The recent emergence of machine-learning based generative models for speech suggests a significant reduction in bit rate for speech codecs is possible. However, the performance of generative models deteriorates significantly with the distortions present in real-world input signals. We argue that this deterioration is due to the sensitivity of the maximum likelihood criterion to outliers and the ineffectiveness of modeling a sum of independent signals with a single autoregressive model. We introduce predictive-variance regularization to reduce the sensitivity to outliers, resulting in a significant increase in performance. We show that noise reduction to remove unwanted signals can significantly increase performance. We provide extensive subjective performance evaluations that show that our system based on generative modeling provides state-of-the-art coding performance at 3 kb/s for real-world speech signals at reasonable computational complexity.