  We consider a discrete-time model of continuous-time distributed optimization
over dynamic directed-graphs (digraphs) with applications to distributed
learning. Our optimization algorithm works over general strongly connected
dynamic networks under switching topologies, e.g., in mobile multi-agent
systems and volatile networks due to link failures. Compared to many existing
lines of work, there is no need for bi-stochastic weight designs on the links.
The existing literature mostly needs the link weights to be stochastic using
specific weight-design algorithms needed both at the initialization and at all
times when the topology of the network changes. This paper eliminates the need
for such algorithms and paves the way for distributed optimization over
time-varying digraphs. We derive the bound on the gradient-tracking step-size
and discrete time-step for convergence and prove dynamic stability using
arguments from consensus algorithms, matrix perturbation theory, and Lyapunov
theory. This work, particularly, is an improvement over existing
stochastic-weight undirected networks in case of link removal or packet drops.
This is because the existing literature may need to rerun time-consuming and
computationally complex algorithms for stochastic design, while the proposed
strategy works as long as the underlying network is weight-symmetric and
balanced. The proposed optimization framework finds applications to distributed
classification and learning.
