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Active Sequential Hypothesis Testing with Non-Homogeneous Costs

George Vershinin, Asaf Cohen, Omer Gurewitz

TL;DR

The paper extends sequential hypothesis testing to a setting with non-homogeneous action costs (NHSHT), formulating an objective to minimize expected total cost under an average error constraint $\delta$. It shows the objective factorizes into the product of the mean sample count and the mean per-action cost, motivating optimization of the ratio of expected information gain to expected cost rather than a per-step bit-per-buck measure. The authors adapt Chernoff’s classic scheme by solving a linear-fractional program to obtain action-mix policies $\boldsymbol{\lambda}_i^*$ that maximize $ \min_{j\neq i} \mathbb{E}_{A}[ D_{KL}( f_i^A \Vert f_j^A ) ] / \mathbb{E}_{A}[ c_A ]$, and implement a Cost-Aware Chernoff policy that selects actions according to the current posterior. The proposed approach preserves the $\Theta(\log(1/\delta))$ scaling and yields substantial finite-regime improvements in simulations (up to 50\% cost reduction vs classic Chernoff and up to 90\% vs bit-per-buck heuristics), highlighting its practical impact for cost-aware, real-time decision systems.

Abstract

We study the Non-Homogeneous Sequential Hypothesis Testing (NHSHT), where a single active Decision-Maker (DM) selects actions with heterogeneous positive costs to identify the true hypothesis under an average error constraint \(δ\), while minimizing expected total cost paid. Under standard arguments, we show that the objective decomposes into the product of the mean number of samples and the mean per-action cost induced by the policy. This leads to a key design principle: one should optimize the ratio of expectations (expected information gain per expected cost) rather than the expectation of per-step information-per-cost ("bit-per-buck"), which can be suboptimal. We adapt the Chernoff scheme to NHSHT, preserving its classical \(\log 1/δ\) scaling. In simulations, the adapted scheme reduces mean cost by up to 50\% relative to the classic Chernoff policy and by up to 90\% relative to the naive bit-per-buck heuristic.

Active Sequential Hypothesis Testing with Non-Homogeneous Costs

TL;DR

The paper extends sequential hypothesis testing to a setting with non-homogeneous action costs (NHSHT), formulating an objective to minimize expected total cost under an average error constraint . It shows the objective factorizes into the product of the mean sample count and the mean per-action cost, motivating optimization of the ratio of expected information gain to expected cost rather than a per-step bit-per-buck measure. The authors adapt Chernoff’s classic scheme by solving a linear-fractional program to obtain action-mix policies that maximize , and implement a Cost-Aware Chernoff policy that selects actions according to the current posterior. The proposed approach preserves the scaling and yields substantial finite-regime improvements in simulations (up to 50\% cost reduction vs classic Chernoff and up to 90\% vs bit-per-buck heuristics), highlighting its practical impact for cost-aware, real-time decision systems.

Abstract

We study the Non-Homogeneous Sequential Hypothesis Testing (NHSHT), where a single active Decision-Maker (DM) selects actions with heterogeneous positive costs to identify the true hypothesis under an average error constraint , while minimizing expected total cost paid. Under standard arguments, we show that the objective decomposes into the product of the mean number of samples and the mean per-action cost induced by the policy. This leads to a key design principle: one should optimize the ratio of expectations (expected information gain per expected cost) rather than the expectation of per-step information-per-cost ("bit-per-buck"), which can be suboptimal. We adapt the Chernoff scheme to NHSHT, preserving its classical scaling. In simulations, the adapted scheme reduces mean cost by up to 50\% relative to the classic Chernoff policy and by up to 90\% relative to the naive bit-per-buck heuristic.

Paper Structure

This paper contains 11 sections, 3 theorems, 14 equations, 2 figures.

Table of Contents

  1. Introduction
  2. System Model
  3. Notation
  4. Model
  5. Reviewing the Model and Problem
  6. The Adapted Chernoff Scheme
  7. Numerical Results
  8. Conclusion
  9. Miscellaneous Proofs
  10. Bounding The Objective Function Using NJ LB dominating term
  11. Proof of Lemma \ref{['lemma: adjusted Chernoff dist']}

Key Result

Proposition 1

For any $L'>1$ and arbitrary $\eta\in(0, 1)$, $\mathbb{E}_{\theta}\left[N\right]$ is lower bounded by

Figures (2)

  • Figure 1: System model. The DM is tasked to identify the correct hypothesis indexed by $\theta\in\mathcal{H}$. By taking action $A_n$ at time step $n$, the DM obtains a sample $X_n\sim f_\theta^{A_n}$. Note that the alphabet of $X_n$ depends on the action $A_n$.
  • Figure 2: The expected total paid costs of the classic Chernoff and Cost-Aware Chernoff schemes. All samples are unit-variance, normally distributed with randomly drawn means.

Theorems & Definitions (3)

  • Proposition 1: Eq. (80) in Naghshvar_Javidi2013_SHT_DynamicProgramming_Supp
  • Lemma 1
  • Theorem 1: Cost-Aware Chernoff Scheme