  Under appropriate cooperation protocols and parameter choices, fully
decentralized solutions for stochastic optimization have been shown to match
the performance of centralized solutions and result in linear speedup (in the
number of agents) relative to non-cooperative approaches in the strongly-convex
setting. More recently, these results have been extended to the pursuit of
first-order stationary points in non-convex environments. In this work, we
examine in detail the dependence of second-order convergence guarantees on the
spectral properties of the combination policy for non-convex multi agent
optimization. We establish linear speedup in saddle-point escape time in the
number of agents for symmetric combination policies and study the potential for
further improvement by employing asymmetric combination weights. The results
imply that a linear speedup can be expected in the pursuit of second-order
stationary points, which exclude local maxima as well as strict saddle-points
and correspond to local or even global minima in many important learning
settings.
