Polynomial-Time Near-Optimal Estimation over Certain Type-2 Convex Bodies

TL;DR

This paper introduces polynomial-time algorithms for near-optimal estimation in Gaussian models constrained by type-2 convex bodies, extending to linear regression and heavy-tailed scenarios, achieving statistical near-optimality efficiently.

Contribution

It presents the first general framework for computationally efficient, near-optimal estimation under broad geometric convex constraints in Gaussian and related models.

Findings

Achieves minimax rates up to poly-logarithmic factors.

Extends methodology to linear regression and heavy-tailed models.

Provides polynomial-time algorithms with near-optimal statistical guarantees.

Abstract

We develop polynomial-time algorithms for near-optimal minimax mean estimation under $\ell_2$-squared loss in a Gaussian sequence model under convex constraints. The parameter space is an origin-symmetric, type-2 convex body $K \subset \mathbb{R}^n$, and we assume additional regularity conditions: specifically, we assume $K$ is well-balanced, i.e., there exist known radii $r, R > 0$ such that $r B_2 \subseteq K \subseteq R B_2$, as well as oracle access to the Minkowski gauge of $K$. Under these and some further assumptions on $K$, our procedures achieve the minimax rate up to small factors, depending poly-logarithmically on the dimension, while remaining computationally efficient. We further extend our methodology to the linear regression and robust heavy-tailed settings, establishing polynomial-time near-optimal estimators when the constraint set satisfies the regularity conditions above. To the best of our knowledge, these results provide the first general framework for attaining statistically near-optimal performance under such broad geometric constraints while preserving computational tractability.