Add prior-free minimax initialization simulation

1 files changed, 25 insertions, 0 deletions

diff --git a/notes/02_experiment_notes.md b/notes/02_experiment_notes.md
index ab69cd8..05b108b 100644
--- a/notes/02_experiment_notes.md
+++ b/notes/02_experiment_notes.md
@@ -259,3 +259,28 @@ Result:
 - chance-level max total cost: `18.3652` nats at width `128`, layers `16`
 
 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.
+
+## Minimax Initialization Run Log
+
+Script:
+
+```bash
+python scripts/minimax_initialization.py --dimension 32 --feedback-samples 20000 --target-samples 10000 --seed 11 --subspace-dim 4 --plot
+```
+
+Result:
+
+- minimax bound \(1/D\): `0.03125`
+- isotropic \(\lambda_{\min}\): `0.029084138`
+- rademacher \(\lambda_{\min}\): `0.028946927`
+- anisotropic \(\lambda_{\min}\): `0.006149976`
+- subspace \(\lambda_{\min}\): `0`
+- axis \(\lambda_{\min}\): `0`
+
+Random-target means remain close to \(1/D\) for all distributions, but worst-case target coverage differs sharply:
+
+- isotropic and rademacher nearly equalize all target directions;
+- anisotropic improves some directions while sacrificing others;
+- subspace and axis initializations leave entire orthogonal directions uncovered.
+
+This empirically illustrates the prior-free minimax theorem: without target or weight prior information, anisotropic feedback cannot improve the worst-case angular bound.