Centre for Discrete and Applicable Mathematics 

CDAM Research Report, LSECDAM9701February 1997 
Shai BenDavid and Eli Dichterman
Abstract
We consider learning tasks in which the learner faces restrictions on the amount of information he can extract from each example he encounters. We introduce a formal framework for the analysis of such scenarios. We call it RFA (Restricted Focus of Attention) learning. While being a natural refinement of the PAC learning model, some of the fundamental PAClearning results and techniques fail in the RFA paradigm; learnability in the RFA model is no longer characterized by the VC dimension, and many PAC learning algorithms are not applicable in the RFA setting. Hence, the RFA formulation reflects the need for new techniques and tools to cope with some fundamental constraints of realistic learning problems. In this work we also present some paradigms and algorithms that may serve as a first step towards answering this need.
Two main types of restrictions are considered here: In the stronger one, called kRFA, only k of the n attributes of each example are revealed to the learner,while in the weakest one, called kwRFA, the restriction is made on the size of each observation (k bits), and no restriction is made on how the observations are extracted from the examples.
For the stronger kRFA restriction we develop a general technique for composing efficient kRFA algorithms, and apply it to deduce, for instance, the efficient kRFA learnability of kDNF formulas, and the efficient 1RFA learnability of axisaligned rectangles in the Euclidean space R^{n}. We also prove the kRFA learnability of richer classes of Boolean functions (such as kdecision lists) with respect to a given distribution, and the efficient (n1)RFA learnability (for fixed n), under product distributions, of classes of subsets of R^{n} which are defined by mild surfaces.
For the weaker kwRFA restriction, we show that for k = O(log n ), efficient kwRFA learning is robust against classification noise. As a straightforward application, we construct a new simple noisetolerant algorithm for the class of kdecision lists by constructing an intuitive kwRFA algorithm for this task.
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