# Learning Implicitly with Noisy Data in Linear Arithmetic

arXiv: https://arxiv.org/abs/2010.12619 · HTML: https://mlsanity.com/paper/2010.12619/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Oct 2020.

## Built on by (most cited first, 2 of 2)

- [Learnability with PAC Semantics for Multi-agent Beliefs](https://mlsanity.com/paper/2306.05490/) (2023)
- [Lifted Relational Probabilistic Inference via Implicit Learning](https://mlsanity.com/paper/2602.14890/) (2026)

## Abstract (arXiv)

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.