Contents 1. Introduction
We consider a network of agents. Each agent is equipped with a local, stochastic cost of the form , where denotes a parameter vector and denotes random data. We consider a global optimization problem of the form:\begin{equation*} \min_{w}J(w),\quad \mathrm{where}\ J(w)\triangleq \sum_{k=1}^{N}p_{k}J_{k}(w) \tag{1} \end{equation*}
where the weights are a function of the graph topology and will be specified further below in (4). Solutions to such problems via distributed strategies can be pursued through a variety of algorithms, including those of the consensus and diffusion type [3]–[9]. We study the diffusion strategy due to its proven enhanced performance in adaptive environments in response to streaming data and drifting conditions [4], [10]. The strategy takes the form:\begin{align*}
\phi_{k,i}=w_{k,i-1}-\mu\widehat{\nabla J}_{k}(w_{k,i-1})\tag{2a}\\
w_{k,i}=\sum_{\ell=1}^{N}a_{\ell k}\phi_{\ell,i}\tag{2b}
\end{align*}
