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 $n\defeq |\mathcal{T}|$.

\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:re}.

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 some $\alpha\in(0, 1)$ and for all $x\in\reals^m$ such that
$f(\inprod{\theta^*}{x})\notin\{0,1\}$.

In the voter model, $\frac{f'}{f}(z) = \frac{1}{z}$ and $\frac{f'}{f}(1-z)
= \frac{1}{1-z}$. Hence (LF) will hold as soon as $\alpha\leq \Theta_{i,j}\leq
1-\alpha$ for all $(i,j)\in E$. Similarly, in the Independent Cascade Model,
$\frac{f'}{f}(z) = \frac{1}{1-e^z}$ and $\frac{f'}{1-f}(z) = -1$ and (LF) holds
if $p_{i, j}\leq 1-\alpha$ for all $(i, j)\in E$. Remember that in this case we
have $\Theta_{i,j} = \log(1-p_{i,j})$.

\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}

Before moving to the proof of Theorem~\ref{thm:main}, note that interpreting it
in the case of the Independent Cascade Model requires one more step. Indeed, to
cast it as a generalized linear cascade model, we had to perform the
transformation $\Theta_{i,j} = \log(1-p_{i,j})$, where $p_{i,j}$ are the
infection probabilities. Fortunately, if we estimate $p_j}$ through
$\hat{p}_{j} = \hat{\theta}_j$, a bound on $\|\hat\theta - \theta^*\|_2$
directly implies a bound on $\|\hat p - p^*\|$. Indeed we have:

\begin{lemma}
    $\|\hat{\theta} - \theta' \|_2 \geq \|\hat{p} - p^*\|_2$.
\end{lemma}
\begin{proof}
Using the inequality $\forall x>0, \; \log x \geq 1 - \frac{1}{x}$, we have
$|\log (1 - p) - \log (1-p')| \geq \max(1 - \frac{1-p}{1-p'},
1 - \frac{1-p'}{1-p}) \geq \max( p-p', p'-p)$.
\end{proof}

Theorem~\ref{thm:main} 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{Restricted Eigenvalue Condition}
\label{sec:re}

In this section, we discuss the main assumption of Theorem~\ref{thm:main}
namely the restricted eigenvalue condition. We then compare to the
irrepresentability condition considered in \cite{Daneshmand:2014}.

\paragraph{Interpretation}

The restricted eigenvalue condition, introduced in \cite{bickel:2009}, is one
of the weakest sufficient condition on the design matrix for successful sparse
recovery \cite{vandegeer:2009}. Several recent papers show that large classes
of correlated designs obey the restricted eigenvalue property with high
probability \cite{raskutti:10} \cite{rudelson:13}. 


\paragraph{(RE) with high probability}

Expressing the minimum restricted eigenvalue $\gamma$ as a function of the
cascade model parameters is highly non-trivial. 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
8 \cite{rudelson:13}\footnote{this result still needs to be confirmed!}, this
  can be reduced to:
\begin{displaymath}
n \geq C s \log m \log^3 \left( \frac{s \log m}{C'} \right)
\end{displaymath}
where $C, C'$ are constants not depending on $(s, m, n)$.

\paragraph{(RE) vs Irrepresentability Condition}

\cite{Daneshmand:2014} rely on an `incoherence' condition on the hessian of the likelihood function. It is in fact easy to see that their condition is equivalent to the more commonly called {\it (S,s)-irrepresentability} condition:

\begin{definition}
Following similar notation, let $Q^* \defeq \nabla^2 f(\theta^*)$. Let $S$ and $S^c$ be the set of indices of all the parents and non-parents respectively and $Q_{S,S}$, $Q_{S^c,S}$, $Q_{S, S^c}$, and $Q_{S^c, S^c}$ the induced sub-matrices. Consider the following constant:
\begin{equation}
\nu_{\text{irrepresentable}}(S) \defeq \max_{\tau \in \mathbb{R}^p \ :\ \| \tau \|_{\infty} \leq 1} \|Q_{S^c, S} Q_{S, S}^{-1} \tau\|_{\infty}
\end{equation}
The (S,s)-irrepresentability holds if $\nu_{\text{irrepresentable}}(S) < 1 - \epsilon$ for $\epsilon > 0$
\end{definition}

This condition has been shown to be essentially necessary for variable selection \cite{Zhao:2006}. Several recent papers \cite{vandegeer:2011}, \cite{Zou:2006}, argue this condition is unrealistic in situations where there is a correlation between variables. Consider the following simplified example from \cite{vandegeer:2011}:
\begin{equation}
\nonumber
\left( 
\begin{array}{cccc}
I_s & \rho J \\
\rho J & I_s \\
\end{array}
\right)
\end{equation}
where $I_s$ is the $s \times s$ identity matrix, $J$ is the all-ones matrix and $\rho \in \mathbb{R}^+$. It is easy to see that $\nu_{\text{irrepresentable}}(S) = \rho s$ and $\lambda_{\min}(Q) \geq 1 - \rho$, such that for any $\rho > \frac{1}{s}$ and $\rho < 1$, the restricted eigenvalue holds trivially but the (S,s)-irrepresentability does not hold.

It is intuitive that the irrepresentability condition is stronger than the {\bf(RE)} assumption. In fact, a slightly modified result from \cite{vandegeer:2009} shows that a `strong' irrepresentability condition directly {\it implies} the {\bf(RE)} condition for $\ell2$-recovery:

\begin{proposition}
\label{prop:irrepresentability}
If the irrepresentability condition holds with $\epsilon > \frac{2}{3}$, then the restricted eigenvalue condition holds with constant  $\gamma_n \geq \frac{ (1 - 3(1 -\epsilon))^2 \lambda_{\min}^2}{4s}n$, where $\lambda_{\min} > 0$ is the smallest eigenvalue of $Q^*_{S,S}$, on which the results of \cite{Daneshmand:2014} also depend.
\end{proposition}

\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}

\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 $p_{i,j}$:

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