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zai-org
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RPC-Bench

@@ -486,7 +486,7 @@ $$
 Thus the whole line segment from $( U , V )$ to $( U , V ^ { * } )$ is contained in $L _ { \alpha }$ . Since $C$ is a connected component of $L _ { \alpha }$ which contains $( U , V )$ , it follows that $( U , V ^ { * } ) \in C$ . Moreover, one has $\Phi ( U , \bar { V } ^ { * } ) = \varphi ( G ^ { * } ) \leq \epsilon .$ , which thus implies that the loss can always be made $\epsilon$ -small inside the set $C$ for every $\epsilon \in ( p ^ { * } , \alpha )$ .
-3. Let $( U _ { 0 } , V _ { 0 } )$ be a strict suboptimal local minimum, then there exists $r > 0$ such that $\Phi ( U , V ) > \Phi ( U _ { 0 } , V _ { 0 } ) > p ^ { * }$ for all $( U , V ) \in { \cal B } ( ( U _ { 0 } , V _ { 0 } ) , r ) \ : \backslash \ : \{ ( U _ { 0 } , V _ { 0 } ) \}$ where $B ( \cdot , r )$ denotes a closed ball of radius r. Let α = min(U,V )∈∂B(U0,V0),r which exists as $\Phi$ is continuous and the boundary $\partial B \big ( ( U _ { 0 } , V _ { 0 } ) , r \big )$ of $B \big ( ( U _ { 0 } , V _ { 0 } ) , r \big )$ is compact. Note that $\Phi ( U _ { 0 } , V _ { 0 } ) < \alpha$ as $( U _ { 0 } , V _ { 0 } )$ is a strict local minimum, and thus $( U _ { 0 } , \dot { V } _ { 0 } ) \in L _ { \alpha }$ . Let $E$ be the connected component of $L _ { \alpha }$ which contains $( U _ { 0 } , V _ { 0 } )$ , that is, $( U _ { 0 } , V _ { 0 } ) \in E \subseteq L _ { \alpha }$ . Since the loss of every point inside $E$ is strictly smaller than $\alpha$ , whereas the loss of every point on the boundary $\mathrm { \widehat { \partial } } D \big ( ( U _ { 0 } , V _ { 0 } ) , r \big )$ is greater than or equal to $\alpha$ , $E$ must be contained in the interior of the ball, that is $E \subset \dot { B } \big ( ( U _ { 0 } , V _ { 0 } ) , r \big )$ . Moreover, $\Phi ( U , V ) \ge \Phi ( U _ { 0 } , V _ { 0 } ) > p ^ { * }$ for every $( U , V ) \in E$ and thus the values of $\Phi$ restricted to $E$ can not be arbitrarily close to $p ^ { * }$ , which means that $E$ is a bad local valley, which contradicts G.3.2.
 4. The proof for cross-entropy loss is similar to Theorem 3.4. For square loss, one has

 Thus the whole line segment from $( U , V )$ to $( U , V ^ { * } )$ is contained in $L _ { \alpha }$ . Since $C$ is a connected component of $L _ { \alpha }$ which contains $( U , V )$ , it follows that $( U , V ^ { * } ) \in C$ . Moreover, one has $\Phi ( U , \bar { V } ^ { * } ) = \varphi ( G ^ { * } ) \leq \epsilon .$ , which thus implies that the loss can always be made $\epsilon$ -small inside the set $C$ for every $\epsilon \in ( p ^ { * } , \alpha )$ .
+3. Let $( U _ { 0 } , V _ { 0 } )$ be a strict suboptimal local minimum, then there exists $r > 0$ such that $\Phi ( U , V ) > \Phi ( U _ { 0 } , V _ { 0 } ) > p ^ { * }$ for all $( U , V ) \in { \cal B } ( ( U _ { 0 } , V _ { 0 } ) , r ) \ : \backslash \ : \{ ( U _ { 0 } , V _ { 0 } ) \}$ where $B ( \cdot , r )$ denotes a closed ball of radius r. Let α = min(U,V )∈∂B(U0,V0),r which exists as $\Phi$ is continuous and the boundary $\partial B \big ( ( U _ { 0 } , V _ { 0 } ) , r \big )$ of $B \big ( ( U _ { 0 } , V _ { 0 } ) , r \big )$ is compact. Note that $\Phi ( U _ { 0 } , V _ { 0 } ) < \alpha$ as $( U _ { 0 } , V _ { 0 } )$ is a strict local minimum, and thus $( U _ { 0 } , \dot { V } _ { 0 } ) \in L _ { \alpha }$ . Let $E$ be the connected component of $L _ { \alpha }$ which contains $( U _ { 0 } , V _ { 0 } )$ , that is, $( U _ { 0 } , V _ { 0 } ) \in E \subseteq L _ { \alpha }$ . Since the loss of every point inside $E$ is strictly smaller than $\alpha$ , whereas the loss of every point on the boundary $\mathrm { \widehat { \partial } } D \big ( ( U _ { 0 } , V _ { 0 } ) , r \big )$ is greater than or equal to $\alpha$ , $E$ must be contained in the interior of the ball, that is $E \subset \dot { B } \big ( ( U _ { 0 } , V _ { 0 } ) , r \big )$ . Moreover, $\Phi ( U , V ) \ge \Phi ( U _ { 0 } , V _ { 0 } ) > p ^ { * }$ for every $( U , V ) \in E$ and thus the values of $\Phi$ restricted to $E$ can not be arbitrarily close to $p ^ { * }$ , which means that $E$ is a bad local valley, which contradicts G.3.2.
 4. The proof for cross-entropy loss is similar to Theorem 3.4. For square loss, one has

parse/train/oh71uL93yay/oh71uL93yay.md CHANGED Viewed

@@ -104,7 +104,7 @@ It is also worth noticing how the optimal $A ^ { * }$ is configured. The princip
 # 2.4 ANALYSIS OF GCN-LPA MODEL BEHAVIOR
-In this subsection, we show benefits of our unified model compared with GCN by analyzing properties of embeddings produced by the two models. We first analyze the update rule of GCN for node $\begin{array} { r } { \mathbf { \Phi } _ { \upsilon i } \colon \mathbf { x } _ { i } ^ { ( k + 1 ) } = \sigma \left( \sum _ { v _ { j } \in \mathcal { N } ( v _ { i } ) } \tilde { a } _ { i j } \mathbf { x } _ { j } ^ { ( k ) } W ^ { ( k ) } \right) } \end{array}$ , where $\tilde { a } _ { i j } = a _ { i j } / d _ { i i }$ is the normalized weight of edge $( j , i )$ . This formula can be decomposed into the following two steps: (1) In aggregation step, we calculate the aggregated representation ${ \bf h } _ { i } ^ { ( k ) }$ of all neighborhoods $\begin{array} { r } { \mathcal { N } ( v _ { i } ) \colon { \mathbf { h } } _ { i } ^ { ( k ) } = \sum _ { v _ { j } \in \mathcal { N } ( v _ { i } ) } \tilde { a } _ { i j } { \mathbf { x } } _ { j } ^ { ( k ) } } \end{array}$ (2) In transformation step, the aggregated representation ${ \bf h } _ { i } ^ { ( k ) }$ is mapped to a new space by a transformation matrix and nonlinear function: x(k+1)i = σh(k)i W (k). We show by the following theorem that the aggregation step reduces the overall distance in the embedding space between the nodes that are connected in the graph:
 Theorem 3 (Shrinking property in GCN) Let $\begin{array} { r } { D ( \mathbf { x } ) = \frac { 1 } { 2 } \sum _ { v _ { i } , v _ { j } } \widetilde { a } _ { i j } \| \mathbf { x } _ { i } - \mathbf { x } _ { j } \| _ { 2 } ^ { 2 } } \end{array}$ be a distance metric over node embeddings x. Then we have $D ( \mathbf { h } ^ { ( k ) } ) \leq D ( \mathbf { x } ^ { ( k ) } )$ .

 # 2.4 ANALYSIS OF GCN-LPA MODEL BEHAVIOR
+In this subsection, we show benefits of our unified model compared with GCN by analyzing properties of embeddings produced by the two models. We first analyze the update rule of GCN for node $\begin{array} { r } { \mathbf { \Phi } _ { \upsilon i } \colon \mathbf { x } _ { i } ^ { ( k + 1 ) } = \sigma \left( \sum _ { v _ { j } \in \mathcal { N } ( v _ { i } ) } \tilde { a } _ { i j } \mathbf { x } _ { j } ^ { ( k ) } W ^ { ( k ) } \right) } \end{array}$ , where $\tilde { a } _ { i j } = a _ { i j } / d _ { i i }$ is the normalized weight of edge $( j , i )$ . This formula can be decomposed into the following two steps: (1) In aggregation step, we calculate the aggregated representation ${ \bf h } _ { i } ^ { ( k ) }$ of all neighborhoods $\begin{array} { r } { \mathcal { N } ( v _ { i } ) \colon { \mathbf { h } } _ { i } ^ { ( k ) } = \sum _ { v _ { j } \in \mathcal { N } ( v _ { i } ) } \tilde { a } _ { i j } { \mathbf { x } } _ { j } ^ { ( k ) } } \end{array}$ (2) In transformation step, the aggregated representation ${ \bf h } _ { i } ^ { ( k ) }$ is mapped to a new space by a transformation matrix and nonlinear function: x(k+1)i = σh(k)i W (k). We show by the following theorem that the aggregation step reduces the overall distance in the embedding space between the nodes that are connected in the graph:
 Theorem 3 (Shrinking property in GCN) Let $\begin{array} { r } { D ( \mathbf { x } ) = \frac { 1 } { 2 } \sum _ { v _ { i } , v _ { j } } \widetilde { a } _ { i j } \| \mathbf { x } _ { i } - \mathbf { x } _ { j } \| _ { 2 } ^ { 2 } } \end{array}$ be a distance metric over node embeddings x. Then we have $D ( \mathbf { h } ^ { ( k ) } ) \leq D ( \mathbf { x } ^ { ( k ) } )$ .