  In decentralized optimization, nodes of a communication network each possess
a local objective function, and communicate using gossip-based methods in order
to minimize the average of these per-node functions. While synchronous
algorithms are heavily impacted by a few slow nodes or edges in the graph (the
\emph{straggler problem}), their asynchronous counterparts are notoriously
harder to parametrize. Indeed, their convergence properties for networks with
heterogeneous communication and computation delays have defied analysis so far.
  In this paper, we use a \emph{ continuized} framework to analyze asynchronous
algorithms in networks with delays. Our approach yields a precise
characterization of convergence time and of its dependency on heterogeneous
delays in the network. Our continuized framework benefits from the best of both
continuous and discrete worlds: the algorithms it applies to are based on
event-driven updates. They are thus essentially discrete and hence readily
implementable. Yet their analysis is essentially in continuous time, relying in
part on the theory of delayed ODEs.
  Our algorithms moreover achieve an \emph{asynchronous speedup}: their rate of
convergence is controlled by the eigengap of the network graph weighted by
local delays, instead of the network-wide worst-case delay as in previous
analyses. Our methods thus enjoy improved robustness to stragglers.
