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

Matey Neykov

TL;DR

These results provide the first general framework for attaining statistically near-optimal performance under such broad geometric constraints while preserving computational tractability.

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.

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

TL;DR

These results provide the first general framework for attaining statistically near-optimal performance under such broad geometric constraints while preserving computational tractability.

Abstract

We develop polynomial-time algorithms for near-optimal minimax mean estimation under -squared loss in a Gaussian sequence model under convex constraints. The parameter space is an origin-symmetric, type-2 convex body , and we assume additional regularity conditions: specifically, we assume is well-balanced, i.e., there exist known radii such that , as well as oracle access to the Minkowski gauge of . Under these and some further assumptions on , 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.
Paper Structure (12 sections, 22 theorems, 143 equations)

This paper contains 12 sections, 22 theorems, 143 equations.

Table of Contents

  1. Introduction
  2. Related Work
  3. Definitions and Notation
  4. Organization
  5. Background and Conditional Algorithm
  6. Quadratic Form Maximization (QFM)
  7. Robust Heavy-Tailed Extension
  8. Linear Regression
  9. Some Remarks
  10. Discussion
  11. Acknowledgments
  12. Supplemental Proofs

Key Result

Lemma 2.1

Let $K$ be a type-2 convex body with type-2 constant $T_2(K)$, such that $rB_2 \subseteq K \subseteq RB_2$ for some known $r,R$. Let $\kappa:= \kappa(K)$ be a given scalar which can depend on $K$. Let $\eta^\star := \eta^\star(K)$ be the minimax rate defined through the equation $\eta^\star = \sup_\ with $c$ being the constant from the definition of local entropy, which is sufficiently large.

Theorems & Definitions (44)

  • Definition 1.1: Kolmogorov width
  • Definition 1.2: Entropy numbers
  • Definition 1.3: Local Metric Entropy
  • Lemma 2.1
  • proof : Proof of Lemma \ref{['Kwidth:bound:lemma']}
  • Lemma 2.2
  • proof : Proof of Lemma \ref{['SDP:relaxation:bound']}
  • Lemma 2.3: Projection onto the capped simplex
  • Lemma 2.4
  • proof
  • ...and 34 more