  Driven by the need to solve increasingly complex optimization problems in
signal processing and machine learning, there has been increasing interest in
understanding the behavior of gradient-descent algorithms in non-convex
environments. Most available works on distributed non-convex optimization
problems focus on the deterministic setting where exact gradients are available
at each agent. In this work and its Part II, we consider stochastic cost
functions, where exact gradients are replaced by stochastic approximations and
the resulting gradient noise persistently seeps into the dynamics of the
algorithm. We establish that the diffusion learning strategy continues to yield
meaningful estimates non-convex scenarios in the sense that the iterates by the
individual agents will cluster in a small region around the network centroid.
We use this insight to motivate a short-term model for network evolution over a
finite-horizon. In Part II [2] of this work, we leverage this model to
establish descent of the diffusion strategy through saddle points in O(1/$\mu$)
steps and the return of approximately second-order stationary points in a
polynomial number of iterations.
