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 local geometry given constraint set $K$ . Our main result shows that risk notation="LaTeX">$\vphantom {_{\int _{\int }}}$ (up to constant factors) squared notation="LaTeX">$\ell _{2}$ loss is by notation="LaTeX">$\varepsilon ^{*2} \wedge \mathrm {diam} (K)^{2}$ with ^{*} = \sup \bigg \{\varepsilon: ({\varepsilon ^{2}}/{\sigma ^{2}}) \leq \log M^{\mathrm {loc}}(\varepsilon)\bigg \}$ , where notation="LaTeX">$\log {loc}}(\varepsilon)$ denotes entropy and notation="LaTeX">$\sigma ^{2}$ variance noise. utilize our abstract re-derive known rates for some special sets such as hyperrectangles, ellipses, more generally quadratically orthosymmetric sets. Finally, we extend results unbounded case show ^{*2}$