Centre for Discrete and Applicable Mathematics

 CDAM Research Report, LSE-CDAM-98-01

February 1998

Learning with Limited Visibility

Eli Dichterman


This paper surveys recent studies of learning problems in which the learner faces restrictions on the amount of information he can extract from each example he encounters. Our main framework for the analysis of such scenarios is the RFA (Restricted Focus of Attention) model. While being a natural refinement of the PAC learning model, some of the fundamental PAC-learning 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. We also present some paradigms and algorithms that may serve as a first step towards answering this need.

Two main types of restrictions can be considered in the general RFA setting: In the more stringent one, called k-RFA, only  k  of the  n  attributes of each example are revealed to the learner, while in the more permissive one, called k-wRFA, 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.

We show an information-theoretic characterization of RFA learnability upon which we build a general tool for proving hardness results. We then apply this and other new techniques for studying RFA learning to two particularly expressive function classes, k-decision-lists (k-DL) and k-TOP, the class of thresholds of parity functions in which each parity function takes at most  k  inputs. Among other results, we show a hardness result for k-RFA learnability of k-DL,  k <= n-2.  In sharp contrast, an (n-1)-RFA algorithm for learning (n-1)-DL is presented. Similarly, we prove that 1-DL is learnable if and only if at least half of the inputs are visible in each instance. In addition, we show that there is a uniform-distribution k-RFA learning algorithm for the class of k-DL. For k-TOP we show weak learnability by a k-RFA algorithm (with efficient time and sample complexity for constant  k) and strong uniform-distribution k-RFA learnability of k-TOP with efficient sample complexity for constant  k.  Finally, by combining some of our k-DL and k-TOP results, we show that, unlike the PAC model, weak learning does not imply strong learning in the k-RFA model.

We also show a general technique for composing efficient k-RFA algorithms, and apply it to deduce, for instance, the efficient k-RFA learnability of k-DNF formulas, and the efficient 1-RFA learnability of axis-aligned rectangles in the Euclidean space  Rn.  We also prove the k-RFA learnability of richer classes of Boolean functions (such as k-decision lists) with respect to a given distribution, and the efficient (n-1)-RFA learnability (for fixed  n), under product distributions, of classes of subsets of  Rn  which are defined by mild surfaces.

For the k-wRFA restriction, we show that for  k = O(log n),  efficient k-wRFA learning is robust against classification noise. As a straightforward application, we obtain a new simple noise-tolerant algorithm for the class of k-decision lists, by constructing an intuitive k-wRFA algorithm for this task.

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