This paper proposes a novel approach to discriminative training ofprobabilistic linear discriminant analysis (PLDA) for speaker veri-fication with x-vectors. The Newton Method is used to discrimi-natively train the PLDA model by minimizing the log loss of ver-ification trials. By diagonalizing the across-class and within-classcovariance matrices as a pre-processing step, the PLDA model canbe trained without relying on approximations, and while maintain-ing important properties of the underlying covariance matrices. Thetraining procedure is extended to allow for efficient domain adapta-tion. When applied to the Speakers in the Wild and SRE16 tasks, theproposed approach provides significant performance improvementsrelative to conventional PLDA.