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.

翻译：运用真实世界数据表达语言的有力学习仍然是一项具有挑战性的任务。许多传统方法都对超自然学有吸引力,而没有任何强健的保证。虽然大概是相当正确的(PAC)语义学提供了有力的保证,但学习明确的表述是无法做到的,即使是在假设逻辑上也是如此。然而,最近关于所谓“隐含”语义的学习工作显示,在获得第一阶逻辑碎片的多元时间结果方面,我们有着巨大的希望。在这项工作中,我们将PAC-语言学的隐含学习扩大到以线性算术语言的间隔和临界不确定性形式处理吵闹的数据。我们证明,我们的扩展框架保留了现有的多时复杂程度的保证。此外,我们提供了迄今为止这一纯理论框架的第一次实验性调查。我们使用基准问题,表明我们学习最佳线性编程目标的隐含方法大大超越了实践中的明确方法。