Active Value Querying to Minimize Additive Error in Subadditive Set Function Learning

TL;DR

This paper introduces active querying methods to efficiently approximate unknown subadditive set functions with minimal additive error, addressing the challenge of incomplete data in complex economic and AI applications.

Contribution

It provides a thorough analysis of set function completions, develops methods to minimize approximation error through active queries, and empirically validates these algorithms.

Findings

Effective algorithms for minimizing additive error in set function approximation.

Active querying significantly reduces the number of value disclosures needed.

Empirical results demonstrate practical applicability in real-world scenarios.

Abstract

Subadditive set functions play a pivotal role in computational economics (especially in combinatorial auctions), combinatorial optimization or artificial intelligence applications such as interpretable machine learning. However, specifying a set function requires assigning values to an exponentially large number of subsets in general, a task that is often resource-intensive in practice, particularly when the values derive from external sources such as retraining of machine learning models. A~simple omission of certain values introduces ambiguity that becomes even more significant when the incomplete set function has to be further optimized over. Motivated by the well-known result about inapproximability of subadditive functions using deterministic value queries with respect to a multiplicative error, we study a problem of approximating an unknown subadditive (or a subclass of thereof) set function with respect to an additive error -- i. e., we aim to efficiently close the distance between minimal and maximal completions. Our contributions are threefold: (i) a thorough exploration of minimal and maximal completions of different classes of set functions with missing values and an analysis of their resulting distance; (ii) the development of methods to minimize this distance over classes of set functions with a known prior, achieved by disclosing values of additional subsets in both offline and online manner; and (iii) empirical demonstrations of the algorithms' performance in practical scenarios.