The Key Theorem of Learning Theory on Uncertainty Space

Abstract

Statistical Learning Theory is commonly regarded as a sound framework within which we handle a variety of learning problems in presence of small size data samples. However, since the theory is based on probability space, it hardly handles statistical learning problems on uncertainty space. In this paper, the Statistical Learning Theory on uncertainty space is investigated. The Khintchine law of large numbers on uncertainty space is proved. The definitions of empirical risk functional, expected risk functional and empirical risk minimization principle on uncertainty space are introduced. On the basis of these concepts, the key theorem of learning theory on uncertainty space is introduced and proved.