Understanding generalization of functional linear regression from non-i.i.d. sample viewpoint

The known works on the generalization of functional linear regression are usually based on the assumption of independently and identically distributed (i.i.d.) sample and this i.i.d. assumption does not hold in many machine learning applications. In this paper, we want to understand the generalization ability of functional regularized least squares regression (FRLSR) and functional Huber linear regression (FHLR) from non-i.i.d. sample perspective. We first establish the generalization bounds of the FRLSR based on exponentially strongly mixing sequence (e.s.m.s.) and uniformly ergodic Markov chain (u.e.M.c.) samples, respectively. Since the solution of FRLSR may suffer from lack of robustness and be spoiled by outliers, fHLR can enhance the robustness of the squared error loss function to outliers, we then research the generalization bounds of FHLR based on e.s.m.s. and u.e.M.c. samples, respectively. These established results show that FRLSR and FHLR based on non-i.i.d. samples are consistent and the corresponding learning rates are same as that of i.i.d. sample.