<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">% !TEX root = main.tex

\appendix
 


\section{Small-world Inter-clique Topology}
 We present a more detailed and precise explanation the algorithm to establish a small-world
 inter-clique topology (Algorithm~\ref{Algorithm:Smallworld}). Algorithm~\ref{Algorithm:Smallworld} instantiates the function 
\textit{interconnect} with a
small-world inter-clique topology as described in Section~\ref{section:interclique-topologies}. It adds a
linear number of inter-clique edges by first arranging cliques on a ring. It then adds a logarithmic number of ``finger'' edges to other cliques on the ring chosen such that there is a constant number of edges added per set, on sets that are exponentially bigger the further away on the ring. ``Finger'' edges are added symmetrically on both sides of the ring to the cliques in each set that are closest to a given set.

\begin{algorithm}[h]
   \caption{$\textit{smallworld}(DC)$:  adds $O(\# N \log(\# N))$ edges}
   \label{Algorithm:Smallworld}
   \begin{algorithmic}[1]
        \STATE \textbf{Require:} set of cliques $DC$ (set of set of nodes)
        \STATE ~~size of neighborhood $ns$ (default 2)
        \STATE ~~function $\textit{least\_edges}(S, E)$ that returns one of the nodes in $S$ with the least number of edges in $E$
        \STATE $E \leftarrow \emptyset$ \COMMENT{Set of Edges}
        \STATE $L \leftarrow [ C~\text{for}~C \in DC ]$ \COMMENT{Arrange cliques in a list}
        \FOR{$i \in \{1,\dots,\#DC\}$} % \COMMENT{For every clique}
%          %\STATE ~\COMMENT{For sets of cliques exponentially further away from $i$}
          \FOR{$\textit{offset} \in \{ 2^x~\text{for}~x~\in \{ 0, \dots, \lceil \log_2(\#DC) \rceil \} \}$} 
%             \STATE \COMMENT{Pick the $ns$ closest}
             \FOR{$k \in \{0,\dots,ns-1\}$}
%                 \STATE \COMMENT{Add inter-clique connections in both directions}
                 \STATE $n \leftarrow \textit{least\_edges}(L_i, E)$
                 \STATE $m \leftarrow \textit{least\_edges}(L_{(i+\textit{offset}+k) \% \#DC}, E)$ %\COMMENT{clockwise in ring}
                 \STATE $E \leftarrow E \cup \{ \{n,m\} \}$
                 \STATE $n \leftarrow \textit{least\_edges}(L_i, E)$
                 \STATE $m \leftarrow \textit{least\_edges}(L_{(i-\textit{offset}-k)\% \#DC} , E)$ %\COMMENT{counter-clockwise in ring}
                 \STATE $E \leftarrow E \cup \{ \{n,m\} \}$
             \ENDFOR
           \ENDFOR
        \ENDFOR
         \RETURN E
   \end{algorithmic}
\end{algorithm}

Algorithm~\ref{Algorithm:Smallworld}  expects a set of cliques $DC$, previously computed by 
Algorithm~\ref{Algorithm:D-Clique-Construction}; a size of neighborhood $ns$,
which is the number of finger edges to add per set of cliques, and a function 
\textit{least\_edges}, which given a set of nodes $S$ and an existing set of
edges $E =  \{\{i,j\}, \dots \}$, returns one of the nodes in $E$ with the least number of edges. It returns a new set of edges $\{\{i,j\}, \dots \}$ with all edges added by the small-world topology.

The implementation first arranges the cliques of $DC$ in a list, which
represents the ring. Traversing the list with increasing indices is equivalent
to traversing the ring in the clockwise direction, and inversely. Then, for every clique $i$ on the ring from which we are computing the distance to others, a number of edges are added. All other cliques are implicitly arranged in mutually exclusive sets, with size and at offset exponentially bigger (doubling at every step). Then for every of these sets, $ns$ edges are added, both in the clockwise and counter-clockwise directions, always on the nodes with the least number of edges in each clique. The ring edges are implicitly added to the cliques at offset $1$ in both directions.
 
 % \section{Additional Experiments}

% \subsection{Effect of Clique Averaging and Uniform Initialization}

% Section~\ref{section:clique-averaging} explained how Clique Averaging reduces bias and showed that Clique Averaging was significantly beneficial on MNIST with fully-connected D-Cliques. In this section, we provide additional results for the ring topology, as well as for CIFAR10. In addition, during our early exploration, we noticed that ensuring \textit{uniform initialization}, i.e. ensuring that all nodes start with the same model, increased convergence speed when connecting two cliques with 1-2 interclique edges. We therefore also verify whether this effect is still significant with 10 cliques (100 nodes), on a ring and with full connections between cliques, as well as on MNIST and CIFAR10. We also verified what interaction this had with Clique Averaging.

% Figure~\ref{fig:d-cliques-mnist-init-clique-avg-effect} shows all the results for MNIST. Comparing Figure~\ref{fig:d-cliques-mnist-init-clique-avg-effect-ring-test-accuracy} to~\ref{fig:d-cliques-mnist-no-init-clique-avg-effect-ring-test-accuracy}, and Figure~\ref{fig:d-cliques-mnist-init-clique-avg-effect-fcc-test-accuracy} to~\ref{fig:d-cliques-mnist-no-init-clique-avg-effect-fcc-test-accuracy} together, we see that Uniform Initialization has imperceptible effects. However, for all four sub-figures, using Clique Averaging has a slightly better average convergence speed, and significantly lower variance between nodes, than not using it. Moreover, the improvement is larger with Fully-Connected D-Cliques.

% \begin{figure}[htbp]
%      \centering
%       % To regenerate the figure, from directory results/mnist   
%       % python ../../../learn-topology/tools/plot_convergence.py clique-ring/all/2021-03-10-18:14:35-CET no-clique-avg/clique-ring/all/2021-03-12-10:40:37-CET --add-min-max --yaxis test-accuracy --labels 'with clique avg.' 'without clique avg.' --legend 'lower right' --ymin 85 --ymax 92.5 --save-figure ../../figures/d-cliques-mnist-init-clique-avg-effect-ring-test-accuracy.png
%       \begin{subfigure}[b]{0.48\textwidth}
%          \centering
%          \includegraphics[width=\textwidth]{figures/d-cliques-mnist-init-clique-avg-effect-ring-test-accuracy}
%          \caption{\label{fig:d-cliques-mnist-init-clique-avg-effect-ring-test-accuracy} D-Cliques (Ring), with Uniform Initialization}
%      \end{subfigure}
%      \quad 
%      % To regenerate the figure, from directory results/mnist
%      % python ../../../learn-topology/tools/plot_convergence.py no-init/clique-ring/all/2021-03-12-10:40:11-CET no-init-no-clique-avg/clique-ring/all/2021-03-12-10:41:03-CET --add-min-max --yaxis test-accuracy --labels 'with clique avg.' 'without clique avg.' --legend 'lower right' --ymin 85 --ymax 92.5 --save-figure ../../figures/d-cliques-mnist-no-init-clique-avg-effect-ring-test-accuracy.png   
%       \begin{subfigure}[b]{0.48\textwidth}
%          \centering
%          \includegraphics[width=\textwidth]{figures/d-cliques-mnist-no-init-clique-avg-effect-ring-test-accuracy}
%          \caption{\label{fig:d-cliques-mnist-no-init-clique-avg-effect-ring-test-accuracy} D-Cliques (Ring), without Uniform Initialization}
%      \end{subfigure}
     
%      % To regenerate the figure, from directory results/mnist
%      %python ../../../learn-topology/tools/plot_convergence.py fully-connected-cliques/all/2021-03-10-10:19:44-CET no-clique-avg/fully-connected-cliques/all/2021-03-12-11:12:26-CET --add-min-max --yaxis test-accuracy --labels 'with clique avg.'    'without clique avg.'  --legend 'lower right' --ymin 85 --ymax 92.5 --save-figure ../../figures/d-cliques-mnist-init-clique-avg-effect-fcc-test-accuracy.png
%        \begin{subfigure}[b]{0.48\textwidth}
%          \centering
%          \includegraphics[width=\textwidth]{figures/d-cliques-mnist-init-clique-avg-effect-fcc-test-accuracy}
%          \caption{\label{fig:d-cliques-mnist-init-clique-avg-effect-fcc-test-accuracy} D-Cliques (Fully-Connected), with Uniform Initialization}
%      \end{subfigure}
%      \quad
%       % To regenerate the figure, from directory results/mnist
%      %python ../../../learn-topology/tools/plot_convergence.py no-init/fully-connected-cliques/all/2021-03-12-11:12:01-CET no-init-no-clique-avg/fully-connected-cliques/all/2021-03-12-11:12:49-CET --add-min-max --yaxis test-accuracy --labels 'with clique avg.' 'without clique avg.' --legend 'lower right' --ymin 85 --ymax 92.5 --save-figure ../../figures/d-cliques-mnist-no-init-clique-avg-effect-fcc-test-accuracy.png
%        \begin{subfigure}[b]{0.48\textwidth}
%          \centering
%          \includegraphics[width=\textwidth]{figures/d-cliques-mnist-no-init-clique-avg-effect-fcc-test-accuracy}
%          \caption{\label{fig:d-cliques-mnist-no-init-clique-avg-effect-fcc-test-accuracy} D-Cliques (Fully-Connected), without Uniform Initialization}
%      \end{subfigure}
     
     
% \caption{\label{fig:d-cliques-mnist-init-clique-avg-effect} MNIST: Effects of Clique Averaging and Uniform Initialization on Convergence Speed. (100 nodes, non-IID, D-Cliques, bsz=128)}
% \end{figure}

% Figure~\ref{fig:d-cliques-cifar10-init-clique-avg-effect} shows all the results for CIFAR10. One the one hand, with D-Cliques arranged in a ring, uniform initialization has a small but positive effect on convergence speed, whether Clique Averaging is used or not. With fully-connected D-Cliques,  the effect is significantly smaller and almost negligible, both with and without Clique Averaging. On the other hand, Clique Averaging is always beneficial, by a significantly larger margin for both interclique topologies and with and without uniform initialization. Moreover, the effect is stronger than for MNIST.

%     \begin{figure}[htbp]
%      \centering
%      % To regenerate the figure, from directory results/cifar10
%      % python ../../../learn-topology/tools/plot_convergence.py clique-ring/all/2021-03-10-11:58:43-CET no-init/clique-ring/all/2021-03-13-18:28:30-CET no-clique-avg/clique-ring/all/2021-03-13-18:27:09-CET  no-init-no-clique-avg/clique-ring/all/2021-03-13-18:29:58-CET --add-min-max --yaxis test-accuracy --labels 'with clique avg., with uniform init.' 'with clique avg., without uniform init.'  'without clique avg., with uniform init.'   'without clique avg., without uniform init.' --legend 'upper left' --ymax 115  --save-figure ../../figures/d-cliques-cifar10-init-clique-avg-effect-ring-test-accuracy.png --font-size 15
%       \begin{subfigure}[b]{0.48\textwidth}
%          \centering
%          \includegraphics[width=\textwidth]{figures/d-cliques-cifar10-init-clique-avg-effect-ring-test-accuracy}
%          \caption{\label{fig:d-cliques-cifar10-init-clique-avg-effect-ring-test-accuracy} D-Cliques (Ring)}
%      \end{subfigure}
%      % To regenerate the figure, from directory results/cifar10
%      %python ../../../learn-topology/tools/plot_convergence.py fully-connected-cliques/all/2021-03-10-13:58:57-CET no-init/fully-connected-cliques/all/2021-03-13-18:32:55-CET no-clique-avg/fully-connected-cliques/all/2021-03-13-18:31:36-CET  no-init-no-clique-avg/fully-connected-cliques/all/2021-03-13-18:34:35-CET --add-min-max --yaxis test-accuracy --labels 'with clique avg., with uniform init.' 'with clique avg., without uniform init.'  'without clique avg., with uniform init.'   'without clique avg., without uniform init.' --legend 'upper left'  --ymax 115 --save-figure ../../figures/d-cliques-cifar10-init-clique-avg-effect-fcc-test-accuracy.png --font-size 15
%        \begin{subfigure}[b]{0.48\textwidth}
%          \centering
%          \includegraphics[width=\textwidth]{figures/d-cliques-cifar10-init-clique-avg-effect-fcc-test-accuracy}
%          \caption{\label{fig:d-cliques-cifar10-init-clique-avg-effect-fcc-test-accuracy} D-Cliques (Fully-Connected)}
%      \end{subfigure}
% \caption{\label{fig:d-cliques-cifar10-init-clique-avg-effect} CIFAR10: Effects of Clique Averaging and Uniform Initialization on Convergence Speed. (100 nodes, non-IID, D-Cliques, bsz=20, momentum=0.9)}
% \end{figure}

% We conclude that Uniform Initialization is not so important for convergence speed but that Clique Averaging is always significantly so.

% \subsection{Comparison to Non-Clustered Topologies}    
%     
%     \begin{figure}
%\centering
%              \begin{subfigure}[htb]{0.48\textwidth}
%% To regenerate the figure, from directory results/mnist/gn-lenet
%% python ../../../../learn-topology/tools/plot_convergence.py no-init/all/2021-03-22-21:39:54-CET no-init-no-clique-avg/all/2021-03-22-21:40:16-CET random-10/all/2021-03-22-21:41:06-CET random-10-diverse/all/2021-03-22-21:41:46-CET random-10-diverse-unbiased-grad/all/2021-03-22-21:42:04-CET --legend 'lower right' --add-min-max --labels 'd-clique (fcc) clique avg.' 'd-clique (fcc) no clique avg.' '10 random edges' '10 random edges (all classes repr.)' '10 random edges (all classes repr.) with unbiased grad.' --ymin 80 --yaxis test-accuracy --save-figure ../../../figures/d-cliques-mnist-lenet-comparison-to-non-clustered-topologies.png
%         \includegraphics[width=\textwidth]{figures/d-cliques-mnist-lenet-comparison-to-non-clustered-topologies}
%         \caption{\label{fig:d-cliques-mnist-lenet-comparison-to-non-clustered-topologies} LeNet Model}
%        \end{subfigure}
%        \hfill
%                      \begin{subfigure}[htb]{0.48\textwidth}
%% To regenerate the figure, from directory results/mnist/gn-lenet
%% python ../../../../learn-topology/tools/plot_convergence.py no-init/all/2021-03-22-21:39:54-CET no-init-no-clique-avg/all/2021-03-22-21:40:16-CET random-10/all/2021-03-22-21:41:06-CET random-10-diverse/all/2021-03-22-21:41:46-CET random-10-diverse-unbiased-grad/all/2021-03-22-21:42:04-CET --legend 'upper right' --add-min-max --labels 'd-clique (fcc) clique avg.' 'd-clique (fcc) no clique avg.' '10 random edges' '10 random edges (all classes repr.)' '10 random edges (all classes repr.) with unbiased grad.' --ymax 0.7 --yaxis scattering --save-figure ../../../figures/d-cliques-mnist-lenet-comparison-to-non-clustered-topologies-scattering.png
%         \includegraphics[width=\textwidth]{figures/d-cliques-mnist-lenet-comparison-to-non-clustered-topologies-scattering}
%         \caption{\label{fig:d-cliques-mnist-lenet-comparison-to-non-clustered-topologies-scattering} LeNet Model (Scattering)}
%        \end{subfigure}
%       
%         \caption{\label{fig:d-cliques-mnist-comparison-to-non-clustered-topologies} MNIST: Comparison to non-Clustered Topologies}
%\end{figure}
%
% \begin{figure}
% \centering
%     % To regenerate the figure, from directory results/cifar10
%% python ../../../learn-topology/tools/plot_convergence.py fully-connected-cliques/all/2021-03-10-13:58:57-CET no-init-no-clique-avg/fully-connected-cliques/all/2021-03-13-18:34:35-CET  random-10/all/2021-03-17-20:30:03-CET  random-10-diverse/all/2021-03-17-20:30:41-CET random-10-diverse-unbiased-gradient/all/2021-03-17-20:31:14-CET random-10-diverse-unbiased-gradient-uniform-init/all/2021-03-17-20:31:41-CET --labels 'd-clique (fcc) clique avg., uniform init.' 'd-clique (fcc) no clique avg. no uniform init.' '10 random edges' '10 random edges (all classes repr.)' '10 random (all classes repr.) with unbiased grad.' '10 random (all classes repr.) with unbiased grad., uniform init.' --add-min-max --legend 'upper right' --yaxis scattering --save-figure ../../figures/d-cliques-cifar10-linear-comparison-to-non-clustered-topologies-scattering.png --ymax 0.7
%        \begin{subfigure}[b]{0.48\textwidth}
%        \centering
%         \includegraphics[width=\textwidth]{figures/d-cliques-cifar10-linear-comparison-to-non-clustered-topologies-scattering}
%         \caption{\label{fig:d-cliques-cifar10-linear-comparison-to-non-clustered-topologies-scattering} LeNet Model: Scattering}
%     \end{subfigure}  
%         
%\caption{\label{fig:d-cliques-cifar10-linear-comparison-to-non-clustered-topologies} CIFAR10: Comparison to non-Clustered Topologies}
%\end{figure} 
%
%
%\begin{itemize}
%  \item Clustering does not seem to make a difference in MNIST, even when using a higher-capacity model (LeNet) instead of a linear model. (Fig.\ref{fig:d-cliques-mnist-comparison-to-non-clustered-topologies})     
%  \item Except for the random 10 topology, convergence speed seems to be correlated with scattering in CIFAR-10 with LeNet model (Fig.\ref{fig:d-cliques-cifar10-linear-comparison-to-non-clustered-topologies}). There is also more difference between topologies both in convergence speed and scattering than for MNIST (Fig.~\ref{fig:d-cliques-mnist-comparison-to-non-clustered-topologies}). Scattering computed similar to Consensus Control for Decentralized Deep Learning~\cite{consensus_distance}.
%\end{itemize}
%
%

% \clearpage

\section{Additional Experiments on Scaling Behavior with Increasing Number of
Nodes}
\label{app:scaling}

Section~\ref{section:interclique-topologies} compares the convergence speed of various inter-clique topologies at a scale of 1000 nodes. In this section, we show the effect of scaling the number of nodes, by comparing the convergence speed with 1, 10, 100, and 1000 nodes, and adjusting the batch size to maintain a constant number of updates per epoch. We present results for Ring, Fractal, Small-world, and Fully-Connected inter-clique topologies.
 
Figure~\ref{fig:d-cliques-mnist-scaling-fully-connected} shows the results for
MNIST. For all topologies, we notice a perfect scaling up to 100 nodes, i.e.
the accuracy curves overlap, with low variance between nodes. Starting at 1000
nodes, there is a significant increase in variance between nodes and the
convergence is slower, only marginally for Fully-Connected but
significantly so for Fractal and Ring. Small-world has higher variance between nodes but maintains a convergence speed close to that of Fully-Connected.

Figure~\ref{fig:d-cliques-cifar10-scaling-fully-connected} shows the results
for CIFAR10. When increasing from 1 to 10 nodes (resulting in a single
fully-connected clique), there is actually a small increase both in final
accuracy and convergence speed. We believe this increase is due to the
gradient being computed with exactly the same number of examples from all
classes with 10 fully-connected non-IID nodes, while the gradient for a single
non-IID node may have a slightly larger bias because the random sampling does
not guarantee the representation of all classes perfectly in each batch. At a
scale of 100 nodes, there is no difference between Fully-Connected and
Fractal, as the connections are the same; however, a Ring already shows a
significantly slower convergence. At 1000 nodes, the convergence significantly
slows down for Fractal and Ring, while remaining close, albeit with a larger
variance, for Fully-Connected. Similar to MNIST, Small-world has
higher variance and slightly lower convergence speed than Fully-Connected but
remains very close.

We therefore conclude that Fully-Connected and Small-world have good scaling
properties in terms of convergence speed, and that the
linear-logarithmic number of edges of Small-world makes it the best compromise
between convergence speed and connectivity, and thus the best choice for
efficient large-scale decentralized learning in practice.

\begin{figure}[htbp]
         \centering     
              % To regenerate the figure, from directory results/scaling
% python ../../../learn-topology/tools/plot_convergence.py ../mnist/1-node-iid/all/2021-03-10-09:20:03-CET 10/mnist/fully-connected-cliques/all/2021-03-12-09:13:27-CET ../mnist/fully-connected-cliques/all/2021-03-10-10:19:44-CET 1000/mnist/fully-connected-cliques/all/2021-03-14-17:56:26-CET --labels '1 node IID bsz=12800' '10 nodes bsz=1280' '100 nodes bsz=128' '1000 nodes bsz=13' --legend 'lower right' --yaxis test-accuracy --save-figure ../../figures/d-cliques-mnist-scaling-fully-connected-cst-updates.png --ymin 80 --add-min-max
      \begin{subfigure}[b]{0.35\textwidth}
         \centering
         \includegraphics[width=\textwidth]{../figures/d-cliques-mnist-scaling-fully-connected-cst-updates}
         \caption{Fully-Connected}
     \end{subfigure}
     \quad
       % To regenerate the figure, from directory results/scaling
% python ../../../learn-topology/tools/plot_convergence.py ../mnist/1-node-iid/all/2021-03-10-09:20:03-CET 10/mnist/fully-connected-cliques/all/2021-03-12-09:13:27-CET ../mnist/smallworld-logn-cliques/all/2021-03-23-21:44:56-CET 1000/mnist/smallworld-logn-cliques/all/2021-03-23-21:45:39-CET --labels '1 node IID bsz=12800' '10 nodes bsz=1280' '100 nodes bsz=128' '1000 nodes bsz=13' --legend 'lower right' --yaxis test-accuracy --save-figure ../../figures/d-cliques-mnist-scaling-smallworld-cst-updates.png --ymin 80 --add-min-max
      \begin{subfigure}[b]{0.35\textwidth}
         \centering
         \includegraphics[width=\textwidth]{../figures/d-cliques-mnist-scaling-smallworld-cst-updates}
         \caption{Small-world}
     \end{subfigure}
     \quad

          % To regenerate the figure, from directory results/scaling
% python ../../../learn-topology/tools/plot_convergence.py ../mnist/1-node-iid/all/2021-03-10-09:20:03-CET 10/mnist/clique-ring/all/2021-03-13-18:22:01-CET ../mnist/fully-connected-cliques/all/2021-03-10-10:19:44-CET 1000/mnist/fractal-cliques/all/2021-03-14-17:41:59-CET --labels '1 node IID bsz=12800' '10 nodes bsz=1280' '100 nodes bsz=128' '1000 nodes bsz=13' --legend 'lower right' --yaxis test-accuracy --save-figure ../../figures/d-cliques-mnist-scaling-fractal-cliques-cst-updates.png --ymin 80 --add-min-max
         \begin{subfigure}[b]{0.35\textwidth}
         \centering
         \includegraphics[width=\textwidth]{../figures/d-cliques-mnist-scaling-fractal-cliques-cst-updates}
         \caption{Fractal}
     \end{subfigure}  
     \quad
     % To regenerate the figure, from directory results/scaling
% python ../../../learn-topology/tools/plot_convergence.py ../mnist/1-node-iid/all/2021-03-10-09:20:03-CET 10/mnist/clique-ring/all/2021-03-13-18:22:01-CET ../mnist/clique-ring/all/2021-03-10-18:14:35-CET 1000/mnist/clique-ring/all/2021-03-13-18:22:36-CET --labels '1 node IID bsz=12800' '10 nodes bsz=1280' '100 nodes bsz=128' '1000 nodes bsz=13' --legend 'lower right' --yaxis test-accuracy --save-figure ../../figures/d-cliques-mnist-scaling-clique-ring-cst-updates.png --ymin 80 --add-min-max
         \begin{subfigure}[b]{0.35\textwidth}
         \centering
         \includegraphics[width=\textwidth]{../figures/d-cliques-mnist-scaling-clique-ring-cst-updates}
         \caption{Ring}
     \end{subfigure}  
     
     \caption{\label{fig:d-cliques-mnist-scaling-fully-connected} MNIST:
     D-Cliques scaling behavior (constant updates per epoch) for different
     inter-clique topologies.}
\end{figure}
     
\begin{figure}[htbp]
         \centering
     
              % To regenerate the figure, from directory results/scaling
% python ../../../learn-topology/tools/plot_convergence.py ../cifar10/1-node-iid/all/2021-03-10-13:52:58-CET 10/cifar10/fully-connected-cliques/all/2021-03-13-19:06:02-CET ../cifar10/fully-connected-cliques/all/2021-03-10-13:58:57-CET 1000/cifar10/fully-connected-cliques/all/2021-03-14-17:41:20-CET --labels '1 node IID bsz=2000' '10 nodes non-IID bsz=200' '100 nodes non-IID bsz=20' '1000 nodes non-IID bsz=2' --legend 'lower right' --yaxis test-accuracy --save-figure ../../figures/d-cliques-cifar10-scaling-fully-connected-cst-updates.png --add-min-max
      \begin{subfigure}[b]{0.35\textwidth}
         \centering
         \includegraphics[width=\textwidth]{../figures/d-cliques-cifar10-scaling-fully-connected-cst-updates}
         \caption{Fully-Connected}
     \end{subfigure}
     \quad
     % python ../../../learn-topology/tools/plot_convergence.py ../cifar10/1-node-iid/all/2021-03-10-13:52:58-CET 10/cifar10/fully-connected-cliques/all/2021-03-13-19:06:02-CET ../cifar10/smallworld-logn-cliques/all/2021-03-23-22:13:23-CET 1000/cifar10/smallworld-logn-cliques/all/2021-03-23-22:13:57-CET --labels '1 node IID bsz=2000' '10 nodes non-IID bsz=200' '100 nodes non-IID bsz=20' '1000 nodes non-IID bsz=2' --legend 'lower right' --yaxis test-accuracy --save-figure ../../figures/d-cliques-cifar10-scaling-smallworld-cst-updates.png --add-min-max
      \begin{subfigure}[b]{0.35\textwidth}
         \centering
         \includegraphics[width=\textwidth]{../figures/d-cliques-cifar10-scaling-smallworld-cst-updates}
         \caption{Small-world}
     \end{subfigure}
     
     
          % To regenerate the figure, from directory results/scaling
% python ../../../learn-topology/tools/plot_convergence.py  ../cifar10/1-node-iid/all/2021-03-10-13:52:58-CET 10/cifar10/fully-connected-cliques/all/2021-03-13-19:06:02-CET ../cifar10/fully-connected-cliques/all/2021-03-10-13:58:57-CET 1000/cifar10/fractal-cliques/all/2021-03-14-17:42:46-CET  --labels '1 node IID bsz=2000' '10 nodes non-IID bsz=200' '100 nodes non-IID bsz=20' '1000 nodes non-IID bsz=2'  --legend 'lower right' --yaxis test-accuracy --save-figure ../../figures/d-cliques-cifar10-scaling-fractal-cliques-cst-updates.png --add-min-max
         \begin{subfigure}[b]{0.35\textwidth}
         \centering
         \includegraphics[width=\textwidth]{../figures/d-cliques-cifar10-scaling-fractal-cliques-cst-updates}
         \caption{Fractal}
     \end{subfigure}  
     \quad
     % To regenerate the figure, from directory results/scaling
% python ../../../learn-topology/tools/plot_convergence.py  ../cifar10/1-node-iid/all/2021-03-10-13:52:58-CET 10/cifar10/fully-connected-cliques/all/2021-03-13-19:06:02-CET ../cifar10/clique-ring/all/2021-03-10-11:58:43-CET 1000/cifar10/clique-ring/all/2021-03-14-09:55:24-CET  --labels '1 node IID bsz=2000' '10 nodes non-IID bsz=200' '100 nodes non-IID bsz=20' '1000 nodes non-IID bsz=2'   --legend 'lower right' --yaxis test-accuracy --save-figure ../../figures/d-cliques-cifar10-scaling-clique-ring-cst-updates.png --add-min-max
         \begin{subfigure}[b]{0.35\textwidth}
         \centering
         \includegraphics[width=\textwidth]{../figures/d-cliques-cifar10-scaling-clique-ring-cst-updates}
         \caption{Ring}
     \end{subfigure}  
     
     \caption{\label{fig:d-cliques-cifar10-scaling-fully-connected} CIFAR10: D-Cliques scaling behavior (constant updates per epoch) for different
     inter-clique topologies.}
\end{figure}


%%         % To regenerate the figure, from directory results/scaling
%%% python ../../../learn-topology/tools/plot_convergence.py 10/mnist/fully-connected-cliques/all/2021-03-10-14:40:35-CET ../mnist/fully-connected-cliques/all/2021-03-10-10:19:44-CET 1000/mnist/fully-connected-cliques/all/2021-03-10-16:44:35-CET --labels '10 nodes bsz=128' '100 nodes bsz=128' '1000 nodes bsz=128 (45)' --legend 'lower right' --yaxis test-accuracy --save-figure ../../figures/d-cliques-mnist-scaling-fully-connected-cst-bsz.png --ymin 80 --add-min-max
%         \begin{figure}[htbp]
%         \centering
%         \includegraphics[width=0.48\textwidth]{figures/d-cliques-mnist-scaling-fully-connected-cst-bsz}
%         \caption{FCC: Constant Batch-Size}
%     \end{figure}

\section{Additional Experiments with Extreme Node Skew}
\label{app:extreme-local-skew} 

In this Section, we present additional results for similar experiments as in Section~\ref{section:evaluation} but in the presence of
 \textit{extreme local class bias}: we consider that each node only has examples from a single class. This extreme partitioning case provides an upper bound on the effect of label distribution skew suggesting that D-Cliques should perform similarly or better in less extreme cases, as long as a small-enough average skew can be obtained on all cliques. In turn, this helps to provide insights on why D-Cliques work well, as well as to quantify the loss in convergence speed
that may result from using construction algorithms that generate cliques with higher skew.

\subsection{Non-IID assumptions.}
\label{section:non-iid-assumptions}

To isolate the effect of local class bias from other potentially compounding
factors, we make the following simplifying assumptions: (1) All classes are
equally represented in the global dataset; (2) All classes are represented on
the same number of nodes; (3) All nodes have the same number of examples.

While less realistic than the assumptions used Section~\ref{section:evaluation}, 
these assumptions are still reasonable because: (1) Global class imbalance equally
affects the optimization process on a single node and is therefore not
specific to the decentralized setting; (2) Our results do not exploit specific
positions in the topology;  (3) Imbalanced dataset sizes across nodes can be
addressed for instance by appropriately weighting the individual loss
functions.

These assumptions do make the construction of cliques actually easier by 
making it easy to build cliques that have zero skew, as shown in 
Section~\ref{section:ideal-cliques}. 

\subsection{Constructing Ideal Cliques}
\label{section:ideal-cliques}
 
 Algorithm~\ref{Algorithm:D-Clique-Construction} shows the overall approach
 for constructing a D-Cliques topology under the assumptions of Section~\ref{section:non-iid-assumptions}.\footnote{An IID
 version of D-Cliques, in which each node has an equal number of examples of
 all classes, can be implemented by picking $\#L$ nodes per clique at random.}
 It expects the following inputs: $L$, the set of all classes present in the global distribution $D = \bigcup_{i \in N} D_i$; $N$, the set of all nodes; a function $classes(S)$, which given a subset $S$ of nodes in $N$ returns the set of classes in their joint local distributions ($D_S = \bigcup_{i \in S} D_i$); a function $intraconnect(DC)$, which given $DC$, a set of cliques (set of set of nodes), creates a set of edges ($\{\{i,j\}, \dots \}$) connecting all nodes within each clique to one another; a function $interconnect(DC)$, which given a set of cliques, creates a set of edges ($\{\{i,j\}, \dots \}$) connecting nodes belonging to different cliques; and a function $weigths(E)$, which given a set of edges, returns the weighted matrix $W_{ij}$.  Algorithm~\ref{Algorithm:D-Clique-Construction} returns both $W_{ij}$, for use in D-SGD (Algorithm~\ref{Algorithm:D-PSGD} and~\ref{Algorithm:Clique-Unbiased-D-PSGD}), and $DC$, for use with Clique Averaging (Algorithm~\ref{Algorithm:Clique-Unbiased-D-PSGD}).
 
   \begin{algorithm}[h]
   \caption{D-Cliques Construction}
   \label{Algorithm:D-Clique-Construction}
   \begin{algorithmic}[1]
        \STATE \textbf{Require:} set of classes globally present $L$, 
        \STATE~~ set of all nodes $N = \{ 1, 2, \dots, n \}$,
        \STATE~~ fn $\textit{classes}(S)$ that returns the classes present in a subset of nodes $S$,
        \STATE~~ fn $\textit{intraconnect}(DC)$ that returns edges intraconnecting cliques of $DC$,
        \STATE~~ fn $\textit{interconnect}(DC)$ that returns edges interconnecting cliques of $DC$ (Sec.~\ref{section:interclique-topologies})
         \STATE~~ fn $\textit{weights}(E)$ that assigns weights to edges in $E$ 
         
        \STATE $R \leftarrow \{ n~\text{for}~n \in N \}$ \COMMENT{Remaining nodes}
        \STATE $DC \leftarrow \emptyset$ \COMMENT{D-Cliques}
        \STATE $\textit{C} \leftarrow \emptyset$ \COMMENT{Current Clique}
        \WHILE{$R \neq \emptyset$}
       \STATE $n \leftarrow \text{pick}~1~\text{from}~\{ m \in R | \textit{classes}(\{m\}) \subsetneq \textit{classes}(\textit{C}) \}$
       \STATE $R \leftarrow R \setminus \{ n \}$
       \STATE $C \leftarrow C \cup \{ n \}$
       \IF{$\textit{classes}(C) = L$}
           \STATE $DC \leftarrow DC \cup \{ C \}$
           \STATE $C \leftarrow \emptyset$
       \ENDIF
        \ENDWHILE
        \RETURN $(weights(\textit{intraconnect}(DC) \cup \textit{interconnect}(DC)), DC)$
   \end{algorithmic}
\end{algorithm}
 
The implementation builds a single clique by adding nodes with different
classes until all classes of the global distribution are represented. Each
clique is built sequentially until all nodes are parts of cliques.
Because all classes are represented on an equal number of nodes, all cliques
will have nodes of all classes. Furthermore, since nodes have examples
of a single class, we are guaranteed a valid assignment is possible in a greedy manner. 
After cliques are created, edges are added and weights are assigned to edges, 
using the corresponding input functions.

\subsection{Evaluation}
\label{section:ideal-cliques-evaluation}

\subsubsection{Convergence Speed of D-Cliques Compared to Fully-Connected}

% From directory 'results-v2':
% MNIST
% python $TOOLS/analyze/filter.py all --dataset:name mnist --topology:name fully-connected d-cliques/ideal --nodes:name max-local-skew --meta:seed 1 --nodes:nb-nodes 100 | python $TOOLS/analyze/diff.py
% python $TOOLS/analyze/diff.py --rundirs all/2021-09-28-12:39:00-CEST-labostrex117 all/2021-09-28-23:18:40-CEST-labostrex118 --pass-through | python $TOOLS/plot/convergence.py --add-min-max --ymin 80 --ymax 92.5 --yaxis test-accuracy --labels 'fully-connected' 'd-cliques (fc) w/ cliq-avg' --save-figure ../mlsys2022style/figures/convergence-speed-mnist-dc-fc-vs-fc-1-class-per-node.png --linestyles 'solid' 'dashed'
% CIFAR10
% python $TOOLS/analyze/filter.py all --dataset:name cifar10 --topology:name fully-connected d-cliques/ideal --nodes:name max-local-skew --meta:seed 1 --nodes:nb-nodes 100 | python $TOOLS/analyze/diff.py
% python $TOOLS/analyze/diff.py --rundirs all/2021-10-03-16:09:21-CEST-labostrex112 all/2021-10-03-19:45:14-CEST-labostrex118 --pass-through | python $TOOLS/plot/convergence.py --add-min-max --ymin 0 --ymax 100 --yaxis test-accuracy --labels 'fully-connected w/ momentum' 'd-cliques (fc) w/ cliq-avg and momentum' --save-figure ../mlsys2022style/figures/convergence-speed-cifar10-dc-fc-vs-fc-1-class-per-node.png --linestyles 'solid' 'dashed' --legend 'lower right'

\begin{figure}[htbp]
    \centering        
    \begin{subfigure}[b]{0.35\textwidth}
    \centering
    \includegraphics[width=\textwidth]{figures/convergence-speed-mnist-dc-fc-vs-fc-1-class-per-node}
    \caption{\label{fig:convergence-speed-mnist-dc-fc-vs-fc-1-class-per-node} MNIST}
    \end{subfigure}
    \hfill
    \begin{subfigure}[b]{0.35\textwidth}
    \centering
    \includegraphics[width=\textwidth]{figures/convergence-speed-cifar10-dc-fc-vs-fc-1-class-per-node}
    \caption{\label{fig:convergence-speed-cifar10-dc-fc-vs-fc-1-class-per-node} CIFAR10}
    \end{subfigure}
\caption{\label{fig:convergence-speed-dc-vs-fc-1-class-per-node} Convergence Speed of D-Cliques Compared to Fully-Connected on 100 Nodes (1 class/node).}
\end{figure}

\subsubsection{Effect of Removing Intra-clique Edges}

% From directory 'results-v2':
% MNIST
% python $TOOLS/analyze/filter.py all --dataset:name mnist --topology:name d-cliques/ideal --nodes:name max-local-skew --meta:seed 1 --nodes:nb-nodes 100 | python $TOOLS/analyze/diff.py

% w/o Clique Gradient
% python $TOOLS/analyze/diff.py --rundirs all/2021-09-29-03:52:47-CEST-labostrex118 all/2021-10-02-21:26:18-CEST-labostrex113 all/2021-10-03-06:33:52-CEST-labostrex113 --pass-through | python $TOOLS/plot/convergence.py --add-min-max --ymin 89 --ymax 92.5 --yaxis test-accuracy --labels 'full intra-connectivity' '-1 edge/clique' '-5 edges/clique' --save-figure ../mlsys2022style/figures/d-cliques-ideal-wo-clique-avg-impact-of-edge-removal.png --linestyles 'solid' 'dashed' 'dotted' --font-size 18
% w/ Clique Gradient
% python $TOOLS/analyze/diff.py --rundirs all/2021-09-28-23:18:40-CEST-labostrex118 all/2021-10-02-16:50:53-CEST-labostrex113  all/2021-10-03-02:00:23-CEST-labostrex113 --pass-through | python $TOOLS/plot/convergence.py --add-min-max --ymin 89 --ymax 92.5 --yaxis test-accuracy  --labels 'full intra-connectivity' '-1 edge/clique' '-5 edges/clique' --save-figure ../mlsys2022style/figures/d-cliques-ideal-w-clique-avg-impact-of-edge-removal.png --linestyles 'solid' 'dashed' 'dotted' --font-size 18

\begin{figure}[t]
     \centering

\begin{subfigure}[htbp]{0.23\textwidth}
     \centering   
         \includegraphics[width=\textwidth]{figures/d-cliques-ideal-wo-clique-avg-impact-of-edge-removal}     
\caption{\label{fig:d-cliques-ideal-wo-clique-avg-impact-of-edge-removal} Without Clique Averaging }
\end{subfigure}
\hfill
\begin{subfigure}[htbp]{0.23\textwidth}
     \centering
         \includegraphics[width=\textwidth]{figures/d-cliques-ideal-w-clique-avg-impact-of-edge-removal}
\caption{\label{fig:d-cliques-ideal-w-clique-avg-impact-of-edge-removal} With Clique Averaging}
\end{subfigure}
\caption{\label{fig:d-cliques-ideal-intra-connectivity} Impact of Intra-clique Connectivity Failures on MNIST (1 class/node). Y axis starts at 89.}
\end{figure}
     
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