Risk reversal for least squares estimators under nested convex constraints

TL;DR

This paper demonstrates that in noisy constrained estimation, imposing stricter convex constraints can paradoxically increase the estimator's risk, challenging common intuition.

Contribution

It provides explicit examples of risk reversal in Gaussian models, showing that tighter constraints can worsen estimator risk in high-noise regimes.

Findings

Risk reversal can occur with nested convex constraints in Gaussian models.

Tighter constraints reduce risk in low-noise regimes but can increase risk in high-noise regimes.

The phenomenon depends on the noise level and geometric properties of the constraint sets.

Abstract

In constrained stochastic optimization, one naturally expects that imposing a stricter feasible set does not increase the statistical risk of an estimator defined by projection onto that set. In this paper, we show that this intuition can fail even in canonical settings. We study the Gaussian sequence model, a deliberately austere test best, where for a compact, convex set $\Theta \subset \mathbb{R}^d$ one observes \[ Y = \theta^\star + \sigma Z, \qquad Z \sim N(0, I_d), \] and seeks to estimate an unknown parameter $\theta^\star \in \Theta$. The natural estimator is the least squares estimator (LSE), which coincides with the Euclidean projection of $Y$ onto $\Theta$. We construct an explicit example exhibiting \emph{risk reversal}: for sufficiently large noise, there exist nested compact convex sets $\Theta_S \subset \Theta_L$ and a parameter $\theta^\star \in \Theta_S$ such that the LSE constrained to $\Theta_S$ has strictly larger risk than the LSE constrained to $\Theta_L$. We further show that this phenomenon can persist at the level of worst-case risk, with the supremum risk over the smaller constraint set exceeding that over the larger one. We clarify this behavior by contrasting noise regimes. In the vanishing-noise limit, the risk admits a first-order expansion governed by the statistical dimension of the tangent cone at $\theta^\star$, and tighter constraints uniformly reduce risk. In contrast, in the diverging-noise regime, the risk is determined by global geometric interactions between the constraint set and random noise directions. Here, the embedding of $\Theta_S$ within $\Theta_L$ can reverse the risk ordering. These results reveal a previously unrecognized failure mode of projection-based estimators: in sufficiently noisy settings, tightening a constraint can paradoxically degrade statistical performance.