Joining Late in Decentralized Federated Learning: A Theoretical Analysis and Two Local Rules

In decentralized federated learning (DFL), a client arriving mid-run changes the objective the network minimizes, and classical reasoning cannot quantify that change, since without a unique minimizer the displacement one would bound is undefined. We ask what that arrival costs, and what any treatment of it can achieve, and answer both without convexity of any kind. We prove two identities, holding under no assumption, that fix its entire geometric effect. It displaces the network average by $D/n$, where $D$ is the arriving client's distance from the incumbents and $n$ the network size, and adds to the consensus error a join shock of order $D^2/n$. Carried into a post-join stationarity bound with every constant explicit, they yield four terms that separate exactly. Two transients, the inherited gap and the join shock, decay as $1/T^{\prime }$; two persistent floors are set by gradient noise and by the spectral gap. The floors involve neither the join time nor the arriving client, so in this bound joining late carries no persistent penalty, only a transient one. The decomposition also fixes what can be acted on, and we act on both terms it leaves open. The shock admits no step size, so only initialization acts on it; averaging with the neighbors' models brings it to the network's existing consensus scale, and to zero at the incumbent average. The bound-minimizing step size depends on an oracle gap no client can evaluate, yet it is computable from a purely local misfit, with $n$ canceling, no schedule beating it in this bound class, and degradation at most $(\sqrt{\kappa }+1/\sqrt{\kappa })/2$ under $\kappa$-fold misestimation. Experiments on CIFAR-10/100 and Tiny-ImageNet test each prediction: the identities hold to machine precision, and networks with and without a late joiner reach the same late-window stationarity. The prescribed step, computed from a single local loss gap, costs a factor $2.6$ to $3.3$ against the grid-optimal constant on the two CIFAR-10 topologies, inside the $4.1$ to $7.0$ its tolerance permits.