On the Minimax Rate of the Gaussian Sequence Model Under Bounded Convex Constraints

Matey Neykov*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

2 Scopus citations

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 languageEnglish (US)
Pages (from-to)1244-1260
Number of pages17
JournalIEEE Transactions on Information Theory
Volume69
Issue number2
DOIs
StatePublished - Feb 1 2023

Keywords

  • Estimation
  • Gaussian distribution
  • minimax techniques

ASJC Scopus subject areas

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

Fingerprint

Dive into the research topics of 'On the Minimax Rate of the Gaussian Sequence Model Under Bounded Convex Constraints'. Together they form a unique fingerprint.

Cite this