Transmit beamforming for minimum outage via stochastic approximation

We consider an outage based approach for transmit beamforming where the downlink channels are modeled as random vectors drawn from an unknown distribution. Our problem model is applicable to both point-to-point transmit beamforming as well as single-group multicasting. Given the lack of channel information, we equivalently reformulate our problem as a stochastic optimization (SO) problem with a discontinuous and non-convex cost function. We design two judicious smooth approximations of the said function, which are amenable to stochastic gradient type methods. Using these, we compute approximate online solutions via streaming first-order methods (FOMs) based on intermittent, delayed, or peer feedback. Simulation results for massive MIMO systems demonstrate the effective performance of our methods.