% !TEX root = main.tex

\section{D-Cliques: Creating Locally Representative Cliques}
\label{section:d-cliques}

In this section, we present the design of D-Cliques. To give an intuition of our approach, let us consider the neighborhood of a single node in a grid similar to that of Figure~\ref{fig:grid-IID-vs-non-IID}, represented on Figure~\ref{fig:grid-iid-vs-non-iid-neighbourhood}.
The colors of a node represent the different classes present in its local
dataset. In the IID setting (Figure~\ref{fig:grid-iid-neighbourhood}), each
node has examples of all classes in equal proportions. In the non-IID setting 
(Figure~\ref{fig:grid-non-iid-neighbourhood}), each node has examples of only
a
single class and nodes are distributed randomly in the grid.

A single training step, from the point of view of the center node, is equivalent to sampling a mini-batch five times larger from the union of the local distributions of all illustrated nodes.
In the IID case, since gradients are computed from examples of all classes,
the resulting averaged gradient  points in a direction that tends to reduce
the loss across all classes. In contrast, in the non-IID case, only a subset
of classes are
represented in the immediate neighborhood of the node, thus the gradients will
be biased towards these classes.
Importantly, as the distributed averaging algorithm takes several steps to
converge, this variance persists across iterations as the locally computed
gradients are far from the global average.\footnote{It is possible, but
very costly, to mitigate this by performing a sufficiently large number of
averaging steps between each gradient step.} This can significantly slow down
convergence speed to the point of making decentralized optimization
impractical.

\begin{figure}[t]
     \centering
     \begin{subfigure}[b]{0.18\textwidth}
         \centering
         \includegraphics[width=\textwidth]{../figures/grid-iid-neighbourhood}
\caption{\label{fig:grid-iid-neighbourhood} IID}
     \end{subfigure}
     \begin{subfigure}[b]{0.18\textwidth}
         \centering
         \includegraphics[width=\textwidth]{../figures/grid-non-iid-neighbourhood}
\caption{\label{fig:grid-non-iid-neighbourhood}  Non-IID}
     \end{subfigure}
        \caption{Neighborhood in an IID and non-IID grid.}
        \label{fig:grid-iid-vs-non-iid-neighbourhood}
\end{figure}

In D-Cliques, we address the issues of non-iidness by carefully designing a
network topology composed of \textit{cliques} and \textit{inter-clique
connections}.

First, D-Cliques recover a balanced representation of classes, close to
 that of the IID case, by constructing a topology such that each node $i \in N$ is
 part of a \textit{clique} $C$ such that the clique distribution $D_C = \bigcup\limits_{\substack{i \in C}} D_i$ is close to that of the global distribution $D = \bigcup\limits_{\substack{i \in N}} D_i$. We measure the closeness of $D_C$ to $D$ using its \textit{skew}, i.e. the sum of the differences in the probabilities that a sample $(x,y)$ belongs to the same class in $L$ in $D_C$ and $D$: 
\begin{equation}
\label{eq:skew}
\begin{split}
        \textit{skew}(C) =\
    \sum\limits_{\substack{l \in L}} | p(y = l~|(x,y) \in D_C) - \\ p(y = l~|(x,y) \in D) |
\end{split}
\end{equation}

Second, to ensure a global consensus and convergence, 
 \textit{inter-clique connections}
 are introduced by connecting a small number of node pairs that are
 part of  different cliques. In the following, we introduce up to one inter-clique 
 connection per node such that each clique has exactly one
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}.

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
decreases the skew of that pair but keeps
the size of the cliques constant (see Algorithm~\ref{Algorithm:D-Clique-Construction}).  
Only swaps that decrease the skew are performed, hence this algorithm can be
seen as a form of randomized greedy algorithm. We note that this algorithm only requires
the knowledge of the label distribution at each node. For the sake of
simplicity, we assume that D-Cliques are constructed from the global
knowledge of these distributions, which can easily be obtained by
decentralized averaging in a pre-processing step. 

\begin{algorithm}[h]
   \caption{D-Cliques Construction: Greedy Swap}
   \label{Algorithm:D-Clique-Construction}
   \begin{algorithmic}[1]
        \STATE \textbf{Require:} Max clique size $M$, Max steps $K$,
        \STATE Set of all nodes $N = \{ 1, 2, \dots, n \}$,
        \STATE $\textit{skew}(S)$: skew of subset $S \subseteq N$ compared to the global distribution (Eq.~\ref{eq:skew}), 
        \STATE $\textit{intra}(DC)$: edges within cliques $C \in DC$,
        \STATE $\textit{inter}(DC)$: edges between $C_1,C_2 \in DC$ (Sec.~\ref{section:interclique-topologies}),
         \STATE $\textit{weights}(E)$: set weights to edges in $E$ (Eq.~\ref{eq:metro}).
         \STATE ~~
         \STATE $DC \leftarrow []$ \COMMENT{Empty list}
         \WHILE {$N \neq \emptyset$}
         \STATE $C \leftarrow$ sample $M$ nodes from $N$ at random
         \STATE $N \leftarrow N \setminus C$; $DC.append(C)$
         \ENDWHILE
         \FOR{$k \in \{1, \dots, K\}$}
        \STATE $C_1,C_2 \leftarrow$ sample 2 from $DC$ at random
        \STATE $\textit{swaps} \leftarrow []$
        \FOR{$n_1 \in C_1, n_2 \in C_2$}
          \STATE $s \leftarrow skew(C_1) + skew(C_2)$
          \STATE $s' \leftarrow \textit{skew}(C_1-n_1+n_2) + \textit{skew}(C_2 -n_2+n_1)$
          \IF {$s' < s$}
            \STATE \textit{swaps}.append($(n_1, n_2)$)
          \ENDIF
        \ENDFOR
        \IF {\#\textit{swaps} $> 0$}
          \STATE $(n_1,n_2) \leftarrow$ sample 1 from $\textit{swaps}$ at random
          \STATE $C_1 \leftarrow C_1 - n_1 + n_2; C_2 \leftarrow C_2 - n_2 + n1$
        \ENDIF
         \ENDFOR
        \RETURN $(weights(\textit{intra}(DC) \cup \textit{inter}(DC)), DC)$
   \end{algorithmic}
\end{algorithm}

The key idea of D-Cliques is that because the clique-level distribution $D_C$
 is representative of the global distribution $D$,
the local models of nodes across cliques remain rather close. Therefore, a
sparse inter-clique topology can be used, significantly reducing the total
number of edges without slowing down the convergence. Furthermore, the degree
of each node in the network remains low and even, making the D-Cliques
topology very well-suited to decentralized federated learning. 

\section{Optimizing with Clique Averaging and Momentum}
\label{section:clique-averaging-momentum}

In this section, we present Clique Averaging. This feature, when added to D-SGD,
removes the bias caused by the inter-cliques edges of
D-Cliques. We also show how it can be used to successfully implement momentum
for non-IID data.

\subsection{Clique Averaging: Debiasing Gradients from Inter-Clique Edges}
\label{section:clique-averaging}

While limiting the number of inter-clique connections reduces the
amount of messages traveling on the network, it also introduces its own
bias.
Figure~\ref{fig:connected-cliques-bias} illustrates the problem on the
simple case of two cliques connected by one inter-clique edge (here,
between the green node of the left clique and the pink node of the right
clique). Let us focus on node A. With weights computed as in \eqref{eq:metro},
node A's self-weight is $\frac{12}
{110}$, the weight between A and the green node connected to B is
$\frac{10}{110}$, and
all other neighbors of A have a weight of $\frac{11}{110}$. Therefore, the
gradient at A is biased towards its own class (pink) and against the green
class. A similar bias holds for all other nodes
without inter-clique edges with respect to their respective classes. For node
B, all its edge weights (including its self-weight) are equal to $\frac{1}
{11}$. However, the green class is represented twice (once as a clique
neighbor and once from the inter-clique edge), while all other classes are
represented only once. This biases the gradient toward the green class. The
combined effect of these two sources of bias is to increase the variance
of the local models across nodes.

\begin{figure}[t]
         \centering
         \includegraphics[width=0.3\textwidth]{../figures/connected-cliques-bias}
\caption{\label{fig:connected-cliques-bias} Illustrating the bias induced by
inter-clique connections (see main text).}
\end{figure}

We address this problem by adding \emph{Clique Averaging} to D-SGD
(Algorithm~\ref{Algorithm:Clique-Unbiased-D-PSGD}), which essentially
decouples gradient averaging from model averaging. The idea is to use only the
gradients of
neighbors within the same clique to compute the average gradient,
providing an equal representation to all classes. In contrast, all neighbors'
models, including those across inter-clique edges, participate in the model
averaging step as in the original version.

\begin{algorithm}[t]
   \caption{D-SGD with Clique Averaging, Node $i$}
   \label{Algorithm:Clique-Unbiased-D-PSGD}
   \begin{algorithmic}[1]
        \STATE \textbf{Require} initial model parameters $\theta_i^{(0)}$, learning
        rate $\gamma$, mixing weights $W$, mini-batch size $m$, number of
        steps $K$
        \FOR{$k = 1,\ldots, K$}
          \STATE $S_i^{(k)} \gets \text{mini-batch of $m$ samples drawn
          from~} D_i$
          \STATE $g_i^{(k)} \gets \frac{1}{|\textit{Clique}(i)|}\sum_{j \in 
          \textit{Clique(i)}}  \nabla F(\theta_j^{(k-1)}; S_j^{(k)})$
          \STATE $\theta_i^{(k-\frac{1}{2})} \gets \theta_i^{(k-1)} - \gamma g_i^{(k)}$ 
          \STATE $\theta_i^{(k)} \gets \sum_{j \in N} W_{ji}^{(k)} \theta_j^{(k-\frac{1}{2})}$
        \ENDFOR
   \end{algorithmic}
\end{algorithm}


\subsection{Implementing Momentum with Clique Averaging}
\label{section:momentum}

Efficiently training high capacity models usually requires additional
optimization techniques. In particular, momentum~\cite{pmlr-v28-sutskever13}
increases the magnitude of the components of the gradient that are shared
between several consecutive steps, and is critical for deep convolutional networks like
LeNet~\cite{lecun1998gradient,quagmire} to converge quickly. However, a direct
application of momentum in a non-IID setting can actually be very detrimental.
As illustrated in Figure~\ref{fig:d-cliques-cifar10-momentum-non-iid-effect}
for the case of LeNet on CIFAR10 with 100 nodes, D-Cliques with momentum
even fails to converge. Not using momentum actually gives a faster
convergence, but there is a significant gap compared to the case of a single
IID node with momentum.

Clique Averaging (Section~\ref{section:clique-averaging})
allows us to compute an unbiased momentum from the
unbiased average gradient $g_i^{(k)}$ of Algorithm~\ref{Algorithm:Clique-Unbiased-D-PSGD}:
\begin{equation}
v_i^{(k)} \leftarrow m v_i^{(k-1)} +  g_i^{(k)}.
\end{equation}
It then suffices to modify the original gradient step to use momentum:
\begin{equation}
\theta_i^{(k-\frac{1}{2})} \leftarrow \theta_i^{(k-1)} - \gamma v_i^{(k)}.
\end{equation}