Approximation and estimation bounds for free knot splines

The fitting to data by splines has long been known to improve dramatically if the knots can be adjusted adaptively. To demonstrate the quality of the obtained free knot spline, it is essential to characterize its generalization ability. By utilizing the powerful techniques of the empirical process and approximation theory to address the estimation and approximation error bounds, respectively, the generalization ability of the free knot spline learning strategy is successfully characterized. We show that the Pseudo-dimension of free knot splines is essentially a linear function of the number of knots. A class of rather general loss functions is considered here and the squared loss is specially treated for its excellent property. We also provide some numerical results to demonstrate the utility of these theoretical results in guiding the process of choosing the appropriate knot numbers through the training data to avoid the overfitting/underfitting problem.