Learning Implicitly with Noisy Data in Linear Arithmetic

TL;DR

This paper extends implicit learning in PAC-Semantics to noisy linear arithmetic data, maintaining polynomial-time guarantees and demonstrating practical superiority over explicit methods in benchmark problems.

Contribution

It introduces a novel extension of implicit PAC learning to handle noisy linear arithmetic data while preserving computational efficiency.

Findings

Implicit learning effectively handles noisy linear data.

The approach outperforms explicit methods in benchmark tests.

Polynomial-time complexity is maintained despite noise handling.

Abstract

Robust learning in expressive languages with real-world data continues to be a challenging task. Numerous conventional methods appeal to heuristics without any assurances of robustness. While probably approximately correct (PAC) Semantics offers strong guarantees, learning explicit representations is not tractable, even in propositional logic. However, recent work on so-called "implicit" learning has shown tremendous promise in terms of obtaining polynomial-time results for fragments of first-order logic. In this work, we extend implicit learning in PAC-Semantics to handle noisy data in the form of intervals and threshold uncertainty in the language of linear arithmetic. We prove that our extended framework keeps the existing polynomial-time complexity guarantees. Furthermore, we provide the first empirical investigation of this hitherto purely theoretical framework. Using benchmark problems, we show that our implicit approach to learning optimal linear programming objective constraints significantly outperforms an explicit approach in practice.