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Quality, diversity, and size of training data are critical factors for learning-based gaze estimators. We create two datasets satisfying these criteria for near-eye gaze estimation under infrared illumination: a synthetic dataset using anatomically-informed eye and face models with variations in face shape, gaze direction, pupil and iris, skin tone, and external conditions (2M images at 1280x960), and a real-world dataset collected with 35 subjects (2.5M images at 640x480). Using these datasets we train neural networks performing with sub-millisecond latency. Our gaze estimation network achieves 2.06(±0.44)° of accuracy across a wide 30°×40° field of view on real subjects excluded from training and 0.5° best-case accuracy (across the same FOV) when explicitly trained for one real subject. We also train a pupil localization network which achieves higher robustness than previous methods.