### Abstract

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}).

Original language | English (US) |
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Article number | 2 |

Journal | Research in Mathematical Sciences |

Volume | 4 |

Issue number | 1 |

DOIs | |

State | Published - Dec 1 2017 |

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Mathematics (miscellaneous)
- Computational Mathematics
- Applied Mathematics

### Keywords

- Empirical Process
- Estimation Error
- Gaussian Average
- Hypothesis Space
- Optimal Convergence Rate

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## Cite this

*Research in Mathematical Sciences*,

*4*(1), [2]. https://doi.org/10.1186/s40687-016-0093-6