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Asymptotically Optimal Procedures for Sequential Joint Detection and Estimation

Dominik Reinhard, Michael Fauß, Abdelhak M. Zoubir

TL;DR

The work addresses sequential joint detection and estimation across multiple hypotheses, aiming to minimize the expected sampling cost while enforcing strict detection and estimation error constraints. It introduces an asymptotically optimal stopping rule parameterized by coefficient vectors and demonstrates that, as the error tolerances shrink, the coefficients diverge to enforce constraints while maintaining near-optimal sample usage. A projected quasi-Newton method is developed to optimally select these coefficients by exploiting a convex duality between performance metrics and coefficient derivatives. Numerical experiments across shift-in-mean, joint decoding with noise estimation, and unknown-noise scenarios confirm that the proposed AO procedure meets all constraints and achieves sample efficiency close to the strictly optimal benchmark, with significant computational advantages over fully optimal solutions.

Abstract

We investigate the problem of jointly testing multiple hypotheses and estimating a random parameter of the underlying distribution in a sequential setup. The aim is to jointly infer the true hypothesis and the true parameter while using on average as few samples as possible and keeping the detection and estimation errors below predefined levels. Based on mild assumptions on the underlying model, we propose an asymptotically optimal procedure, i.e., a procedure that becomes optimal when the tolerated detection and estimation error levels tend to zero. The implementation of the resulting asymptotically optimal stopping rule is computationally cheap and, hence, applicable for high-dimensional data. We further propose a projected quasi-Newton method to optimally choose the coefficients that parameterize the instantaneous cost function such that the constraints are fulfilled with equality. The proposed theory is validated by numerical examples.

Asymptotically Optimal Procedures for Sequential Joint Detection and Estimation

TL;DR

The work addresses sequential joint detection and estimation across multiple hypotheses, aiming to minimize the expected sampling cost while enforcing strict detection and estimation error constraints. It introduces an asymptotically optimal stopping rule parameterized by coefficient vectors and demonstrates that, as the error tolerances shrink, the coefficients diverge to enforce constraints while maintaining near-optimal sample usage. A projected quasi-Newton method is developed to optimally select these coefficients by exploiting a convex duality between performance metrics and coefficient derivatives. Numerical experiments across shift-in-mean, joint decoding with noise estimation, and unknown-noise scenarios confirm that the proposed AO procedure meets all constraints and achieves sample efficiency close to the strictly optimal benchmark, with significant computational advantages over fully optimal solutions.

Abstract

We investigate the problem of jointly testing multiple hypotheses and estimating a random parameter of the underlying distribution in a sequential setup. The aim is to jointly infer the true hypothesis and the true parameter while using on average as few samples as possible and keeping the detection and estimation errors below predefined levels. Based on mild assumptions on the underlying model, we propose an asymptotically optimal procedure, i.e., a procedure that becomes optimal when the tolerated detection and estimation error levels tend to zero. The implementation of the resulting asymptotically optimal stopping rule is computationally cheap and, hence, applicable for high-dimensional data. We further propose a projected quasi-Newton method to optimally choose the coefficients that parameterize the instantaneous cost function such that the constraints are fulfilled with equality. The proposed theory is validated by numerical examples.

Paper Structure

This paper contains 16 sections, 7 theorems, 57 equations, 3 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. State-of-the-Art
  3. Contributions and Outline
  4. Problem Formulation
  5. Notations and Assumptions
  6. Fundamentals of Sequential Joint Detection and Estimation
  7. Sequential Joint Detection and Estimation as an Optimization Problem
  8. Reduction to an Optimal Stopping Problem
  9. Asymptotically Optimal Procedures
  10. Optimal Choice of the Cost Coefficients
  11. Numerical Results
  12. Benchmarking Methods
  13. Shift-in-Mean Scenario
  14. Joint Symbol Decoding and Noise Power Estimation
  15. Estimating a Random Parameter in Unknown Noise
  16. ...and 1 more sections

Key Result

Theorem 1

Let $\bar{g}(\mathbf{x}_{1:n})$ be as defined in manuscript:eq:defModG. Then, when $\lambda_{m},\mu_{m}>0$, $m\in\{1,\ldots,M\}$, and the assumptions stated in manuscript:sec:assumptions are fulfilled, it holds that

Figures (3)

  • Figure 1: Shift-in-Mean scenario: Comparison of the policy and the strictly optimal policy.
  • Figure 2: Constellation diagram of the 16- symbols.
  • Figure 3: Joint symbol decoding and noise power estimation: Simulation results.

Theorems & Definitions (10)

  • Definition 1: Bickel and Yahav bickel1968asymptotically
  • Theorem 1
  • Lemma 1
  • Lemma 2
  • proof : Proof of \ref{['manuscript:theo:prop']}
  • Lemma 3
  • Theorem 2
  • proof
  • Theorem 3
  • Theorem 4