论文标题
在有限凸约束下，高斯序列模型的最小值速率
On the minimax rate of the Gaussian sequence model under bounded convex constraints
论文作者
论文摘要
我们仅根据给定约束集合$ k $的局部几何形状来确定高斯序列模型的确切最小速率。我们的主要结果表明，在平方$ \ ell_2 $损失下的minimax风险（达到常数因素）由$ε^{*2} \ wedge \ wedge \ pereratatorName {diam}（k）^2 $带有\ begin {align*} ε^* = \ sup \ big {ε：\ frac {ε^2} {σ^2} \ leq \ log m^{\ operatorName {loc}}（loc}}（ε）\ bigG \}集合$ K $的本地熵和$σ^2 $是噪声的差异。我们将抽象结果用于一些特殊集合$ k $的重新衍生的已知最小值，例如超矩形，椭圆形和更通常四边形的凸出正式对称套件。最后，我们将结果扩展到无限的情况，并以已知的$σ^2 $扩展，以表明在这种情况下的最小值为$ε^{*2} $。
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 (up to constant factors) under the squared $\ell_2$ loss is given by $ε^{*2} \wedge \operatorname{diam}(K)^2$ with \begin{align*} ε^* = \sup \bigg\{ε: \frac{ε^2}{σ^2} \leq \log M^{\operatorname{loc}}(ε)\bigg\}, \end{align*} where $\log M^{\operatorname{loc}}(ε)$ denotes the local entropy of the set $K$, and $σ^2$ 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 $σ^2$ to show that the minimax rate in that case is $ε^{*2}$.
