  We consider expected risk minimization in multi-agent systems comprised of
distinct subsets of agents operating without a common time-scale. Each
individual in the network is charged with minimizing the global objective
function, which is an average of sum of the statistical average loss function
of each agent in the network. Since agents are not assumed to observe data from
identical distributions, the hypothesis that all agents seek a common action is
violated, and thus the hypothesis upon which consensus constraints are
formulated is violated. Thus, we consider nonlinear network proximity
constraints which incentivize nearby nodes to make decisions which are close to
one another but not necessarily coincide. Moreover, agents are not assumed to
receive their sequentially arriving observations on a common time index, and
thus seek to learn in an asynchronous manner. An asynchronous stochastic
variant of the Arrow-Hurwicz saddle point method is proposed to solve this
problem which operates by alternating primal stochastic descent steps and
Lagrange multiplier updates which penalize the discrepancies between agents.
This tool leads to an implementation that allows for each agent to operate
asynchronously with local information only and message passing with neighbors.
Our main result establishes that the proposed method yields convergence in
expectation both in terms of the primal sub-optimality and constraint violation
to radii of sizes $\mathcal{O}(\sqrt{T})$ and $\mathcal{O}(T^{3/4})$,
respectively. Empirical evaluation on an asynchronously operating wireless
network that manages user channel interference through an adaptive
communications pricing mechanism demonstrates that our theoretical results
translates well to practice.
