small changes

1 files changed, 6 insertions, 6 deletions

diff --git a/paper/sections/appendix.tex b/paper/sections/appendix.tex
index 80692d0..65576bb 100644
--- a/paper/sections/appendix.tex
+++ b/paper/sections/appendix.tex
@@ -1,9 +1,9 @@
-\subsection*{Proof of support recovery lemma}
+\subsection{Proof of support recovery lemma}
 
 
-\subsection*{Upper-bound for $\|\nabla f(\theta^*)\|_\infty$}
+\subsection{Upper-bound for $\|\nabla f(\theta^*)\|_\infty$}
 
-We show 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:
+We show 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 for a node $\alpha$:
 
 \begin{align}
 \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 \nonumber \\
@@ -19,7 +19,7 @@ $\nabla f(\theta^*)$ is a $N/p_{\min}$-subgaussian variable, where $p_{\min}$ is
 We show that its expectation is $0$ by conditioning on the seed set $S$:
 
 \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 \\
+\mathbb{E} \left( \nabla_k f(\theta) \right) & = \sum_{i=1}^N \mathbb{E} \left[ x^i_k \left( \frac{b^i}{\mathbb{P}(\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}(\alpha \text { infected})} - 1\right) \middle| S \right] \right] \nonumber \\
 & = \sum_{i=1}^N  \mathbb{E}\left[x^i_k \left( \frac{ \mathbb{E}_S \left[ b^i \middle| S \right]}{\mathbb{P}(\alpha \text { infected})} - 1\right)  \right] \nonumber \\
 & = 0 \nonumber
@@ -30,7 +30,7 @@ Therefore, $\nabla f(\theta^*)$ is the sum of zero-mean variables, upper-bounded
 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)
+\mathbb{P}(\| \nabla f(\theta) \|_{\infty} > \lambda) \leq 2 \exp \left( -\frac{\lambda^2 p_{\min}}{2n} + \log m \right)
 \end{equation}
 
-In conclusion, for $\delta>0$, $\lambda := 2 \sqrt{\frac{n^{\delta + 1} \log p}{p_{\min}}}$ is a valid regularizer with probability $1 - \exp(-n^\delta \log p )$
+In conclusion, for $\delta>0$, $\lambda := 2 \sqrt{\frac{n^{\delta + 1} \log m}{p_{\min}}}$ is a valid regularizer with probability $1 - \exp(-n^\delta \log m )$