We consider a convex constrained Gaussian sequence model and characterize necessary and sufficient conditions for the least squares estimator (LSE) to be optimal in a minimax sense. For a closed convex set $K\subset \mathbb{R}^n$ we observe $Y=\mu+\xi$ for $\xi\sim N(0,\sigma^2\mathbb{I}_n)$ and $\mu\in K$ and aim to estimate $\mu$. We characterize the worst case risk of the LSE in multiple ways by analyzing the behavior of the local Gaussian width on $K$. We demonstrate that optimality is equivalent to a Lipschitz property of the local Gaussian width mapping. We also provide theoretical algorithms that search for the worst case risk. We then provide examples showing optimality or suboptimality of the LSE on various sets, including $\ell_p$ balls for $p\in[1,2]$, pyramids, solids of revolution, and multivariate isotonic regression, among others.

翻译：我们考虑一个凸约束高斯序列模型，并刻画了最小二乘估计器（LSE）在极小极大意义下最优的充分必要条件。对于一个闭凸集 $K\subset \mathbb{R}^n$，我们观测到 $Y=\mu+\xi$，其中 $\xi\sim N(0,\sigma^2\mathbb{I}_n)$ 且 $\mu\in K$，目标是估计 $\mu$。我们通过分析 $K$ 上局部高斯宽度的行为，以多种方式刻画了 LSE 的最坏情况风险。我们证明了最优性等价于局部高斯宽度映射的 Lipschitz 性质。我们还提供了搜索最坏情况风险的理论算法。随后，我们通过示例展示了 LSE 在不同集合上的最优性或次优性，这些集合包括 $p\in[1,2]$ 时的 $\ell_p$ 球、棱锥体、旋转体以及多元保序回归等。