Gaze estimation for ordinary smart phone, e.g. estimating where the user is looking at on the phone screen, can be applied in various applications. However, the widely used appearance-based CNN methods still have two issues for practical adoption. First, due to the limited dataset, gaze estimation is very likely to suffer from over-fitting, leading to poor accuracy at run time. Second, the current methods are usually not robust, i.e. their prediction results having notable jitters even when the user is performing gaze fixation, which degrades user experience greatly. For the first issue, we propose a new tolerant and talented (TAT) training scheme, which is an iterative random knowledge distillation framework enhanced with cosine similarity pruning and aligned orthogonal initialization. The knowledge distillation is a tolerant teaching process providing diverse and informative supervision. The enhanced pruning and initialization is a talented learning process prompting the network to escape from the local minima and re-born from a better start. For the second issue, we define a new metric to measure the robustness of gaze estimator, and propose an adversarial training based Disturbance with Ordinal loss (DwO) method to improve it. The experimental results show that our TAT method achieves state-of-the-art performance on GazeCapture dataset, and that our DwO method improves the robustness while keeping comparable accuracy.