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md/dev/-70L8lpp9DF/-70L8lpp9DF.md

@@ -110,7 +110,7 @@ $$
 
 and $\begin{array} { r } { \mathbb { E } \left[ K \right] = \frac { 1 / \gamma - 1 } { \log \left( 1 / \gamma \right) } } \end{array}$ .
 
-This is called the “negative binomial distribution,” since $\begin{array} { r } { \prod _ { \ell = 0 } ^ { k - 1 } \left( \frac { \ell + \eta } { \ell + 1 } \right) = { \binom { k + \eta - 1 } { k } } } \end{array}$ k+η−1k 
 if we extend the definition of binomial coefficients to non-integer $\eta$ . The distribution is called “truncated” because $\mathbb { P } \left[ K = 0 \right] = 0$ , whereas the standard negative binomial distribution includes 0 in its support. The $\eta = 0$ case $\mathcal { D } _ { 0 , \gamma }$ is known as the “logarithmic distribution”. The $\eta = 1$ case $\mathcal { D } _ { 1 , \gamma }$ is simply the geometric distribution. Next, we state our main privacy result for this distribution.
+This is called the “negative binomial distribution,” since $\begin{array} { r } { \prod _ { \ell = 0 } ^ { k - 1 } \left( \frac { \ell + \eta } { \ell + 1 } \right) = { \binom { k + \eta - 1 } { k } } } \end{array}$ k+η−1k 
 if we extend the definition of binomial coefficients to non-integer $\eta$ . The distribution is called “truncated” because $\mathbb { P } \left[ K = 0 \right] = 0$ , whereas the standard negative binomial distribution includes 0 in its support. The $\eta = 0$ case $\mathcal { D } _ { 0 , \gamma }$ is known as the “logarithmic distribution”. The $\eta = 1$ case $\mathcal { D } _ { 1 , \gamma }$ is simply the geometric distribution. Next, we state our main privacy result for this distribution.
 
 Theorem 2 (Main Privacy Result – Truncated Negative Binomial). Let $Q : { \mathcal { X } } ^ { n }  { \mathcal { Y } }$ be a randomized algorithm satisfying $( \lambda , \varepsilon )$ -RDP and $( \hat { \lambda } , \hat { \varepsilon } )$ -RDP for some $\varepsilon , \hat { \varepsilon } \geq 0$ , $\lambda \in ( 1 , \infty )$ , and $\hat { \lambda } \in [ 1 , \infty )$ . 4 Assume $\mathcal { V }$ is totally ordered.
 

md/dev/4p6_5HBWPCw/4p6_5HBWPCw.md

@@ -153,7 +153,7 @@ Figure 4: Loss curves on CPF datasets show GLNN distillation can help to regular
 
 Intuitively, the addition of neighbor information makes GNNs more powerful than MLPs when classifying nodes. Thus, a natural question regarding KD from GNNs to MLPs is whether MLPs are expressive enough to represent graph data as well as GNNs. Many recent works studied GNN model expressiveness (Xu et al., 2018; Chen et al., 2021). The latter analyzed GNNs and GA-MLPs for node classification and characterized expressiveness as the number of equivalence classes of rooted graphs induced by the model (formal definitions in Appendix D). The conclusion is that GNNs are more powerful than GA-MLPs, but in most real-world cases their expressiveness is indistinguishable.
 
-We adopt the analysis framework from Chen et al. (2021) and show in Appendix D that the number of equivalence classes induced by GNNs and MLPs are |X |+m−2m−1 
2L−1 a nd $| \mathcal { X } |$ respectively. Here $m$ denotes the max node degree, $L$ denotes the number of GNN layers, and $\mathcal { X }$ denotes the set of all possible node features. The former is apparently larger which concludes that GNNs are more expressive. Empirically, however, the gap makes little difference when $| \mathcal { X } |$ is large. In real applications, node features can be high dimensional like bag-of-words, or even word embeddings, thus making $| \mathcal { X } |$ enormous. Like for bag-of-words, $| \mathcal { X } |$ is in the order of $\mathcal { O } ( p ^ { D } )$ , where $D$ is the vocabulary size, and $p$ is the max word frequency. The expressiveness of a L-layer GNN is lower bounded by |X |+m−2
2L−1 $\left( \stackrel { | \mathcal { X } | + m - 2 } { m - 1 } \right) ^ { 2 ^ { L } - 1 } = \mathcal { O } ( p ^ { D ( m - 1 ) ( 2 ^ { L } - 1 ) } )$ , but empirically, both MLPs and GNNs should have enough expressiveness given $D$ is usually hundreds or bigger (see Table 5).
+We adopt the analysis framework from Chen et al. (2021) and show in Appendix D that the number of equivalence classes induced by GNNs and MLPs are |X |+m−2m−1 
2L−1 a nd $| \mathcal { X } |$ respectively. Here $m$ denotes the max node degree, $L$ denotes the number of GNN layers, and $\mathcal { X }$ denotes the set of all possible node features. The former is apparently larger which concludes that GNNs are more expressive. Empirically, however, the gap makes little difference when $| \mathcal { X } |$ is large. In real applications, node features can be high dimensional like bag-of-words, or even word embeddings, thus making $| \mathcal { X } |$ enormous. Like for bag-of-words, $| \mathcal { X } |$ is in the order of $\mathcal { O } ( p ^ { D } )$ , where $D$ is the vocabulary size, and $p$ is the max word frequency. The expressiveness of a L-layer GNN is lower bounded by |X |+m−2
2L−1 $\left( \stackrel { | \mathcal { X } | + m - 2 } { m - 1 } \right) ^ { 2 ^ { L } - 1 } = \mathcal { O } ( p ^ { D ( m - 1 ) ( 2 ^ { L } - 1 ) } )$ , but empirically, both MLPs and GNNs should have enough expressiveness given $D$ is usually hundreds or bigger (see Table 5).
 
 # 5.8 WHEN WILL GLNNS FAIL TO WORK?
 
@@ -389,7 +389,7 @@ Assuming the statement holds for $L$ , we show it holds for $L + 1$ by construct
 
 We first consider all paths from the root to leaves in $T$ . Each path consists of a sequence of nodes where the node features form a one-to-one mapping to an $\mathrm { L }$ -tuple $\tau \in \{ ( x _ { 1 } , \ldots , x _ { L } ) : x _ { i } \in \mathcal { X } \}$ . Leaf nodes are called node under $\tau$ if the path from the root to it corresponds to $\tau$ . The children of nodes under different $\tau \mathbf { S }$ are always distinguishable, and thus any assignments lead to distinct rooted aggregation trees of depth $L + 1$ . The assignment of children of nodes under the same $\tau$ , on the other hand, could be overcounted. Therefore, to lower bound $\tau _ { L + 1 , \mathcal { X } , m }$ , we only consider a special way of assignments to avoid over counting, which is that children of all nodes under the same $\tau$ are assigned the same set of features.
 
-Since we assumed that at least two nodes of $T$ have different features, there are at least $2 ^ { L }$ different $\tau \mathbf { S }$ corresponding to the path from the root to leaves. For a leaf node $j$ under a fixed $\tau$ , one of its children needs to have the same feature as $j$ ’s parent node. This restriction is due to the definition of rooted aggregation trees. Therefore, we only pick features for the other $m - 1$ nodes, which will be $\binom { | \mathcal { X } | + m - 2 } { m - 1 }$ cases for each $j$ . Then through this construction, the total number of depth- $\mathrm { . L } { + } 1$ trees from T can be lower bounded by |X |+m−2m−1 
2L. $T \in \mathcal { T } _ { L , \mathcal { X } , m }$ , so we derive $\begin{array} { r } { \mathcal { T } _ { L + 1 , \mathcal { X } , m } \geq \binom { | \mathcal { X } | + m - 2 } { m - 1 } ^ { 2 ^ { L } } \mathcal { T } _ { L , \mathcal { X } , m } } \end{array}$ hav, and $\begin{array} { r } { \mathcal { T } _ { L , \mathcal { X } , m } \geq \binom { | \mathcal { X } | + m - 2 } { m - 1 } ^ { \sum _ { l = 1 } ^ { L } 2 ^ { l } } = } \end{array}$ $\binom { | \mathcal { X } | + m - 2 } { m - 1 } ^ { 2 ^ { L } - 1 }$
+Since we assumed that at least two nodes of $T$ have different features, there are at least $2 ^ { L }$ different $\tau \mathbf { S }$ corresponding to the path from the root to leaves. For a leaf node $j$ under a fixed $\tau$ , one of its children needs to have the same feature as $j$ ’s parent node. This restriction is due to the definition of rooted aggregation trees. Therefore, we only pick features for the other $m - 1$ nodes, which will be $\binom { | \mathcal { X } | + m - 2 } { m - 1 }$ cases for each $j$ . Then through this construction, the total number of depth- $\mathrm { . L } { + } 1$ trees from T can be lower bounded by |X |+m−2m−1 
2L. $T \in \mathcal { T } _ { L , \mathcal { X } , m }$ , so we derive $\begin{array} { r } { \mathcal { T } _ { L + 1 , \mathcal { X } , m } \geq \binom { | \mathcal { X } | + m - 2 } { m - 1 } ^ { 2 ^ { L } } \mathcal { T } _ { L , \mathcal { X } , m } } \end{array}$ hav, and $\begin{array} { r } { \mathcal { T } _ { L , \mathcal { X } , m } \geq \binom { | \mathcal { X } | + m - 2 } { m - 1 } ^ { \sum _ { l = 1 } ^ { L } 2 ^ { l } } = } \end{array}$ $\binom { | \mathcal { X } | + m - 2 } { m - 1 } ^ { 2 ^ { L } - 1 }$
 
 # F ADVANCED GNN ARCHITECTURES AS THE TEACHER
 

md/dev/eYciPrLuUhG/eYciPrLuUhG.md

@@ -441,7 +441,7 @@ $$
 $$
 
 ![](images/2ab158e12eab41ad145dc2084bfe715fd90c381507af0f9f22745544b61e819a.jpg)  
-LX26!X3 (X3)LX1!X3(X3) ! (XX X (XFigure 5: Visualizing the gradient calculation for the incoming edges of $X _ { 2 }$ @ L˜ =  @ L in an example graph with (X )LX3!X2 2three variables. The intervention is being performed on $X _ { 1 }$ LX26!X3 (X3)LX26!X3 (X3)LX26!X3 326! 3 @✓13 @✓31, and the data is used to calculate the log16! 3 LX36!X2 (X2)likelihood estimates under the three randomly sampled graphs: $\mathcal { L } _ { C ^ { 1 } } ( X _ { 2 } ) , \mathcal { L } _ { C ^ { 2 } } ( X _ { 2 } )$ and $\mathcal { L } _ { C ^ { 3 } } ( X _ { 2 } )$ . LX2!X3 (X3)LX1!X3 (X3)Those terms are assigned to the Monte-Carlo estimators for $\mathscr { L } _ { X _ { i } \to X _ { 2 } } ( X _ { 2 } )$ and $\mathcal { L } _ { X _ { i } \not \to X _ { 2 } } ( X _ { 2 } )$ , and LX26!X3 (X3)(X )finally used to determine the gradients for $\gamma$ and $\pmb \theta$ . The same process is performed for $X _ { 3 }$ as well.
+LX26!X3 (X3)LX1!X3(X3) ! (XX X (XFigure 5: Visualizing the gradient calculation for the incoming edges of $X _ { 2 }$ @ L˜ =  @ L in an example graph with (X )LX3!X2 2three variables. The intervention is being performed on $X _ { 1 }$ LX26!X3 (X3)LX26!X3 (X3)LX26!X3 326! 3 @✓13 @✓31, and the data is used to calculate the log16! 3 LX36!X2 (X2)likelihood estimates under the three randomly sampled graphs: $\mathcal { L } _ { C ^ { 1 } } ( X _ { 2 } ) , \mathcal { L } _ { C ^ { 2 } } ( X _ { 2 } )$ and $\mathcal { L } _ { C ^ { 3 } } ( X _ { 2 } )$ . LX2!X3 (X3)LX1!X3 (X3)Those terms are assigned to the Monte-Carlo estimators for $\mathscr { L } _ { X _ { i } \to X _ { 2 } } ( X _ { 2 } )$ and $\mathcal { L } _ { X _ { i } \not \to X _ { 2 } } ( X _ { 2 } )$ , and LX26!X3 (X3)(X )finally used to determine the gradients for $\gamma$ and $\pmb \theta$ . The same process is performed for $X _ { 3 }$ as well.
 
 where $p _ { \gamma , \theta } ( C _ { k l } ) = \sigma ( \gamma _ { k l } ) \cdot \sigma ( \theta _ { k l } )$ . The first expectation over $p _ { \gamma , \theta } ( C _ { - k l } )$ is independent of $\gamma _ { k l }$ as we have defined the adjacency matrix distribution to be a product of independent edge probabilities.
 

md/dev/siCt4xZn5Ve/siCt4xZn5Ve.md

@@ -1169,7 +1169,7 @@ In the following subsections, we provide the proofs of all the components used i
 
 # D.2 PROOF OF LEMMA 6.2
 
-Recall that for each $i \in [ n ] f _ { i } ( x ) = f ( u , v ) = z _ { i } ^ { \top } ( u ^ { \odot 2 } - v ^ { \odot 2 } ) , \nabla f _ { i } ( x ) = 2 \binom { z _ { i } \odot u } { z _ { i } \odot v }$ 2ziuziv
, and K(x) = $( K _ { i j } ( x ) ) _ { i , j \in [ n ] }$ where each $K _ { i j } ( x ) = \langle \nabla f _ { i } ( x ) , \nabla f _ { j } ( x ) \rangle$ . Then
+Recall that for each $i \in [ n ] f _ { i } ( x ) = f ( u , v ) = z _ { i } ^ { \top } ( u ^ { \odot 2 } - v ^ { \odot 2 } ) , \nabla f _ { i } ( x ) = 2 \binom { z _ { i } \odot u } { z _ { i } \odot v }$ 2ziuziv
, and K(x) = $( K _ { i j } ( x ) ) _ { i , j \in [ n ] }$ where each $K _ { i j } ( x ) = \langle \nabla f _ { i } ( x ) , \nabla f _ { j } ( x ) \rangle$ . Then
 
 $$
 \begin{array} { r } { \nabla ^ { 2 } \ell _ { i } ( x ) = 2 \left( \begin{array} { l } { z _ { i } \odot u } \\ { - z _ { i } \odot v } \end{array} \right) \left( ( z _ { i } \odot u ) ^ { \top } \quad - ( z _ { i } \odot v ) ^ { \top } \right) + ( f _ { i } ( u , v ) - y _ { i } ) \cdot \mathrm { d i a g } ( z _ { i } , z _ { i } ) . } \end{array}
@@ -1798,7 +1798,7 @@ $$
 \nabla \varphi ( \boldsymbol { x } ) = \sum _ { j = q + 1 } ^ { d } 2 w _ { j } ^ { * } \left[ \frac { \mathbb { 1 } \{ w _ { j } ^ { * } > 0 \} } { u _ { j } } \cdot e _ { j } - \frac { \mathbb { 1 } \{ w _ { j } ^ { * } < 0 \} } { v _ { j } } \cdot e _ { D + j } \right]
 $$
 
-where $e _ { j }$ is the $j$ -th canonical base of $\mathbb { R } ^ { d }$ . Recall that $\nabla f _ { i } ( x ) = 2 { \left( \begin{array} { l l } { z _ { i } \odot u } \\ { - z _ { i } \odot v } \end{array} \right) }$  ziu−ziv
, and we further have
+where $e _ { j }$ is the $j$ -th canonical base of $\mathbb { R } ^ { d }$ . Recall that $\nabla f _ { i } ( x ) = 2 { \left( \begin{array} { l l } { z _ { i } \odot u } \\ { - z _ { i } \odot v } \end{array} \right) }$  ziu−ziv
, and we further have
 
 $$
 \begin{array} { l } { \displaystyle \mathcal { Z } _ { 2 } = \sum _ { i = q ^ { \prime } + 1 } ^ { n } \lambda _ { i } ( x ( t ) ) \sum _ { j = q + 1 } ^ { d } w _ { j } ^ { * } \left[ \frac { \mathbb { 1 } \{ w _ { j } ^ { * } > 0 \} } { u _ { j } } \langle e _ { j } , z _ { i } \odot u \rangle + \frac { \mathbb { 1 } \{ w _ { j } ^ { * } < 0 \} } { v _ { j } } \langle e _ { j } , z _ { i } \odot v \rangle \right] } \\ { \displaystyle = \sum _ { i = q ^ { \prime } + 1 } ^ { n } \lambda _ { i } ( x ( t ) ) \sum _ { j = q + 1 } ^ { d } w _ { j } ^ { * } \left[ \frac { \mathbb { 1 } \{ w _ { j } ^ { * } > 0 \} } { u _ { j } } z _ { i , j } u _ { j } + \frac { \mathbb { 1 } \{ w _ { j } ^ { * } < 0 \} } { v _ { j } } z _ { i , j } v _ { j } \right] } \\ { \displaystyle = \sum _ { i = q ^ { \prime } + 1 } ^ { n } \lambda _ { i } ( x ( t ) ) \sum _ { j = q + 1 } ^ { d } w _ { j } ^ { * } z _ { i , j } = \sum _ { i = q ^ { \prime } + 1 } ^ { n } \lambda _ { i } ( x ( t ) ) \langle z _ { i , ( q + 1 ) : d } , w _ { ( q + 1 ) : d } ^ { * } \rangle = 0 } \end{array}

md/dev/uCXNOeL0TG/uCXNOeL0TG.md

@@ -306,9 +306,9 @@ Output: balanced allocation $\{ D _ { j } \} _ { j \in [ M ] }$ where $D _ { j }
 6: for $j = 1$ to $M$ do   
 7: if $\tau _ { \sigma _ { j } } \cap L \neq \emptyset$ then   
 8: $x ^ { ' }  \mathrm { t o p } ( \tau _ { j } ) \cap L$   
-9: while cxj + P h2D chj > B do   
-10: ⌧j ⌧j \ {x}   
-11: if ⌧j \ L = ; then   
+9: while cxj + P h2D chj > B do   
+10: ⌧j ⌧j \ {x}   
+11: if ⌧j \ L = ; then   
 12: break   
 13: else   
 14 $\begin{array} { r l r } { } & { \colon   { \mathrm { t o p } } ( \tau _ { \sigma _ { j } } ) \cap L } & \\ { \vdots } & { \quad  { \mathrm { i f } } \tau _ { \sigma _ { j } } \cap L \neq \emptyset  { \mathrm { t h e n } } } \\ { } & { \quad \quad D _ { \sigma _ { j } }  D _ { \sigma _ { j } } \cup \{ x \} ; } & { L  L \setminus \{ x \} ; } & { \tau _ { \sigma _ { j } }  \tau _ { \sigma _ { j } } \setminus \{ x \} } \\ { } & { \colon  { \mathrm { r e t u r n } } \{ D _ { j } \} _ { j \in [ M ] } } & \end{array}$