Supervised learning has always been attached great significance in the machine learning society where various algorithms are established commonly based on the assumption that data follow an i.i.d. distribution. In this manner, Empirical Risk Minimization usually serves as a beneficial theoretical framework for analysis on consistency, learning rates, etc. With the blossom of dynamical systems in a wide range of applications, focus now turns to the non-i.i.d. data. Here, we establish a new concentration inequality, namely a Bernstein-type inequality, that can be utilized as a powerful tool for learning on dynamical systems. When applying this inequality to support vector machines with Gaussian Kernels, we prove the consistency and also achieve almost optimal learning rates.

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