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Generalized Linear Models (GLMs) and Single Index Models (SIMs) providepowerful generalizations of linear regression, where the target variable isassumed to be a (possibly unknown) 1-dimensional function of a linearpredictor. In general, these problems entail non-convex estimation procedures,and, in practice, iterative local search heuristics are often used. Kalai andSastry (2009) recently provided the first provably efficient method forlearning SIMs and GLMs, under the assumptions that the data are in factgenerated under a GLM and under certain monotonicity and Lipschitz constraints.However, to obtain provable performance, the method requires a fresh sampleevery iteration. In this paper, we provide algorithms for learning GLMs andSIMs, which are both computationally and statistically efficient. We alsoprovide an empirical study, demonstrating their feasibility in practice.

Efficient Learning of Generalized Linear and Single Index Models with Isotonic Regression