## Abstract

We determine the exact minimax rate of a Gaussian sequence model under bounded convex constraints, purely in terms of the local geometry of the given constraint set $K$. Our main result shows that the minimax risk $\vphantom {_{\int _{\int }}}$ (up to constant factors) under the squared $\ell _{2}$ loss is given by $\varepsilon ^{*2} \wedge \mathrm {diam} (K)^{2}$ with $\varepsilon ^{*} = \sup \bigg \{\varepsilon: ({\varepsilon ^{2}}/{\sigma ^{2}}) \leq \log M^{\mathrm {loc}}(\varepsilon)\bigg \}$ , where $\log M^{\mathrm {loc}}(\varepsilon)$ denotes the local entropy of the set $K$ , and $\sigma ^{2}$ is the variance of the noise. We utilize our abstract result to re-derive known minimax rates for some special sets $K$ such as hyperrectangles, ellipses, and more generally quadratically convex orthosymmetric sets. Finally, we extend our results to the unbounded case with known $\sigma ^{2}$ to show that the minimax rate in that case is $\varepsilon ^{*2}$.

Original language | English (US) |
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Pages (from-to) | 1244-1260 |

Number of pages | 17 |

Journal | IEEE Transactions on Information Theory |

Volume | 69 |

Issue number | 2 |

DOIs | |

State | Published - Feb 1 2023 |

## Keywords

- Estimation
- Gaussian distribution
- minimax techniques

## ASJC Scopus subject areas

- Information Systems
- Computer Science Applications
- Library and Information Sciences