From 48f43ffa66c8af1b5aec123ebe9877bc3852ecd3 Mon Sep 17 00:00:00 2001
From: YurenHao0426 <Blackhao0426@gmail.com>
Date: Thu, 28 May 2026 22:54:53 -0500
Subject: Add prior-free minimax initialization simulation

---
 scripts/README.md | 31 +++++++++++++++++++++++++++++++
 1 file changed, 31 insertions(+)

(limited to 'scripts/README.md')

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/`.
-- 
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