  We consider a decentralized optimization problem for networks affected by
communication delays. Examples of such networks include collaborative machine
learning, sensor networks, and multi-agent systems. To mimic communication
delays, we add virtual non-computing nodes to the network, resulting in
directed graphs. This motivates investigating decentralized optimization
solutions on directed graphs. Existing solutions assume nodes know their
out-degrees, resulting in limited applicability. To overcome this limitation,
we introduce a novel gossip-based algorithm, called DT-GO, that does not need
to know the out-degrees. The algorithm is applicable in general directed
networks, for example networks with delays or limited acknowledgment
capabilities. We derive convergence rates for both convex and non-convex
objectives, showing that our algorithm achieves the same complexity order as
centralized Stochastic Gradient Descent. In other words, the effects of the
graph topology and delays are confined to higher-order terms. Additionally, we
extend our analysis to accommodate time-varying network topologies. Numerical
simulations are provided to support our theoretical findings.
