Learning probabilities (p-concepts [13]) and other real-valued concepts (regression) is an important role of machine learning. For example, a doctor may need to predict the probability of getting a disease P[y|x], which depends on a number of risk factors. Generalized additive models [9] are a well-studied nonparametric model in the statistics literature, usually with monotonic link functions. However, no known efficient algorithms exist for learning such a general class, We show that regression graphs efficiently learn such real-valued concepts, while regression trees inefficiently learn them. One corollary is that any function E[y|x] = u(w | x) for u monotonic can be learned to arbitrarily small squared error ε in time polynomial in 1/ε, |w\1, and the Lipschitz constant of u (analogous to a margin). The model includes, as special cases, linear and logistic regression, as well as learning a noisy half-space with a margin [5,4]. Kearns, Mansour, and McAllester [12,15], analyzed decision trees and decision graphs as boosting algorithms for classification accuracy. We extend their analysis and the boosting analogy to the case of real-valued predictors, where a small positive correlation coefficient can be boosted to arbitrary accuracy. Viewed as a noisy boosting algorithm [3,10], the algorithm learns both the target function and the asymmetric noise.
CITATION STYLE
Kalai, A. (2004). Learning monotonic linear functions. In Lecture Notes in Artificial Intelligence (Subseries of Lecture Notes in Computer Science) (Vol. 3120, pp. 487–501). Springer Verlag. https://doi.org/10.1007/978-3-540-27819-1_34
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