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Multi-Stage Active Sequential Hypothesis Testing with Clustered Hypotheses

George Vershinin, Asaf Cohen, Omer Gurewitz

TL;DR

This work addresses active sequential hypothesis testing with many hypotheses by proposing a deterministic, adaptive multi-stage elimination strategy that leverages per-action clustering to maximize separation among remaining hypotheses. The method proceeds by selecting actions to create and exploit cluster separations, then discards entire clusters of hypotheses to progressively isolate the true one. The authors prove asymptotic optimality: ABR vanishes as the target error δ goes to zero and remains bounded as the number of hypotheses grows, supported by bounds on mean sample complexity and per-stage error. Numerical results show orders-of-magnitude reductions in observation counts compared with a recent competing method, highlighting the practical impact of cluster-based elimination for efficient sequential testing in multi-hypothesis settings.

Abstract

We consider the problem where an active Decision-Maker (DM) is tasked to identify the true hypothesis using as few as possible observations while maintaining accuracy. The DM collects observations according to its determined actions and knows the distributions under each hypothesis. We propose a deterministic and adaptive multi-stage hypothesis-elimination strategy where the DM selects an action, applies it repeatedly, and discards hypotheses in light of its obtained observations. The DM selects actions based on maximal separation expressed by the distance between the parameter vectors of each distribution under each hypothesis. Close distributions can be clustered, simplifying the search and significantly reducing the number of required observations. Our algorithms achieve vanishing Average Bayes Risk (ABR) as the error probability approaches zero, i.e., the algorithm is asymptotically optimal. Furthermore, we show that the ABR is bounded when the number of hypotheses grows. Simulations are carried out to evaluate the algorithm's performance compared to another multi-stage hypothesis-elimination algorithm, where an improvement of several orders of magnitude in the mean number of observations required is observed.

Multi-Stage Active Sequential Hypothesis Testing with Clustered Hypotheses

TL;DR

This work addresses active sequential hypothesis testing with many hypotheses by proposing a deterministic, adaptive multi-stage elimination strategy that leverages per-action clustering to maximize separation among remaining hypotheses. The method proceeds by selecting actions to create and exploit cluster separations, then discards entire clusters of hypotheses to progressively isolate the true one. The authors prove asymptotic optimality: ABR vanishes as the target error δ goes to zero and remains bounded as the number of hypotheses grows, supported by bounds on mean sample complexity and per-stage error. Numerical results show orders-of-magnitude reductions in observation counts compared with a recent competing method, highlighting the practical impact of cluster-based elimination for efficient sequential testing in multi-hypothesis settings.

Abstract

We consider the problem where an active Decision-Maker (DM) is tasked to identify the true hypothesis using as few as possible observations while maintaining accuracy. The DM collects observations according to its determined actions and knows the distributions under each hypothesis. We propose a deterministic and adaptive multi-stage hypothesis-elimination strategy where the DM selects an action, applies it repeatedly, and discards hypotheses in light of its obtained observations. The DM selects actions based on maximal separation expressed by the distance between the parameter vectors of each distribution under each hypothesis. Close distributions can be clustered, simplifying the search and significantly reducing the number of required observations. Our algorithms achieve vanishing Average Bayes Risk (ABR) as the error probability approaches zero, i.e., the algorithm is asymptotically optimal. Furthermore, we show that the ABR is bounded when the number of hypotheses grows. Simulations are carried out to evaluate the algorithm's performance compared to another multi-stage hypothesis-elimination algorithm, where an improvement of several orders of magnitude in the mean number of observations required is observed.
Paper Structure (11 sections, 4 theorems, 13 equations, 2 figures, 1 algorithm)

This paper contains 11 sections, 4 theorems, 13 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. System Model
  3. Notation
  4. Model
  5. Sample Acquisition Policy
  6. Preprocessing Step - Per-Action Hypothesis Clustering
  7. Multi-Staged Algorithm
  8. Analysis
  9. Numerical Results
  10. Miscellaneous Proofs
  11. Proof of Lemma \ref{['lemma: pe bound']}

Key Result

Theorem 1

Let $\tau_r$ be the number of samples used in stage $r$ when using action $a_r$, $H_{\mathrm{alive}}^{(r)}$ be the alive hypotheses in stage $r$, and $k$ be the representative from $H_i$’s cluster. Then $\expVal{\tau_r | H_i} = (1+o(1)) \max_{ j\in \operatorname{repr}\left( H_{\mathrm{alive}}^{(r)}

Figures (2)

  • Figure 1: System model. The DM is tasked to identify the correct hypothesis (say $H_i$) out of $H$ possible hypotheses. By taking action $a_n$ at time step $n$, the DM obtains a sample $x_n\sim f_{a_n} ( \cdot\ ; \myVec{\theta}_i(a_n) )$. The alphabet and size of $x_n$ and $\myVec{\theta}_i(a_n)$ may depend on the action $a_n$.
  • Figure 2: Comparison of the ABR (equation \ref{['eq: Bayes Risk']}) for $H = 16$ using our algorithm (with proximity parameter $\varepsilon\in\{0, 0.1\}$) against GJL. The samples obtained are either normally distributed (dashed) or exponentially distributed (non-dashed). The ABR vanishes as $\delta\to 0$ for all instances, but GJL’s greedy selection has a slower decay.

Theorems & Definitions (4)

  • Theorem 1
  • Lemma 1
  • Corollary 1: Asymptotic Optimality
  • Proposition 1