In this section, we apply the sparse recovery framework to analyze under which
assumptions our program~\eqref{eq:pre-mle} recovers the true parameter
$\theta_i$ of the cascade model. Furthermore, if we can estimate $\theta_i$ to
a sufficiently good accuracy, it is then possible to recover the support of
$\theta_i$ by simple thresholding, which provides a solution to the standard
Graph Inference problem.

We will first give results in the exactly sparse setting in which $\theta_i$
has a support of size exactly $s$. We will then relax this sparsity constraint
and give results in the \emph{stable recovery} setting where $\theta_i$ is
approximately $s$-sparse.

As mentioned in Section~\ref{sec:mle}, the maximum likelihood estimation
program is decomposable. We will henceforth focus on a single node $i\in V$ and
omit the subscript $i$ in the notations when there is no ambiguity. The
recovery problem is now the one of estimating a single vector $\theta^*$ from
a set $\mathcal{T}$ of observations. We will write $n\defeq |\mathcal{T}|$.

\begin{comment}
As mentioned previously, our objective is twofold:
\begin{enumerate}
    \item We seek to recover the edges of the graph. In other words, for each
        node $j$, we wish to identify the set of indices $\{i: \Theta_{ij} \neq
        0\}$ . 
    \item We seek to recover the edge weights $\Theta$ of ${\cal G}$ i.e.
        upper-bound the error $\|\hat \Theta - \Theta \|_2$.
\end{enumerate}
It is easy to see that solving the second objective allows us to solve the
first. If $\|\hat \Theta - \Theta \|_2$ is sufficiently small, we can recover
all large coordinates of $\Theta$ by keeping only the coordinates above
a certain threshold.
\end{comment}

\subsection{Main Theorem}
\label{sec:main_theorem}

In this section, we analyze the case where $\theta^*$ is exactly sparse. We
write $S\defeq\text{supp}(\theta^*)$ and $s=|S|$. Our main theorem will rely
on the now standard \emph{restricted eigenvalue condition}.

\begin{definition}
    Let $\Sigma\in\mathcal{S}_m(\reals)$ be a real symmetric matrix and $S$ be
    a subset of $\{1,\ldots,m\}$. Defining $\mathcal{C}(S)\defeq
    \{X\in\reals^m\,:\,\|X\|_1\leq 1\text{ and } \|X_{S^c}\|_1\leq
    3\|X_S\|_1\}$. We say that $\Sigma$ satisfies the
    $(S,\gamma)$-\emph{restricted eigenvalue condition} iff:
\begin{equation}
    \forall X \in {\cal C(S)}, \| \Sigma X \|_2^2 \geq \gamma \|X\|_2^2 
\tag{RE}
\label{eq:re}
\end{equation}
\end{definition}

A discussion of this assumption in the context of generalized linear cascade
models can be found in Section~\ref{sec:}.

We will also need the following assumption on the inverse link function $f$ of
the generalized linear cascade model:
\begin{equation}
    \tag{LF}
    \max\left(\left|\frac{f'}{f}(\inprod{\theta^*}{x})\right|,
    \left|\frac{f'}{1-f}(\inprod{\theta^*}{x})\right|\right)\leq
    \frac{1}{\alpha}
\end{equation}
for all $x\in\reals^m$ such that $f(\inprod{\theta^*}{x})\notin\{0,1\}$.

It is easy to verify that this will be true if for all
$(i,j)\in E$, $\delta\leq p_{i,j}\leq 1-\delta$ in the Voter model and when
$p_{i,j}\leq 1-\delta$ in the Independent Cascade model.

\begin{theorem}
\label{thm:main}
Assume the Hessian $\nabla^2\mathcal{L}(\theta^*)$ satisfies the
$(S,\gamma)$-{\bf(RE)} for some $\gamma > 0$ and that {\bf (LF)} holds for
some $\alpha > 0$. For any $\delta\in(0,1)$, let $\hat{\theta}$ be the solution
of \eqref{eq:pre-mle} with $\lambda \defeq 2\sqrt{\frac{\log m}{\alpha n^{1
- \delta}}}$ then:
\begin{equation}
    \label{eq:sparse}
    \|\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{equation}
\end{theorem}

This theorem is a consequence of Theorem~1 in \cite{Negahban:2009} which gives
a bound on the convergence rate of regularized estimators. We state their
theorem in the context of $\ell_1$ regularization in Lemma~\ref{lem:negahban}.

\begin{lemma}
    \label{lem:negahban}
Let ${\cal C}(S) \defeq \{ \Delta \in \mathbb{R}^m\,|\,\|\Delta_S\|_1 \leq
3 \|\Delta_{S^c}\|_1 \}$. Suppose that:
\begin{multline}
    \label{eq:rc}
    \forall \Delta \in {\cal C}(S), \;
    {\cal L}(\theta^* + \Delta) - {\cal L}(\theta^*)\\
    - \inprod{\Delta {\cal L}(\theta^*)}{\Delta}
    \geq \kappa_{\cal L} \|\Delta\|_2^2 - \tau_{\cal L}^2(\theta^*)
\end{multline}
for some $\kappa_{\cal L} > 0$ and function $\tau_{\cal L}$. Finally suppose
that $\lambda \geq 2 \|\nabla {\cal L}(\theta^*)\|_{\infty}$, then if
$\hat{\theta}_\lambda$ is the solution of \eqref{eq:pre-mle}:
\begin{displaymath}
\|\hat \theta_\lambda - \theta^* \|_2^2
\leq 9 \frac{\lambda^2 s}{\kappa_{\cal L}}
+ \frac{\lambda}{\kappa_{\cal L}^2} 2 \tau^2_{\cal L}(\theta^*)
\end{displaymath}
\end{lemma}

To prove Theorem~\ref{thm:main}, we apply Lemma~\ref{lem:negahban} with
$\tau_{\mathcal{L}}=0$. Since $\mathcal{L}$ is twice differentiable and convex,
assumption \eqref{eq:rc} is implied by the restricted eigenvalue condition
\eqref{eq:re}. The upper bound on the $\ell_{\infty}$ norm of
$\nabla\mathcal{L}(\theta^*)$ is given by Lemma~\ref{lem:ub}.

\begin{lemma}
\label{lem:ub}
Assume {\bf(LF)} holds for some $\alpha>0$. For any $\delta\in(0,1)$:
\begin{displaymath}
    \|\nabla {\cal L}(\theta^*)\|_{\infty}
    \leq 2 \sqrt{\frac{\log m}{\alpha n^{1 - \delta}}}
    \quad
    \text{w.p.}\; 1-\frac{1}{e^{n^\delta \log m}}
\end{displaymath}
\end{lemma}

\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}

\begin{corollary}
\label{cor:variable_selection}
Assume that ${\bf (RE)}$ holds with $\gamma > 0$ and that $\theta$ is s-sparse. Consider the set $\hat {\cal S}_\eta \defeq \{ i \in [1..p] : \hat \theta_i > \eta\}$ for $\eta > 0$. For $\epsilon>0$ and $\epsilon < \eta$, let ${\cal S}^*_{\eta + \epsilon} \defeq \{ i \in [1..p] :\theta_i > \eta +\epsilon \}$ be the set of all true `strong' parents. Suppose the number of measurements verifies:
\begin{equation}
n > \frac{36}{p_{\min}\gamma^2 \epsilon^2} s \log m
\end{equation}
Then with probability $1-\frac{1}{m}$, ${\cal S}^*_{\eta + \epsilon} \subset
\hat {\cal S}_\eta \subset {\cal S}^*$. In other words we recover all `strong'
parents and no `false' parents. 
\end{corollary}

\begin{proof}
Suppose $\theta$ is exactly s-sparse. By choosing $\delta = 0$, if $n>\frac{36}{p_{\min}\gamma^2 \epsilon^2} s \log m$, then $\|\hat \theta-\theta\|_2 < \epsilon < \eta$ with probability $1-\frac{1}{m}$. If $\theta_i = 0$ and $\hat \theta > \eta$, then $\|\hat \theta - \theta\|_2 \geq |\hat \theta_i-\theta_j| > \eta$, which is a contradiction. Therefore we get no false positives. If $\theta_i = \eta + \epsilon$, then $|\hat \theta_i - (\eta+\epsilon)| < \epsilon/2 \implies \theta_j > \eta + \epsilon/2$. Therefore, we get all strong parents.
\end{proof}

Note that if we want \emph{perfect recovery} of all edges, we must estimate $p_{\min}$ within $\epsilon'$ and set $\eta \defeq p_{\min} - \epsilon'$

\subsection{Relaxing the Sparsity Constraint}
\label{sec:relaxing_sparsity}

In practice, exact sparsity is rarely verified. For social networks in particular, it is more realistic to assume that each node has few strong `parents' and many `weak' parents. Rather than obtaining an impossibility result in this situation, we show that we pay a small price for relaxing the sparsity constraint. If we let $\theta^*_{\lfloor s \rfloor}$ be the best s-sparse approximation to $\theta^*$ defined as 
$$\theta^*_{\lfloor s \rfloor} \defeq \min_{\|\theta\|_0 \leq s} \|\theta - \theta^*\|_1$$
then we pay a price proportional to $\|\theta^*_s\|_1$ for recovering the weights of non-exactly sparse vectors, i.e. the sum of the coefficients of $\theta$ save for the $s$ largest. In other words, the closer $\theta^*$ is to being sparse, the smaller the price. These results are formalized in the following theorem, which is also a consequence of Theorem 1 in \cite{Negahban:2009} and lemma~\ref{lem:icc_lambda_upper_bound}.

\begin{theorem}
\label{thm:approx_sparse}
Let $\theta^*_{\lfloor s \rfloor}$ be the best s-sparse approximation to the true vector $\theta^*$. Suppose the {\bf(RE)} assumption holds for the Hessian $\nabla^2 f(\theta^*)$ and for the following set 
\begin{align}
\nonumber
{\cal C}' \defeq & \{X \in \mathbb{R}^p : \|X_{S^c}\|_1 \leq 3 \|X_S\|_1 + 4 \|\theta^*_{\lfloor s \rfloor}\|_1 \} \\ \nonumber
& \cap \{ \|X\|_1 \leq 1 \}
\end{align}
By solving \eqref{eq:pre-mle} for $\lambda \defeq 2\sqrt{\frac{\log m}{n^{1 - \delta} p_{\min}}}$ we have:
\begin{align}
\|\hat p - p^* \|_2 \leq 3 \frac{\sqrt{s}\lambda_n}{\gamma_n} + 2 \sqrt{\frac{\lambda_n}{\gamma_n}} \|p^*_{\lfloor s \rfloor}\|_1
\end{align}
\end{theorem}

As before, edge recovery is a consequence of upper-bounding $\|\theta - \hat \theta\|_2$

\begin{corollary}
If $\theta$ is not exactly s-sparse and the number of measurements verifies:
\begin{equation}
n > \frac{36 \| p^*_{\lfloor s\rfloor}\|_1}{p_{\min}\gamma^2 \epsilon^4} s \log m
\end{equation}
then similarly: ${\cal S}^*_{\eta + \epsilon} \subset \hat {\cal S}_\eta \subset {\cal S}^*$ w.h.p.
\end{corollary}


\subsection{The Independent Cascade Model}

Recall that to cast the Independent Cascade model as a Generalized Linear Cascade, we performed the following change of variables: $\forall i,j \ \Theta_{i,j} = \log(1 - p_{i,j})$. The previous results hold \emph{a priori} for the ``effective'' parameter $\Theta$. The following lemma shows they also hold for the original infection probabilities $p_{i,j}$:
\begin{lemma}
\label{lem:theta_p_upperbound}
$\|\theta - \theta' \|_2 \geq \|p - p^*\|_2$ and $\|\theta_{\lfloor s \rfloor}\|_1 \geq \| p_{\lfloor s \rfloor} \|_1$
\end{lemma}
\begin{proof}
Using the inequality $\forall x>0, \; \log x \geq 1 - \frac{1}{x}$, we have $|\log (1 - p) - \log (1-p')| \geq \max(1 - \frac{1-p}{1-p'}, 1 - \frac{1-p'}{1-p}) \geq \max( p-p', p'-p)$. The second result follows easily from the monotonicity of $\log$ and $\forall x > 0, | \log (1 -x) | \geq x$
\end{proof}

The results of sections~\ref{sec:main_theorem} and \ref{sec:relaxing_sparsity} therefore hold for the original transition probabilities $p$.

\begin{comment}
\begin{lemma}
Let ${\cal C}({\cal M}, \bar {\cal M}^\perp, \theta^*) \defeq \{ \Delta \in \mathbb{R}^p | {\cal R}(\Delta_{\bar {\cal M}^\perp} \leq 3 {\cal R}(\Delta_{\bar {\cal M}} + 4 {\cal R}(\theta^*_{{\cal M}^\perp}) \}$, where $\cal R$ is a \emph{decomposable} regularizer with respect to $({\cal M}, \bar {\cal M}^\perp)$, and $({\cal M}, \bar {\cal M})$ are two subspaces such that  ${\cal M} \subseteq \bar {\cal M}$. Suppose that $\exists \kappa_{\cal L} > 0, \; \exists \tau_{\cal L}, \; \forall \Delta \in {\cal C}, \; {\cal L}(\theta^* + \Delta) - {\cal L}(\theta^*) - \langle \Delta {\cal L}(\theta^*), \Delta \rangle \geq \kappa_{\cal L} \|\Delta\|^2 - \tau_{\cal L}^2(\theta^*)$. Let $\Psi({\cal M}) \defeq \sup_{u \in {\cal M} \backslash \{0\}} \frac{{\cal R}(u)}{\|u\|}$. Finally suppose that $\lambda \geq 2 {\cal R}(\nabla {\cal L}(\theta^*))$, where ${\cal R}^*$ is the conjugate of ${\cal R}$. Then: $$\|\hat \theta_\lambda - \theta^* \|^2 \leq 9 \frac{\lambda^2}{\kappa_{\cal L}}\Psi^2(\bar {\cal M}) + \frac{\lambda}{\kappa_{\cal L}}\{2 \tau^2_{\cal L}(\theta^*) + 4 {\cal R}(\theta^*_{{\cal M}^\perp}\}$$
\end{lemma}
\end{comment}
