Prove a bound on LD(hS)− LD(h∗

1. Let be some hypothesis class. For any ∈ H, let |h| denote the description length of h, according to some fixed description language. Consider the MDL learning paradigm in which the algorithm returns: hS ∈ argmin hH LS (h)+ _ |h|+ln(2) 2 , where is a sample of size m. For any 0, let HB = {∈ : |h| ≤ B}, and define h∗ B = arg min hHB LD(h). Prove a bound on LD(hS)− LD(h

B) in terms of B, the confidence parameter δ, and the size of the training set m. _ Note: Such bounds are known as oracle inequalities in the literature: We wish to estimate how good we are compared to a reference classifier (or “oracle”) h∗ B.

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