The Key Theorem of Statistical Learning Theory with Rough Samples

A key theorem of statistical learning theory with rough samples is proposed. The theorem provides a theoretical basis for the applied research of supporting vector machine etc. and therefore plays an important role in statistical learning theory. In view of the uncertainty of the real world, this paper combines the trust theory and statistical learning theory to generalize the key theorem of learning theory. Random samples are replaced with rough samples and rough empirical risk minimization principle is proposed. The theorem is proven in detail.