A Further Discussion on Bounds on the Rate of Uniform Convergence of Learning Theory Based on Fuzzy Samples

Statistical Learning Theory (SLT) based on random samples formed in probability space is considered at present as one of the fundamental theories about small samples statistical learning. It has become a novel and important field of machine learning along with other concepts and architectures such as neural networks. However, many problems involve fuzzy samples in real word and the theory hardly handles statistical learning problems for them. Being motivated by some applications and the ambiguity of real world, in this study, we further develop an SLT based on fuzzy samples. Firstly, we prove a certain law of Hoeffding inequality for fuzzy random variables. Secondly, we further present the bounds on the rate of uniform convergence of learning theory based on fuzzy samples in probability space, which become cornerstones of the theoretical fundamentals of the SLT for fuzzy samples.