\subsection{Proof for different lemmas}

\subsubsection{Bounded gradient}

\begin{proof}
The gradient of $\mathcal{L}$ is given by:
\begin{multline*}
    \nabla \mathcal{L}(\theta^*) =
    \frac{1}{|\mathcal{T}|}\sum_{t\in \mathcal{T}}x^t\bigg[
        x_i^{t+1}\frac{f'}{f}(\inprod{\theta^*}{x^t})\\
    - (1-x_i^{t+1})\frac{f'}{1-f}(\inprod{\theta^*}{x^t})\bigg]
\end{multline*}

Let $\partial_j \mathcal{L}(\theta)$ be the $j$-th coordinate of
$\nabla\mathcal{L}(\theta^*)$.  Writing
$\partial_j\mathcal{L}(\theta^*)
= \frac{1}{|\mathcal{T}|}\sum_{t\in\mathcal{T}} Y_t$ and since
$\E[x_i^{t+1}|x^t]= f(\inprod{\theta^*}{x^t})$, we have that $\E[Y_{t+1}|Y_t]
= 0$. Hence $Z_t = \sum_{k=1}^t Y_k$ is a martingale.

Using assumption (LF), we have almost surely $|Z_{t+1}-Z_t|\leq
\frac{1}{\alpha}$ and we can apply Azuma's inequality to $Z_t$:
\begin{displaymath}
    \P\big[|Z_{\mathcal{T}}|\geq \lambda\big]\leq
    2\exp\left(\frac{-\lambda^2\alpha}{2n}\right)
\end{displaymath}

Applying a union bound to have the previous inequality hold for all coordinates
of $\nabla\mathcal{L}(\theta)$ implies:
\begin{align*}
    \P\big[\|\nabla\mathcal{L}(\theta^*)\|_{\infty}\geq \lambda \big]
    &\leq 2m\exp\left(\frac{-\lambda^2n\alpha}{2}\right)
\end{align*}
Choosing $\lambda\defeq 2\sqrt{\frac{\log m}{\alpha n^{1-\delta}}}$ concludes
the proof.
\end{proof}

\subsubsection{Approximate sparsity proof}
\subsubsection{RE with high probability}

\subsection{Other continuous time processes binned to ours: proportional
hazards model}

\subsection{Irrepresentability vs. Restricted Eigenvalue Condition}
In the words and notation of Theorem 9.1 in \cite{vandegeer:2009}:
\begin{lemma}
\label{lemm:irrepresentability_proof}
Let $\phi^2_{\text{compatible}}(L,S) \defeq \min \{ \frac{s
\|f_\beta\|^2_2}{\|\beta_S\|^2_1} \ : \ \beta \in {\cal R}(L, S) \}$, where
$\|f_\beta\|^2_2 \defeq \{ \beta^T \Sigma \beta \}$ and ${\cal R}(L,S) \defeq
\{\beta : \|\beta_{S^c}\|_1 \leq L \|\beta_S\|_1 \neq 0\}$. If
$\nu_{\text{irrepresentable}(S,s)} < 1/L$, then $\phi^2_{\text{compatible}}(L,S)
\geq (1 - L \nu_{\text{irrepresentable}(S,s)})^2 \lambda_{\min}^2$.
\end{lemma}

Since ${\cal R}(3, S) = {\cal C}$, $\|\beta_S\|_1 \geq \|\beta_S\|_2$, and
$\|\beta_S\|_1 \geq \frac{1}{3} \|\beta_{S^c}\|_1$ it is easy to see that
$\|\beta_S\|_1 \geq \frac{1}{4} \|\beta\|_2$ and therefore that: $\gamma_n \geq
\frac{n}{4s}\phi^2_{\text{compatible}}(3,S)$

Consequently, if $\epsilon > \frac{2}{3}$, then
$\nu_{\text{irrepresentable}(S,s)} < 1/3$ and the conditions of
Lemma~\ref{lemm:irrepresentability_proof} hold.



\subsection{Lower bound for restricted eigenvalues (expected hessian) for
different graphs}

\subsection{Better asymptotic w.r.t expected hessian}

\subsection{Confidence intervals?}

\subsection{Active learning}