Risk reversal for least squares estimators under nested convex constraints

תקציר מקורי באנגליתarXiv:2601.16041v2 Announce Type: replace-cross Abstract: In constrained stochastic optimization, one expects that restricting the feasible set, provided it still contains the true parameter, should not increase the statistical risk of the corresponding projection estimator. We show that this intuition can fail, even in basic settings. We investigate this phenomenon in the Gaussian sequence model. Given a compact, convex set $\Theta \subseteq \mathbb{R}^d$, one observes \[ Y = \theta^\star + \sigma Z, \qquad Z \sim N(0, I_d), \] and seeks to estimate an unknown $\theta^\star \in \Theta$. Here, the maximum likelihood estimator over $\Theta$ coincides with the least squares estimator (LSE), given by the Euclidean projection of $Y$ onto $\Theta$. We construct an explicit example exhibiting \e