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

Martin Černý, David Sychrovský, Filip Úradník, Jakub Černý

TL;DR

A problem of approximating an unknown subadditive set function with respect to an additive error is studied and methods to minimize this distance over classes of set functions with a known prior are developed.

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.

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

TL;DR

A problem of approximating an unknown subadditive set function with respect to an additive error is studied and methods to minimize this distance over classes of set functions with a known prior are developed.

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.
Paper Structure (29 sections, 34 theorems, 52 equations, 3 figures, 3 tables, 5 algorithms)

This paper contains 29 sections, 34 theorems, 52 equations, 3 figures, 3 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries and Problem Formulation
  3. Upper and Lower Completions for Classes of Subadditive Functions
  4. Subadditive functions
  5. Subadditive monotone functions
  6. Fractionally subadditive functions
  7. SCMM functions
  8. Symmetric submodular functions
  9. Concave additive functions
  10. Computing Optimal Subsets to Query
  11. Offline algorithms
  12. Online algorithm
  13. Empirical Evaluation
  14. Conclusion
  15. Proofs of Results
  16. ...and 14 more sections

Key Result

Proposition 1

Let $f\in\mathbb{S}^n$, $\mathbb{C}^n \subseteq \mathbb{S}^n$ and $(\alpha f + \beta)(S) \coloneqq \alpha f(S) + \sum_{i \in S} \beta_i$. The $\mathbb{C}^n$-divergence is

Figures (3)

  • Figure 1: Comparison of divergence across algorithmic steps for various algorithms, showcasing (left) submod-neg($n$), (center) xos-6($n$), and (right) sam-covg($n$) distributions, $n\in\{5,10\}$.
  • Figure 2: An illustration of lower and upper completions for a function $f(T) = g(\left\lvert T \right\rvert )$ with an unknown subset $S$, such that $\left\lvert S \right\rvert = 1$.
  • Figure 3: Comparison of divergence across algorithmic steps for various algorithms, showcasing the $k$-budget($n$) distribution for $n\in\{5,10\}$.

Theorems & Definitions (68)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Proposition 1
  • Definition 5: Value Querying Problems
  • Proposition 2
  • Definition 6: Subadditive monotone function
  • Proposition 3
  • Proposition 4
  • ...and 58 more