We consider uncertainty quantification for high frequency waves. A crucial assumption for stochastic collocation methods to converge rapidly is the stochastic regularity of the QoI. In the high frequency regime, the derivatives in the stochastic variable should preferably be bounded independently of the wavelength. We show that, despite the highly oscillatory character of the waves, QoIs defined as local averages of the squared modulus of the wave solution approximated by Gaussian beams, indeed have this property.