  This work examines a stochastic formulation of the generalized Nash
equilibrium problem (GNEP) where agents are subject to randomness in the
environment of unknown statistical distribution. We focus on fully-distributed
online learning by agents and employ penalized individual cost functions to
deal with coupled constraints. Three stochastic gradient strategies are
developed with constant step-sizes. We allow the agents to use heterogeneous
step-sizes and show that the penalty solution is able to approach the Nash
equilibrium in a stable manner within $O(\mu_\text{max})$, for small step-size
value $\mu_\text{max}$ and sufficiently large penalty parameters. The operation
of the algorithm is illustrated by considering the network Cournot competition
problem.
