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

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
$\varepsilon ^{*2}$
.