Optimal convergence rate of the universal estimation error

We study the optimal convergence rate for the universal estimation error. Let F be the excess loss class associated with the hypothesis space and n be the size of the data set, we prove that if the Fat-shattering dimension satisfies fat _ϵ(F) = O(ϵ^{- p}) , then the universal estimation error is of O(n^{- 1 / 2}) for p< 2 and O(n^{- 1 / p}) for p> 2. Among other things, this result gives a criterion for a hypothesis class to achieve the minimax optimal rate of O(n^{- 1 / 2}). We also show that if the hypothesis space is the compact supported convex Lipschitz continuous functions in R^d with d> 4 , then the rate is approximately O(n^{- 2 / d}).