Consider the Learning Kernel Optimization Based on Svm

The objective of this problem is to derive a learning bound for cross-validation comparing its performance to that of SRM. Let (Hk)k∈N be a countable sequence of hypothesis sets with increasing complexities. The cross-validation (CV) solution is obtained as follows. Suppose the learner receives an i.i.d. sample S of size m ≥ 1. He randomly divides S into a sample S1 of size (1−α)m and a sample S2 of size αm, where α is in (0, 1), with α typically small. S1 is used for training, S2 for validation. For any k ∈ N, let ĥk denote the solution of ERM run on S1 using hypothesis set Hk. The learner then uses sample S2 to return the CV solution fCV = argmink∈N R̂S2(ĥk).