move metropolis to Sec 2

mlsys2022style/d-cliques.tex

@@ -66,19 +66,6 @@ edge with all other cliques, see Figure~\ref{fig:d-cliques-figure} for the
 corresponding D-Cliques network in the case of $n=100$ nodes and $c=10$
 classes. We will explore sparser inter-clique topologies in Section~\ref{section:interclique-topologies}.
 
-The mixing matrix $W$ required by D-SGD is obtained from standard
-Metropolis-Hasting weights~\cite{xiao2004fast} computed from the above
-topology, namely:
-\begin{equation}
-  W_{ij} = \begin{cases}
-    \frac{1}{\max(\text{degree}(i), \text{degree}(j)) + 1} & \text{if}~i \neq
-    j \text{ and } \{i,j\}\in E,\\
-   1 - \sum_{j \neq i} W_{ij} & \text{if}~$i = j$, \\
-   0 & \text{otherwise}.
-  \end{cases}
-  \label{eq:metro}
-\end{equation}
-
 We construct D-Cliques by initializing cliques at random, using at most $M$
 nodes to limit the intra-clique communication costs, then we 
 swap nodes between pairs of cliques chosen at random such that the swap

mlsys2022style/setting.tex

 % !TEX root = main.tex
 
-\section{Problem Statement}
+\section{Problem Setting}
 
 \label{section:problem}
 
@@ -45,6 +45,19 @@ of Problem
 must be doubly
 stochastic ($\sum_{j \in N} W_{ij} = 1$ and $\sum_{j \in N} W_{ji} = 1$) and
 symmetric, i.e. $W_{ij} = W_{ji}$~\cite{lian2017d-psgd}.
+In our experiments, $W$ is given by
+standard
+Metropolis-Hasting weights~\cite{xiao2004fast} computed from the network
+topology $G$, namely:\todo{AB: if we need space we can remove this equation}
+\begin{equation}
+  W_{ij} = \begin{cases}
+    \frac{1}{\max(\text{degree}(i), \text{degree}(j)) + 1} & \text{if}~i \neq
+    j \text{ and } \{i,j\}\in E,\\
+   1 - \sum_{j \neq i} W_{ij} & \text{if } i = j, \\
+   0 & \text{otherwise}.
+  \end{cases}
+  \label{eq:metro}
+\end{equation}
 
 \begin{algorithm}[t]
    \caption{D-SGD, Node $i$}