Sharp Generalization of Transductive Learning: A Transductive Local Rademacher Complexity Approach

We introduce a new tool, Transductive Local Rademacher Complexity (TLRC), to analyze the generalization performance of transductive learning methods and motivate new transductive learning algorithms. Our work extends the idea of the popular Local Rademacher Complexity (LRC) to the transductive setting with considerable changes compared to the analysis of typical LRC methods in the inductive setting. We present a localized version of Rademacher complexity based tool wihch can be applied to various transductive learning problems and gain sharp bounds under proper conditions. Similar to the development of LRC, we build TLRC by starting from a sharp concentration inequality for independent variables with variance information. The prediction function class of a transductive learning model is then divided into pieces with a sub-root function being the upper bound for the Rademacher complexity of each piece, and the variance of all the functions in each piece is limited. A carefully designed variance operator is used to ensure that the bound for the test loss on unlabeled test data in the transductive setting enjoys a remarkable similarity to that of the classical LRC bound in the inductive setting. We use the new TLRC tool to analyze the Transductive Kernel Learning (TKL) model, where the labels of test data are generated by a kernel function. The result of TKL lays the foundation for generalization bounds for two types of transductive learning tasks, Graph Transductive Learning (GTL) and Transductive Nonparametric Kernel Regression (TNKR). When the target function is low-dimensional or approximately low-dimensional, we design low rank methods for both GTL and TNKR, which enjoy particularly sharper generalization bounds by TLRC which cannot be achieved by existing learning theory methods, to the best of our knowledge.