Add prior-free minimax initialization simulation

1 files changed, 31 insertions, 0 deletions

diff --git a/scripts/README.md b/scripts/README.md
index 3124261..94f9fff 100644
--- a/scripts/README.md
+++ b/scripts/README.md
@@ -53,3 +53,34 @@ The default run compares two regimes:
 - `chance`: \(q=1/D\), where \(C_{\mathrm{all}}\) grows mostly with \(L\).
 
 Outputs are written under `outputs/capacity_scaling/`.
+
+## Minimax Initialization Bound
+
+Run:
+
+```bash
+python scripts/minimax_initialization.py --dimension 32 --feedback-samples 20000 --target-samples 10000 --seed 11 --subspace-dim 4 --plot
+```
+
+This estimates the feedback second-moment matrix:
+
+\[
+M_\mu=\mathbb E_\mu[\hat b\hat b^\top]
+\]
+
+for several initialization distributions. The worst-case expected squared alignment is:
+
+\[
+\inf_{\|a\|=1}
+\mathbb E_\mu[(a^\top \hat b)^2]
+=
+\lambda_{\min}(M_\mu).
+\]
+
+The prior-free minimax theorem says:
+
+\[
+\sup_\mu \lambda_{\min}(M_\mu)=\frac1D,
+\]
+
+with equality for isotropic feedback. Outputs are written under `outputs/minimax_initialization/`.