Submodular Information Selection for Hypothesis Testing with Misclassification Penalties

TL;DR

This paper develops a submodular optimization framework for selecting information sources in hypothesis testing, balancing misclassification penalties and resource constraints, with theoretical guarantees and numerical validation.

Contribution

It introduces a novel misclassification penalty framework and demonstrates submodularity properties, enabling efficient greedy algorithms with performance guarantees for information source selection.

Findings

Submodularity of the proposed optimization objectives is established.

Greedy algorithms achieve near-optimal solutions with high probability.

Numerical simulations validate theoretical guarantees.

Abstract

We consider the problem of selecting an optimal subset of information sources for a hypothesis testing/classification task where the goal is to identify the true state of the world from a finite set of hypotheses, based on finite observation samples from the sources. In order to characterize the learning performance, we propose a misclassification penalty framework, which enables nonuniform treatment of different misclassification errors. In a centralized Bayesian learning setting, we study two variants of the subset selection problem: (i) selecting a minimum cost information set to ensure that the maximum penalty of misclassifying the true hypothesis is below a desired bound and (ii) selecting an optimal information set under a limited budget to minimize the maximum penalty of misclassifying the true hypothesis. Under certain assumptions, we prove that the objective (or constraints) of these combinatorial optimization problems are weak (or approximate) submodular, and establish high-probability performance guarantees for greedy algorithms. Further, we propose an alternate metric for information set selection which is based on the total penalty of misclassification. We prove that this metric is submodular and establish near-optimal guarantees for the greedy algorithms for both the information set selection problems. Finally, we present numerical simulations to validate our theoretical results over several randomly generated instances.