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Estimation and inference for minimizer and minimum of convex functions: optimality, adaptivity and uncertainty principles

T. Tony Cai, Ran Chen, Yuancheng Zhu

TL;DR

A novel Uncertainty Principle is established that provides a fundamental limit on how well the minimizer and minimum can be estimated simultaneously for any convex regression function.

Abstract

Optimal estimation and inference for both the minimizer and minimum of a convex regression function under the white noise and nonparametric regression models are studied in a nonasymptotic local minimax framework, where the performance of a procedure is evaluated at individual functions. Fully adaptive and computationally efficient algorithms are proposed and sharp minimax lower bounds are given for both the estimation accuracy and expected length of confidence intervals for the minimizer and minimum. The nonasymptotic local minimax framework brings out new phenomena in simultaneous estimation and inference for the minimizer and minimum. We establish a novel uncertainty principle that provides a fundamental limit on how well the minimizer and minimum can be estimated simultaneously for any convex regression function. A similar result holds for the expected length of the confidence intervals for the minimizer and minimum.

Estimation and inference for minimizer and minimum of convex functions: optimality, adaptivity and uncertainty principles

TL;DR

A novel Uncertainty Principle is established that provides a fundamental limit on how well the minimizer and minimum can be estimated simultaneously for any convex regression function.

Abstract

Optimal estimation and inference for both the minimizer and minimum of a convex regression function under the white noise and nonparametric regression models are studied in a nonasymptotic local minimax framework, where the performance of a procedure is evaluated at individual functions. Fully adaptive and computationally efficient algorithms are proposed and sharp minimax lower bounds are given for both the estimation accuracy and expected length of confidence intervals for the minimizer and minimum. The nonasymptotic local minimax framework brings out new phenomena in simultaneous estimation and inference for the minimizer and minimum. We establish a novel uncertainty principle that provides a fundamental limit on how well the minimizer and minimum can be estimated simultaneously for any convex regression function. A similar result holds for the expected length of the confidence intervals for the minimizer and minimum.
Paper Structure (26 sections, 13 theorems, 73 equations, 3 figures, 2 algorithms)

This paper contains 26 sections, 13 theorems, 73 equations, 3 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Function-specific Benchmarks and Uncertainty Principle
  3. Adaptive Procedures
  4. Related Literature
  5. Organization of the Paper
  6. Notation
  7. Benchmarks and Uncertainty Principle
  8. Local Moduli of Continuity
  9. Uncertainty Principle
  10. Penalty for Super-efficiency
  11. Adaptive Procedures and Optimality
  12. The Construction
  13. Sample Splitting
  14. Localization
  15. Stopping Rule
  16. ...and 11 more sections

Key Result

Theorem 2.1

Let $0<\alpha < 0.3$. Then where the constants $a_1, A_1, b_\alpha, B_\alpha$ can be taken as $a_1=\Phi(-0.5)\approx 0.309$, $A_1=1.5$, $b_\alpha=0.6 - 2\alpha$, and $B_{\alpha} = 3(1-2\alpha)z_{\alpha}$.

Figures (3)

  • Figure 1: Water filling process.
  • Figure 2: Illustration of the localization step. At level $j$, the middle two intervals are the two subintervals of the selected interval at level $j-1$. One adjacent interval of the same length on each side is added and the interval at level $j$ is selected among these four intervals.
  • Figure 3: Illustration of the stopping rule.

Theorems & Definitions (19)

  • Theorem 2.1
  • Proposition 2.1
  • Proposition 2.2
  • Example 2.1
  • Remark 2.1
  • Theorem 2.2: Uncertainty Principle
  • Remark 2.2
  • Theorem 2.3: Penalty for super-efficiency
  • Remark 2.3
  • Remark 3.1
  • ...and 9 more