1 files changed, 24 insertions, 11 deletions
diff --git a/paper/sections/results.tex b/paper/sections/results.tex
index 6c8a35a..01a33e9 100644
--- a/paper/sections/results.tex
+++ b/paper/sections/results.tex
@@ -325,20 +325,33 @@ $\theta$ conveyed by the random observations. Therefore, under an assumption
 which only involves the probabilistic model and not the actual observations, we
 can reformulate Theorem~\ref{thm:main}.
 
-We will need the following additional assumptions on the inverse link
-function $f$:
+We will need the following additional assumptions on the inverse link function $f$:
+\begin{equation}
+    \tag{LF2}
+    \|f'\|_{\infty} \leq M
+    \text{ and }
+    \max\left(\left|\frac{f''}{1-f}\right|,
+    \left|\frac{f''}{f}\right|\right)
+    \leq\frac{1}{\alpha}
+\end{equation}
+whenever $f(\inprod{theta^*}{x})\notin\{0,1\}$.
+
+\begin{theorem}
+    If $\E[\nabla^2\mathcal{L}(\theta^*)]$ verifies the $(S,\gamma)$-{\bf (RE)}
+    condition. Assuming {\bf (LF)} and {\bf (LF2)}, let $\hat\theta$ be
+    a solution to \eqref{eq:pre-mle}, then if $n\geq $ we have:
+    \begin{displaymath}
+         \|\hat \theta - \theta^* \|_2 \leq \frac{3}{\gamma} \sqrt{\frac{s \log
+    m}{\alpha n^{1-\delta}}} \quad \text{w.p.}\;1-\frac{1}{e^{n^\delta \log m}}
+    \end{displaymath}
+\end{theorem}
+
+\begin{proof}
+\end{proof}
 
 
 
-Yet, the restricted eigenvalue
-property is however well behaved in the following sense: under reasonable
-assumptions, if the population matrix of the hessian $\mathbb{E} \left[\nabla^2
-{\cal L}(\theta) \right]$, corresponding to the \emph{Fisher Information
-Matrix} of the Cascade Model as a function of $\Theta$, verifies the restricted
-eigenvalue property, then the finite sample hessian also verifies the
-restricted eigenvalue property with overwhelming probability. It is
-straightforward to show this holds when $n \geq C s^2 \log m$
-\cite{vandegeer:2009}, where $C$ is an absolute constant. By adapting Theorem
+By adapting Theorem
 8 \cite{rudelson:13}, this
   can be reduced to:
 \begin{displaymath}