Learning Aggregate Queries Defined by First-Order Logic with Counting

TL;DR

This paper extends logical classification frameworks to multiclass problems using aggregate queries with counting, demonstrating efficient learnability of such classifiers on sparse databases.

Contribution

It introduces a new approach for learning multiclass classifiers defined by first-order logic with counting, beyond Boolean classification, with efficiency guarantees.

Findings

Efficient index structure construction for polylogarithmic degree databases

Polylogarithmic time learning of FOC1-definable classifiers

Applicable to multiclass classification with aggregate queries

Abstract

In the logical framework introduced by Grohe and Tur\'an (TOCS 2004) for Boolean classification problems, the instances to classify are tuples from a logical structure, and Boolean classifiers are described by parametric models based on logical formulas. This is a specific scenario for supervised passive learning, where classifiers should be learned based on labelled examples. Existing results in this scenario focus on Boolean classification. This paper presents learnability results beyond Boolean classification. We focus on multiclass classification problems where the task is to assign input tuples to arbitrary integers. To represent such integer-valued classifiers, we use aggregate queries specified by an extension of first-order logic with counting terms called FOC1. Our main result shows the following: given a database of polylogarithmic degree, within quasi-linear time, we can build an index structure that makes it possible to learn FOC1-definable integer-valued classifiers in time polylogarithmic in the size of the database and polynomial in the number of training examples.