAE. Policy-gradient variants such as PGA-MAP-Elites relax the requirement — there the gradient drives fitness only — but need a reinforcement-learning contract with a critic and a replay buffer, which is a different problem shape.

This decision should be revisited if a problem with differentiable measures enters the repository. A Lennard-Jones cluster study is the one identified candidate: its energy differentiates analytically, and so do its natural descriptors, radius of gyration and a smoothed Steinhardt Q6.

Supported interoperation pattern

The closure accepted by retry is the project’s optimizer adapter point. An adapter should:

1. seed the external solver from RetryContext::run_seed when replay must be independent of worker scheduling;
2. use bounds, guess, and sdev where the external API supports them;
3. treat max_evaluations as the campaign target and document any unavoidable complete-batch rounding;
4. count every real objective call in RetryRunResult::evaluations; and
5. return a finite decoded point whose score matches the reported objective.

Keep the adapter and its dependency in an application, tutorial, or a small integration crate. Create a shared adapter crate only after several real uses expose stable common code. This preserves the fast, dependency-light core while allowing local, surrogate, gradient, and domain solvers to participate in the same retry and result-accounting workflow.

Decision rule for users

• Start with a structured or gradient solver when the complete problem exposes valid structure or derivatives.
• Consider BO when calls are scarce, sequential, and costly enough to amortize surrogate work; benchmark it at equal wall time.
• Use fcmaes population methods when the objective is bounded, irregular, and parallel evaluations are available.
• Add local refinement only after a paired held-out comparison shows that it beats spending the same resources on the global method.

How expensive is “costly enough”?

Cost alone does not decide it. Per-call cost multiplies a sequential surrogate loop and a parallel population equally, so it largely cancels. Refining the same lattice-Boltzmann room model from 14 ms to 980 ms per call — a 71-fold span — produced no wall-time win for sequential EGO at any deadline by which parallel DE had already finished, and none at all at the most expensive fidelity.

Cost matters relative to the optimizer’s own overhead. EGO spent 1.7–5.5 s of modelling across 120 calls: significant at 14 ms per call, negligible at 1 s. Sample efficiency behaved as expected — at equal call counts EGO reached the better value at three of the four fidelities — but never converted, because the parallel arm completed its whole campaign before the sequential one had made ten calls.

The deciding factor is concurrency. With W workers a sequential loop concedes roughly a factor of W, and raising the per-call cost cannot recover it: at the most expensive fidelity the measured wall-time ratio was 12.3 against W = 16. Ask whether evaluations can run concurrently before asking whether they are expensive.

The broader problem-selection sequence is in Choosing an optimizer. This page records why the library itself stops where it does.