AF: Small: Collaborative Research: The Polynomial Method for Learning

这一系列研究的总体目标是对“什么样的数据可以有效学习，通过什么算法？”这个问题给出原则性的答案。当前机器学习的最新技术是，有大量的可能算法可以尝试解决新的机器学习问题，但尚不清楚哪些技术可以解决哪些问题。 此外，通常情况下，“理论上”运行良好的机器学习算法在“实践中”表现不佳，反之亦然。 PI 概述了解决这些困难的计划，通过多项式方法找到不同方法的统一，并研究该方法的效率。在更直接的层面上，PI 将致力于通过为研究生提供建议和指导以及广泛传播研究成果来产生广泛影响。具体来说，PI 将研究“多项式方法”在机器学习理论中的有效性。 PI 观察到，几乎所有学习算法，无论在理论上还是在实践中，都可以被视为对数据进行低次多项式拟合。 PI 计划通过以下三方面的研究来系统地开发这种多项式学习方法：1．了解在各种数据分布和噪声率下，低次多项式可以在多大程度上拟合不同自然类型的目标函数。本研究涉及近似理论和分析的新方法。 2．开发新的算法方法来找到拟合良好的多项式（当多项式存在时）。 在此，PI 将努力调整几何和概率结果，以识别和消除不相关数据。3．界定多项式方法的有效性。 PI 将展示关于学习线性分离器交集的计算困难性以及学习带有噪声的线性分离器的新结果。

The broad goal of this line of research is to give a principled answer to the question, "What sort of data is efficiently learnable, and by what algorithms?" The current state-of-the-art in machine learning is that there is an overwhelming number of possible algorithms that can be tried on a new machine learning problem, with no clear understanding of which techniques can be expected to work on which problems. Further, it is often the case that machine learning algorithms that work well "in theory" do not perform as well "in practice," and vice versa. The PIs have outlined a plan for resolving these difficulties, finding a unification of disparate methods via the Polynomial Method, and investigating how efficient this method can be. On a more immediate level the PIs will aim for broad impact through advising and guiding graduate students and widely disseminating research results.Specifically, the PIs will investigate the effectiveness of the "Polynomial Method" in machine learning theory. The PIs observe that nearly all learning algorithms, in theory and in practice, can be viewed as fitting a low-degree polynomial to data. The PIs plan to systematically develop this Polynomial Method of learning by working on the following three strands of research:1. Understand the extent to which low-degree polynomials can fit different natural types of target functions, under various data distributions and noise rates. This research involves novel methods from approximation theory and analysis.2. Develop new algorithmic methods for finding well-fitting polynomials when they exist. Here the PIs will work to adapt results in geometry and probability for the purposes of identifying and eliminating irrelevant data.3. Delimit the effectiveness of the Polynomial Method. The PIs will show new results on the computational intractability of learning intersections of linear separators, and on learning linear separators with noise.