broke something w/ references + changed RE paragraph

1 files changed, 36 insertions, 0 deletions

diff --git a/paper/sections/appendix.tex b/paper/sections/appendix.tex
index afcf186..79baf2f 100644
--- a/paper/sections/appendix.tex
+++ b/paper/sections/appendix.tex
@@ -38,6 +38,42 @@ the proof.
 \subsubsection{Approximate sparsity proof}
 \subsubsection{RE with high probability}
 
+\begin{proof}Writing $H\defeq \nabla^2\mathcal{L}(\theta^*)$, if
+    $ \forall\Delta\in C(S),\;
+        \|\E[H] - H]\|_\infty\leq \lambda $
+    and $\E[H]$ verifies the $(S,\gamma)$-(RE)
+    condition then:
+    \begin{equation}
+        \label{eq:foo}
+    \forall \Delta\in C(S),\;
+    \Delta H\Delta \geq
+    \Delta \E[H]\Delta(1-32s\lambda/\gamma)
+    \end{equation}
+    Indeed, $
+    |\Delta(H-E[H])\Delta| \leq 2\lambda \|\Delta\|_1^2\leq
+    2\lambda(4\sqrt{s}\|\Delta_s\|_2)^2
+    $.
+    Writing
+    $\partial^2_{i,j}\mathcal{L}(\theta^*)=\frac{1}{|\mathcal{T}|}\sum_{t\in
+    T}Y_t$ and using $(LF)$ and $(LF2)$ we have $\big|Y_t - \E[Y_t]\big|\leq
+    \frac{4(M+2)}{\alpha}$.
+    Applying Azuma's inequality as in the proof of Lemma~\ref{lem:ub}, this
+    implies:
+    \begin{displaymath}
+        \P\big[\|\E[H]-H\|_{\infty}\geq\lambda\big] \leq
+        2\exp\left(-\frac{n\alpha\lambda^2}{4(M+2)} + 2\log m\right)
+    \end{displaymath}
+    Thus, if we take $\lambda=\sqrt{\frac{12(M+2)\log m}{\alpha
+    n^{1-\delta}}}$, $\|E[H]-H\|_{\infty}\leq\lambda$ w.p at least
+    $1-e^{-n^{\delta}\log m}$. When $n^{1-\delta}\geq
+    \frac{M+2}{21\gamma\alpha}s^2\log m$, \eqref{eq:foo} implies
+    $
+    \forall \Delta\in C(S),\;
+    \Delta H\Delta \geq \frac{1}{2} \Delta \E[H]\Delta,
+    $ w.p. at least $1-e^{-n^{\delta}\log m}$ and the conclusion of
+    Proposition~\ref{prop:fi} follows.
+\end{proof}
+
 \subsection{Other continuous time processes binned to ours: proportional
 hazards model}