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

这一研究的广泛目标是给出一个原则性的答案，“什么样的数据是可以有效学习的，通过什么样的算法？”目前机器学习的最先进水平是，有压倒性的可能算法可以在一个新的机器学习问题上试验，但没有明确的理解哪些技术可以预期对哪些问题起作用。此外，经常出现的情况是，在理论上工作得很好的机器学习算法“在实践中”表现得不那么好，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了不同方法的统一，并调查了这种方法的效率。在更直接的层面上，个人投资促进计划旨在通过建议和指导研究生以及广泛传播研究成果来产生广泛的影响。具体地说，个人投资促进计划将调查机器学习理论中“多项式方法”的有效性。PI观察到，在理论和实践中，几乎所有的学习算法都可以被视为适合数据的低次多项式。PI计划通过以下三个方面的研究来系统地发展这种多项式学习方法：1.了解在不同数据分布和噪声率下，低次多项式可以在多大程度上适应不同自然类型的目标函数。本研究从逼近理论和分析两个方面提出了新的研究方法。开发新的算法方法来寻找合适的多项式，如果它们存在的话。在这里，PI将致力于调整几何和概率结果，以识别和消除不相关的数据。界定了多项式方法的有效性。PI将展示关于学习线性分隔符的交集的计算困难以及关于学习带噪声的线性分隔符的新结果。

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.