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

这一系列研究的总体目标是对“什么样的数据是可有效学习的，通过什么算法？”这个问题给出一个原则性的答案。目前机器学习领域的最新技术是，在一个新的机器学习问题上，有大量可能的算法可以尝试，但对于哪些技术可以解决哪些问题却没有明确的认识。此外，通常情况下，“理论上”工作良好的机器学习算法在“实践中”表现不佳，反之亦然。pi已经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，并研究这种方法的效率。在更直接的层面上，ppi将通过建议和指导研究生以及广泛传播研究成果来实现广泛的影响。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观察到几乎所有的学习算法，无论是在理论上还是在实践中，都可以看作是对数据的低次多项式的拟合。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：了解在不同数据分布和噪声率下，低次多项式可以拟合不同自然类型的目标函数的程度。本研究涉及近似理论和分析的新方法。当多项式存在时，开发新的算法方法来寻找良好拟合的多项式。在这里，pi将调整几何和概率的结果，以识别和消除不相关的数据。划分多项式方法的有效性。pi将在学习线性分离的交叉点的计算难易性和学习有噪声的线性分离上显示新的结果。

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.