  This paper addresses a distributed optimization problem in a communication
network where nodes are active sporadically. Each active node applies some
learning method to control its action to maximize the global utility function,
which is defined as the sum of the local utility functions of active nodes. We
deal with stochastic optimization problem with the setting that utility
functions are disturbed by some non-additive stochastic process. We consider a
more challenging situation where the learning method has to be performed only
based on a scalar approximation of the utility function, rather than its
closed-form expression, so that the typical gradient descent method cannot be
applied. This setting is quite realistic when the network is affected by some
stochastic and time-varying process, and that each node cannot have the full
knowledge of the network states. We propose a distributed optimization
algorithm and prove its almost surely convergence to the optimum. Convergence
rate is also derived with an additional assumption that the objective function
is strongly concave. Numerical results are also presented to justify our claim.
