The robustness of the p -norm algorithms

Abstract

We consider two on-line learning frameworks: binary classification through linear threshold functions and linear regression. We study a family of on-line algorithms, called p-norm algorithms, introduced by Grove, Littlestone and Schuurmans in the context of deterministic binary classification. We show how to adapt these algorithms for use in the regression setting, and prove worst-case bounds on the square loss, using a technique from Kivinen and Warmuth. As pointed out by Grove, et al., these algorithms can be made to approach a version of the classification algorithm Winnow as p goes to infinity; similarly they can be made to approach the corresponding regression algorithm EG in the limit. Winnow and EG are notable for having loss bounds that grow only logarithmically in the dimension of the instance space. Here we describe another way to use the p-norm algorithms to achieve this logarithmic behavior. With the way to use them that we propose, it is less critical than with Winnow and EG to retune the parameters of the algorithm as the learning task changes. Since the correct setting of the parameters depends on characteristics of the learning task that are not typically known a priori by the learner, this gives the p-norm algorithms a desireable robustness. Our elaborations yield various new loss bounds in these on-line settings. Some of these bounds improve or generalize known results. Others are incomparable.

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“…Grove, Littlestone, and Schuurmans (1997) have generalized this by giving for each p > 2 a classification algorithm that has a loss bound in terms of on the p-norm of the instances and the q-norm of the correct weight vector, where 1/ p + 1/q = 1. This result for general norm pairs has also been extended to linear regression (Gentile & Littlestone, 1999).…”

“…In work parallel to this, the general additive update (9) in the context of linear classification, i.e., with a thresholded transfer function, has recently been developed and analyzed by Grove, Littlestone, and Schuurmans (1997) with methods and results very similar to ours; see also Gentile and Littlestone (1999). Gentile and Warmuth (1999) have shown how the notion of matching loss can be generalized to thresholded transfer functions.…”

Learning to assign degrees of belief in relational domains

“…For example, a conjunctive query such as ∃x∃z (Mother(Ann, x) ∧ Gene(x, z, a)) can be transformed into a linear number of non-overlapping decomposable conjunctions; model counting can hence be evaluated in polynomial time. A second direction is to explore other learning strategies such as, for example, quasi-additive algorithms (Grove et al 2001;Gentile 2003) which might open the door to new learnability results. A third direction is to examine other types of learning interfaces.…”

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“…(This choice of p was also suggested by Gentile (2001) in the context of p-norm perceptron algorithms.) A similar bound can be obtained, under the same assumption on the r t 's, by setting η = √ 2 ln N /t in the exponential potential.…”