The runtime wins in specific, stateable conditions — not everywhere. This page maps it against each classical baseline across budget regimes, from benchmarks against fairly-tuned methods. Every cell is a regime, not a slogan: where the runtime leads, where a baseline leads, and the condition that flips it. We state the losses as plainly as the wins.
| Baseline | Where the runtime wins | Where the baseline wins |
|---|---|---|
| Gradient / SPSA / finite-difference | Every tight-budget regime, at all sizes — and the margin is large. These methods must spend measurements estimating a direction; under a tight budget at high dimension they cannot assemble a step and stall. In benchmarks at n=256 under a starved budget the runtime reaches ~0.52 recovery where SPSA reaches ~0.14. | When measurements are abundant and the objective is smooth and stationary, a well-tuned gradient method is fine. If your current loop already converges comfortably, keep it — the client says so. |
| Simulated annealing (SA) | The starved-budget regime at scale. When a classical solver gets too few evaluations per variable to search (≈ 1–2 per variable) it stalls at a fixed quality floor regardless of dimension; the runtime keeps improving as the problem grows and overtakes SA past a crossover (≈ n=256–512 in benchmarks), the margin widening with size. | Given an ample budget (≥ ~5 evaluations per variable), SA remains competitive and often edges the runtime on final solution quality. We say so plainly. SA is a strong classical optimizer; the runtime's edge over it is the starved regime, not a universal win. |
| Random / equal-sample search | Everywhere it matters. The runtime reaches a usable solution while equal-sample random search stays near the noise floor (≈ 0.19–0.26 at n=256 across budgets). The substrate-as-optimizer beats equal-sample random search by an order of magnitude in benchmarks. | On trivially small instances random search can be competitive simply because the space is tiny. The advantage appears and grows with dimension. |
| Best-tuned PI controller (regulation) | Nonlinear and sign-varying plants under drift. The runtime re-estimates each channel's local sensitivity from measurement every round, tracking 2–4× tighter than a gains-swept PI on nonlinear plants, and the margin grows with how nonlinear the plant is. | On a linear, stationary, well-characterized plant a tuned PI is excellent and simpler. The runtime is for plants that move, are nonlinear, or whose polarity is unknown. |
Mean recovery (fraction of the achievable optimum), coupled QUBO at n=256, 8 seeds, identical total measurement budget per column. The budget is expressed as evaluations per variable. SWC uses the score-weighted estimator. This single experiment is the source for the cells above.
| Budget regime | SWC | SA | SPSA | Random |
|---|---|---|---|---|
| Starved (1 / variable) | 0.52 | 0.54 | 0.14 | 0.19 |
| Tight (8 / variable) | 0.94 | 0.96 | 0.15 | 0.23 |
| Ample (40 / variable) | 0.96 | 1.00 | 0.16 | 0.26 |
Read it honestly: against SPSA and random search the runtime dominates in every regime. Against SA at a single size it is close and SA edges it given budget — the runtime’s win over SA is a scaling effect that appears at larger n under starvation, shown next.
At exactly 1 evaluation per variable (a starved budget held as a fixed ratio), SA stalls — its quality flat regardless of dimension — while the runtime’s quality climbs with dimension and overtakes it. 12 seeds, coupled QUBO, bootstrap-significant past the crossover.
| n | SWC | SA | Margin |
|---|---|---|---|
| 128 | 0.45 | 0.53 | −0.08 |
| 256 | 0.52 | 0.54 | −0.02 |
| 512 | 0.58 | 0.54 | +0.05 |
| 1024 | 0.64 | 0.54 | +0.10 |
SA is flat (~0.54) across an 8× increase in dimension — it is out of budget to search. The runtime climbs from 0.45 to 0.64 and passes SA near n=512, the margin widening from there. This is the precise, defensible form of “the advantage grows with dimension” in optimization: it holds against a stalled classical solver under a starved budget, and we do not claim it outside that regime.