\documentclass[11pt]{article}
\usepackage{fullpage, amsmath, amssymb, amsthm}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{remark}{Remark}
\newtheorem{proposition}{Proposition}

\title{Maximum Likelihood Approach}
\author{Jean Pouget-Abadie}

\begin{document}

\maketitle

We consider the node $\alpha$. We index the measurements by $i \in [1, n]$. Let $b^i$ be the indicator variable for node $\alpha$ active at the round following measurememt $i$ and let $x^i$ be the vector of active nodes for measurement $i$. Recall that:

\begin{equation}
\label{eq:probability_of_infection}
1 - \exp(\langle x^i, \theta \rangle) = \mathbb{P}(\text{node } \alpha \text{ is active at the following round})
\end{equation}

The likelihood problem can be formulated as such:

\begin{equation}
\label{eq:main_formulation}
\min_{\theta \in \mathbb{R}^p} \quad \lambda_n \| \theta \|_1 + \sum^n_{i=1} - b^i \log \left(e^{-\langle x^i, \theta \rangle} - 1 \right) - \langle x^i, \theta \rangle
\end{equation}

We define $f(\theta):= \sum^n_{i=1} - b^i \log \left(\exp(-\langle x^i, \theta \rangle) \right) - \langle x^i, \theta \rangle$ such that Eq.~\ref{eq:main_formulation} can be rewritten as:

\begin{equation}
\label{eq:small_formulation}
\min_{\theta \in \mathbb{R}^p} \quad f(\theta) + \lambda_n \| \theta \|_1
\end{equation}

We cite the following theorem from \cite{Negahban:2009} (roughly, because the statements of the theorem are either slightly wrong or unclear):

\begin{proposition}
\label{thm:cited_theorem}
Let ${\cal C}:=\{\Delta \in \mathbb{R}^p : \exists S \subset [1, n] \ s.t. \ \|\Delta_{S^c}\|_1 \leq 3 \| \Delta_S \|_1 \}$. Suppose that $\theta^*$ is s-sparse, and the following two conditions are met: 
\begin{equation}
\lambda_n \geq 2 \|\nabla f(\theta^*) \|_\infty
\label{eq:lambda_condition}
\end{equation}
\begin{equation}
\forall \Delta \in {\cal C}, \ \Delta^T \cdot \nabla^2 f(\theta^*) \cdot \Delta \geq \gamma_n \| \Delta \|_2^2
\label{eq:RSC_condition}
\end{equation}
then:
\begin{equation}
\| \theta - \theta^* \|_2 \leq \frac{\sqrt{s} \lambda_n}{\gamma_n}
\end{equation}
\end{proposition}

It remains to show the two conditions for Proposition~\ref{thm:cited_theorem} are met.

\section*{Condition~\ref{eq:lambda_condition}}
Condition~\ref{eq:lambda_condition} requires us to find an upper-bound for $ 2 \|\nabla f(\theta^*) \|_\infty$. If we only consider the first measurement of every cascade, this can be done easily. Let $N$ be the number of cascades (to distinguish from $n$ number of total measurements). Begin by noting that:

\begin{equation}
\nabla_k f(\theta) = \sum^n_{i=1} \frac{b^i x^i_k}{1 - e^{\langle x^i, \theta \rangle}} - \sum^n_{i=1} x^i_k = \sum_{i=1}^n x^k_i \left( \frac{b^i}{\mathbb{P}(\text{node } \alpha \text { infected})} - 1\right)
\end{equation}

\begin{lemma}
\label{lem:subgaussian_variable}
$\nabla f(\theta^*)$ is a $N/p_{\min}$-subgaussian variable, where $p_{\min}$ is the smallest non-zero link to node $\alpha$.
\end{lemma}

\begin{proof}
\begin{align}
\mathbb{E} \left( \nabla_k f(\theta) \right) & = \sum_{i=1}^N \mathbb{E} \left[ x^i_k \left( \frac{b^i}{\mathbb{P}(\text{node } \alpha \text { infected})} - 1\right) \right] \nonumber \\
& = \sum_{i=1}^N \mathbb{E}_S \left[ \mathbb{E}\left[x^i_k \left( \frac{b^i}{\mathbb{P}(\text{node } \alpha \text { infected})} - 1\right) \middle| S \right] \right] \quad \text{where S is the seed set} \nonumber \\
& = \sum_{i=1}^N  \mathbb{E}\left[x^i_k \left( \frac{ \mathbb{E}_S \left[ b^i \middle| S \right]}{\mathbb{P}(\text{node } \alpha \text { infected})} - 1\right)  \right] \nonumber \\
& = 0
\end{align}
Therefore, $\nabla f(\theta^*)$ is the sum of zero-mean variables, upper-bounded by $1/p_{\min}$. It follows that $\nabla f(\theta^*)$ is $N/p_{\min}$-subgaussian.
\end{proof}

By union bound and characterization of sub-gaussian variables:

\begin{equation}
\mathbb{P}(\| \nabla f(\theta) \|_{\infty} > \lambda) \leq 2 \exp \left( -\frac{\lambda^2 p_{\min}}{2n} + \log p \right)
\end{equation}

In conclusion, for $\delta>0$, $\lambda := 2 \sqrt{\frac{n^{\delta + 1} \log p}{p_{\min}}}$ meets Condition~\ref{eq:lambda_condition} with probability $1 - \exp(-n^\delta \log p )$


\section*{Condition~\ref{eq:RSC_condition}}

Note that:
\begin{equation}
\nabla_{kj} f(\theta) = \sum_{i=1}^n \frac{b^i x_k^i x_j^i e^{\langle x^i, \theta \rangle}}{\left(1 - e^{\langle x^i, \theta \rangle} \right)^2} = \sum_{i=1}^n b^i x_k^i x_j^i \frac{\mathbb{P}(\text{node } \alpha \text { not infected})}{\mathbb{P}(\text{node } \alpha \text { infected})^2}
\end{equation}


We are going to explicitate a constant $\gamma$ such that: $\forall \Delta \in {\cal C}, \Delta^T \cdot \nabla^2 f(\theta^*) \cdot \Delta \geq \gamma n \|\Delta\|_2^2$. 

\paragraph{Notation} Let $p_i := \mathbb{P}(\text{node } \alpha \text { infected})$. Let $Z^i_k := b^i x^i_k \frac{1-p_i}{p_i^2}$and let $Z^i_{k,j} := b^i x^i_k x^i_j \frac{1-p_i}{p_i^2}$. We also define $Z_k := \sum_i Z^i_k$ and $Z_{k,j} := \sum_i Z^i_{k,j}$.

\begin{align}
\Delta^T \cdot \nabla^2 f(\theta^*) \cdot \Delta & = \sum_k \Delta_k^2 \left[ \sum_i b^i x_k^i \frac{1 - p_i}{p_i^2} \right] + 2 \sum_{k< j} \Delta_k \Delta_j \left[ \sum_i b^i x^i_k x^i_j \frac{1 - p_i}{p_i^2}\right] \nonumber \\ 
& = \sum_k \Delta_k^2 Z_k + 2 \sum_{k< j} \Delta_k \Delta_j Z_{k,j} \nonumber
\end{align}



\begin{lemma}
\label{lem:first_term}
Suppose that $\forall k$, $\mathbb{E}_{S(k)} \left[ \frac{1}{p_i} \right] \geq 1 + \alpha$, then with probability greater than $1 - 2 p e^{-2 \alpha^2 (1-c)^2p_{\min}^2 p_{\text{init}}^2 N}$, $$\forall k, \ Z_k > c \alpha p_{\text{init}} N$$
\end{lemma}

\begin{proof}
Let $S(k)$ denote the active set conditioned on the fact that node $k$ is active.
\begin{align}
\nonumber
\mathbb{E}[Z^i_k] & = \mathbb{P}(x^i_k = 1) \mathbb{E}_{S(k)} \left[ \mathbb{E}[b^i | S(k)] \frac{1 - p_i}{p_i^2} \right] \\ \nonumber
& = p_{\text{init}} \left( \mathbb{E}_{S(k)} \left[ \frac{1}{p_i} \right] - 1 \right) \\ \nonumber
& \geq \alpha p_{\text{init}} \quad \text{by assumption}
\end{align}

Note that $|Z^i_k| < \frac{1}{p_{\text{min}}^2}$ {\it a.s.}. By Hoeffding's first inequality, for $0<c <1$,

\begin{align}
\nonumber
\mathbb{P}\left(Z_k < c \alpha p_{\text{init}} N \right) & < 2 e^{- \frac{2(1-c)^2}{Nb^2} \left( \mathbb{E}_{S(k)} \left[ \frac{1}{p_i} \right] - 1 \right)^2} \\ \nonumber
& < 2 e^{-2 \alpha^2 (1-c)^2p_{\min}^2 p_{\text{init}}^2 N}
\end{align}

We conclude by union bound.
\end{proof}

\begin{lemma}
Suppose that $\forall k,j$, $1 + \alpha \leq \mathbb{E}_{S(k, j)} \left[ \frac{1}{p_i} \right] \leq 1 + \beta$, then with probability greater than $1 - pe^{-2 \alpha^2 (1-c)^2p_{\min}^2 p_{\text{init}}^4 N}$, $$\forall k,j, \ Z_{k,j} < c \beta p_{\text{init}}^2 N$$
\end{lemma}

\begin{proof}
We follow the same reasoning as Lemma~\ref{lem:first_term}:
\begin{align}
\nonumber
\mathbb{E}[Z^i_{k,j}] & = p_{\text{init}}^2 \left( \mathbb{E}_{S(k,j)} \left[ \frac{1}{p_i} \right] - 1 \right) \\ \nonumber
& \leq \beta p_{\text{init}}^2 \quad \text{by assumption}
\end{align}

By Hoeffding's second inequality, for $0 < c < 1$,
\begin{align}
\nonumber
\mathbb{P}\left(Z_{k,j} > c \beta p_{\text{init}}^2 N \right) & \leq e^{- \frac{2(1-c)^2}{Nb^2} \left( \mathbb{E}_{S(k,j)} \left[ \frac{1}{p_i} \right] - 1 \right)^2} \\ \nonumber
& \leq e^{-2 \alpha^2 (1-c)^2p_{\min}^2 p_{\text{init}}^4 N} 
\end{align}
We conclude by union bound.
\end{proof}

\begin{proposition}
Suppose that $\forall k,j$, $1 + \alpha \leq \mathbb{E}_{S(k, j)} \left[ \frac{1}{p_i} \right] \leq 1 + \beta$, then with probability greater than $1 - XXX$, condition~\ref{eq:RSC_condition} is met with $\gamma_n = \gamma n$ where $\gamma := p_{\text{init}}(\alpha - \beta p_{\text{init}})$
\end{proposition}


\paragraph{Hoeffding's inequality}

For $t \in \mathbb{R}$ and independent variables $Z_i$ such that $|Z_i|<b$ {\it a.s.}, we have:
\begin{align}
\label{eq:hoeffding_inequality}
\mathbb{P} \left( \middle| \sum_{i=1}^N Z_i - \mathbb{E}\left[\sum_{i=1}^N Z_i \right] \middle| \geq t \right) \leq 2 \exp\left(- \frac{2 t^2}{N b^2} \right) \nonumber \\
\mathbb{P} \left(\sum_{i=1}^N Z_i \geq \mathbb{E}\left[\sum_{i=1}^N Z_i \right] + t \right) \leq \exp\left(- \frac{2 t^2}{N b^2} \right) \nonumber
\end{align}


% \subsection*{First term}

% We are first going to find a lower-bound for $\sum_i b^i x_k^i \frac{1 - p_i}{p_i^2}$ by computing a lower-bound on its expectation and concluding with Hoeffding's inequality. If we only consider the first measurement of every cascade and we further suppose that $p_i < 1 - \eta$ no matter the configuration of active nodes (slightly less strong than correlation decay).

% \begin{align}
% \mathbb{E}\left(\sum_i b^i x_k^i \frac{1 - p_i} {p_i}^2 \right) & = \sum_i \mathbb{E} \left(x_k^i \frac{1 - p_i} {p_i}^2  \right) \nonumber \\
% & = \sum_i \mathbb{P}(b^i =1 | x_k^i =1) \mathbb{P}(x^i_k =1) \mathbb{E}\left(\frac{1 - p_i}{p_i^2} \middle| b^i =1 = x_k^i \right) \nonumber \\
% & \geq \sum_{i=1}^n p_{\min} \cdot \min \left(1 , 1 - (1 - p_{init})^s \right)\cdot p_{init} \cdot \frac{\eta}{(1 - \eta)^2} \nonumber \\
% & \geq s p_{init}^2 p_{\min} \frac{\eta}{(1 - \eta)^2} \quad \text{si }s < \frac{1}{p_{init}} \nonumber \\
% & \geq p_{init} p_{\min} \frac{\eta}{(1 - \eta)^2} \quad \text{si }s > \frac{1}{p_{init}} \nonumber
% \end{align}

% We can conclude using the following Hoeffding inequality for independent random variables bounded by $[0, b_i]$ by noticing that our variables are bounded by above by $\frac{1 - p_{\min}}{p_{\min}^2}$

% \paragraph{Hoeffding's inequality}
% \begin{equation}
% \label{eq:hoeffding_inequality}
% \mathbb{P} \left(\sum Z_i \geq \mathbb{E}[\sum Z_i] - t \right) \leq \exp\left(- \frac{2 N t^2}{b^2} \right)
% \end{equation}

% It follows that for $c<1$ with probability $1 - \exp \left( - n^3 c^2 s^2 p_{init}^4 p_{\min}^6 \frac{\eta^2}{(1 - \eta)^4} \right)$, we have that $$\sum_k \Delta_k^2 \left[ \sum_i b^i x_k^i \frac{1 - p_i}{p_i^2} \right] \geq  \gamma N =: (1 -c) s p_{init}^2 p_{\min} \frac{\eta}{(1 - \eta)^2} N$$

% \begin{remark}
% Would it be possible to extend this result using Azuma's inequality on Martingales to not just the first measurement of every cascade?
% \end{remark}

% \subsection*{Second term}
% We are now going to find an upper-bound on the term $\sum_i b^i x^i_k x^i_j \frac{1 - p_i}{p_i^2}$.

\section*{Conclusion}

Suppose we show that Condition~\ref{eq:RSC_condition} is met for $\gamma_n = \gamma N$, then we have the following theorems:

\begin{theorem}
\label{thm:l2_bound}
Suppose that $\theta^* \in \mathbb{R}^p$ is s-sparse and that we choose $\lambda_n = 2 \sqrt{\frac{n^{\delta + 1} \log p}{p_{\min}}}$ for $\delta >0$, then with probability $1 - \exp(-n^\delta \log p )$, we have
\begin{equation}
\|\hat \theta - \theta^* \|_2 \leq \frac{2}{\gamma} \sqrt{\frac{s \log p}{p_{\min} N^{1 - \delta}}}
\end{equation}
\end{theorem}

Note that we can choose $\delta = 0$ in high-dimensions since the probability of success will then be $1 - \frac{1}{p} \approx 1$. We can also conclude on support recovery with the following reasoning. 

\begin{theorem}
\label{thm:support_recovery}
Suppose that $N$ is chosen such that $\frac{2}{\gamma}\sqrt{\frac{s \log p}{p_{\min} N^{1 -\delta}}} < \eta$ and suppose we only keep as elements of the support of $\theta^*$ the coordinates $\hat \theta_i > \eta$. Then no wrong parent will be included, and all `strong' parents will be included, where `strong' signifies: $\theta^*_i > 2 \eta$.
\end{theorem}

It follows that we have found an ${\cal O}(s \log p)$ algorithm for recovering the graph, with better constants and fewer assumptions than any previous work.

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\end{document}