保序回归下广义线性和单指标模型的有效学习
Efficient Learning of Generalized Linear and Single Index Models with Isotonic Regression
摘要：
广义线性模型（GLM）和单指数模型（SIM）提供了线性回归的强大概括，其中假设目标变量是线性预测因子的一维函数（可能未知）。在一般情况下，这些问题需要非凸估计程序，并在实践中，经常使用迭代局部搜索算法。Kalai和Sastry（2009）最近提供了第一个可证明有效的学习SIM和GLM的方法，假设数据实际上是在GLM下生成的，并且在某些单调性和Lipschitz约束下。然而，为了获得可证明的性能，该方法每次迭代都需要一个新的样本。在本文中，我们提供了学习GLM和SIM，这是计算和统计效率的算法。我们还提供了一个实证研究，证明他们在实践中的可行性。
英文摘要：
Generalized Linear Models (GLMs) and Single Index Models (SIMs) provide powerful generalizations of linear regression, where the target variable is assumed to be a (possibly unknown) 1-dimensional function of a linear predictor. In general, these problems entail non-convex estimation procedures, and, in practice, iterative local search heuristics are often used. Kalai and Sastry (2009) recently provided the first provably efficient method for learning SIMs and GLMs, under the assumptions that the data are in fact generated under a GLM and under certain monotonicity and Lipschitz constraints. However, to obtain provable performance, the method requires a fresh sample every iteration. In this paper, we provide algorithms for learning GLMs and SIMs, which are both computationally and statistically efficient. We also provide an empirical study, demonstrating their feasibility in practice.
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