Learning Rates of Tikhonov Regularized Regressions Based on Sample Dependent RKHS

It is known that the learning algorithm with sample hypothesis spaces is essentially different from the algorithms with hypothesis spaces independent of the sample. In the present paper, we consider the error bounds for norm square regularized regressions associated with Lipschitz loss and sample depending reproducing kernel spaces. By giving the unique solution with subgradients of the loss functions, we estimate the learning rates with the regularization parameters lambda and the sample number m. The sample error rates obtained are O(1/lambda root m) and the approximation error rates are O(1/root m + lambda).