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| -rw-r--r-- | notes/02_experiment_notes.md | 51 | ||||
| -rw-r--r-- | notes/03_paper_outline.md | 1 |
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diff --git a/notes/02_experiment_notes.md b/notes/02_experiment_notes.md index 896961b..dcf797a 100644 --- a/notes/02_experiment_notes.md +++ b/notes/02_experiment_notes.md @@ -260,6 +260,57 @@ Result: This cleanly separates the fixed-threshold regime, where total cost scales like \(Ln^2\), from the chance-level regime, where per-layer cost is nearly width-independent. +## Large Empirical Capacity Validation Run Log + +Script: + +```bash +python scripts/capacity_empirical_validation.py --dimensions 64 128 256 512 1024 2048 4096 --samples 100000 --batch-size 2048 --seed 123 --plot +``` + +Setup: + +- dimensions \(D\): `64, 128, 256, 512, 1024, 2048, 4096` +- samples per dimension: `100000` +- total \(Q\) samples: `700000` +- sampler: fixed-target Gaussian direction, using rotational invariance +- chance thresholds: \(q=c/D\), \(c\in\{0.5,1,2,5,10\}\) +- fixed thresholds: \(q\in\{0.001,0.002,0.005,0.01\}\) +- multilayer all-event validation: \(c\in\{0.5,1\}\), layers \(L\in\{1,2,4,8\}\) + +Distribution calibration: + +| \(D\) | empirical mean | theory mean | KS statistic | empirical q99 | theory q99 | +|---:|---:|---:|---:|---:|---:| +| `64` | `0.0156793` | `0.0156250` | `0.0034949` | `0.1007135` | `0.1007080` | +| `128` | `0.0078528` | `0.0078125` | `0.0019669` | `0.0511649` | `0.0510968` | +| `256` | `0.0039289` | `0.0039062` | `0.0029576` | `0.0259898` | `0.0257333` | +| `512` | `0.0019553` | `0.0019531` | `0.0016418` | `0.0129654` | `0.0129127` | +| `1024` | `0.0009706` | `0.0009766` | `0.0031219` | `0.0063694` | `0.0064679` | +| `2048` | `0.0004882` | `0.0004883` | `0.0031095` | `0.0032339` | `0.0032368` | +| `4096` | `0.0002444` | `0.0002441` | `0.0024262` | `0.0016322` | `0.0016191` | + +Capacity-tail calibration: + +- chance-threshold rows: `35/35` had expected hits \(\ge 20\) +- max absolute smoothed cost error over all chance rows: `0.2023` nats +- mean absolute smoothed cost error over all chance rows: `0.01915` nats +- for chance rows with expected hits \(\ge 500\): max error `0.02771` nats, mean error `0.00607` nats + +Multilayer product-capacity calibration: + +- rows with expected hits \(\ge 20\): `49` +- max absolute smoothed total-cost error: `0.45788` nats +- mean absolute smoothed total-cost error: `0.03880` nats +- for rows with expected hits \(\ge 500\): max error `0.04319` nats, mean error `0.00826` nats + +Interpretation: + +- The beta-law distribution calibration is very tight across two orders of magnitude in dimension. +- Empirical \(C(q)=-\log P(Q\ge q)\) matches the theoretical incomplete-beta cost when the expected tail count is large enough. +- Multilayer capacity accumulation matches the product law until the all-event becomes too rare for the finite sample budget. +- Fixed-threshold high-dimensional tails quickly become too rare for direct Monte Carlo, which is itself consistent with the exponential/geometric volume-collapse interpretation. + ## Minimax Initialization Run Log Script: diff --git a/notes/03_paper_outline.md b/notes/03_paper_outline.md index 65a5c47..7e2e5d4 100644 --- a/notes/03_paper_outline.md +++ b/notes/03_paper_outline.md @@ -164,6 +164,7 @@ Scaling: - depth and width sweeps. - threshold regimes. +- large empirical tail calibration for \(C(q)=-\log P(Q\ge q)\). Trajectory: |