In supervised learning, learning from queries rather than from random examples can improve generalization performance significantly. Westudy the performance of query learning for problems where the student cannot learn the teacher perfectly, which occur frequently in practice. As a prototypical scenario of this kind, we consider a linear perceptron student learning a binary perceptron teacher. Two kinds of queries for maximum information gain, i.e., minimum entropy, are investigated: Minimum student space entropy (MSSE)queries, which are appropriate if the teacher space is unknown, and minimum teacher space entropy (MTSE) queries, which can be used if the teacher space is assumed to be known, but a student of a simpler form has deliberately been chosen. We find that for MSSE queries, the structure of the student space determines theefficacy of query learning, whereas MTSE queries lead to a higher generalization error than random examples, due to a lack of feedback about the progress of the student in the way queries are selected.

Using a statistical mechanical formalism we calculate the evidence, generalisation error and consistency measure for a linear perceptron trained and tested on a set of examples generated by a non linear teacher. The teacher is said to be unrealisable because the student can never model it without error. Our model allows us to interpolate between the known case of a linear teacher, and an unrealisable, nonlinear teacher. A comparison of the hyperparameters which maximise the evidence with those that optimise the performance measures reveals that, in the nonlinear case, the evidence procedure is a misleading guide to optimising performance. Finally, we explore the extent to which the evidence procedure is unreliable and find that, despite being sub-optimal, in some circumstances it might be a useful method for fixing the hyperparameters. 1 INTRODUCTION The analysis of supervised learning or learning from examples is a major field of research within neural networks.

We develop a principled strategy to sample a function optimally for function approximation tasks within a Bayesian framework. Using ideas from optimal experiment design, we introduce an objective function (incorporating both bias and variance) to measure the degree of approximation, and the potential utility of the data points towards optimizing this objective. We show how the general strategy can be used to derive precise algorithms to select data for two cases: learning unit step functions and polynomial functions. In particular, we investigate whether such active algorithms can learn the target with fewer examples. We obtain theoretical and empirical results to suggest that this is the case. 1 INTRODUCTION AND MOTIVATION Learning from examples is a common supervised learning paradigm that hypothesizes a target concept given a stream of training examples that describes the concept. In function approximation, example-based learning can be formulated as synthesizing an approximation function for data sampled from an unknown target function (Poggio and Girosi, 1990). Active learning describes a class of example-based learning paradigms that seeks out new training examples from specific regions of the input space, instead of passively accepting examples from some data generating source.

Random errors and insufficiencies in databases limit the performance of any classifier trained from and applied to the database. In this paper we propose a method to estimate the limiting performance of classifiers imposed by the database. We demonstrate this technique on the task of predicting failure in telecommunication paths. 1 Introduction Data collection for a classification or regression task is prone to random errors, e.g.

Instead of "ground truth" one may only have the subjective opinion(s) of one or more experts. For example, medical data or image data may be collected off-line and some time later a set of experts analyze the data and produce a set of class labels. The central problem is that of trying to infer the "ground truth" given the noisy subjective estimates of the experts. When one wishes to apply a supervised learning algorithm to the data, the problem is primarily twofold: (i) how to evaluate the relative performance of experts and algorithms, and (ii) how to train a pattern recognition system in the absence of absolute ground truth. In this paper we focus on problem (i), namely the performance evaluation issue, and in particular we discuss the application of a particular modelling technique to the problem of counting volcanoes on the surface of Venus.

In the area of inductive learning, generalization is a main operation, and the usual definition of induction is based on logical implication. Recently there has been a rising interest in clausal representation of knowledge in machine learning. Almost all inductive learning systems that perform generalization of clauses use the relation theta-subsumption instead of implication. The main reason is that there is a well-known and simple technique to compute least general generalizations under theta-subsumption, but not under implication. However generalization under theta-subsumption is inappropriate for learning recursive clauses, which is a crucial problem since recursion is the basic program structure of logic programs. We note that implication between clauses is undecidable, and we therefore introduce a stronger form of implication, called T-implication, which is decidable between clauses. We show that for every finite set of clauses there exists a least general generalization under T-implication. We describe a technique to reduce generalizations under implication of a clause to generalizations under theta-subsumption of what we call an expansion of the original clause. Moreover we show that for every non-tautological clause there exists a T-complete expansion, which means that every generalization under T-implication of the clause is reduced to a generalization under theta-subsumption of the expansion.

Learning and reasoning are both aspects of what is considered to be intelligence. Their studies within AI have been separated historically, learning being the topic of machine learning and neural networks, and reasoning falling under classical (or symbolic) AI. However, learning and reasoning are in many ways interdependent. This paper discusses the nature of some of these interdependencies and proposes a general framework called FLARE, that combines inductive learning using prior knowledge together with reasoning in a propositional setting. Several examples that test the framework are presented, including classical induction, many important reasoning protocols and two simple expert systems.

This paper presents a method for inducing logic programs from examples that learns a new class of concepts called first-order decision lists, defined as ordered lists of clauses each ending in a cut. The method, called FOIDL, is based on FOIL (Quinlan, 1990) but employs intensional background knowledge and avoids the need for explicit negative examples. It is particularly useful for problems that involve rules with specific exceptions, such as learning the past-tense of English verbs, a task widely studied in the context of the symbolic/connectionist debate. FOIDL is able to learn concise, accurate programs for this problem from significantly fewer examples than previous methods (both connectionist and symbolic).

In a companion paper it was shown that the class of constant-depth determinate k-ary recursive clauses is efficiently learnable. In this paper we present negative results showing that any natural generalization of this class is hard to learn in Valiant's model of pac-learnability. In particular, we show that the following program classes are cryptographically hard to learn: programs with an unbounded number of constant-depth linear recursive clauses; programs with one constant-depth determinate clause containing an unbounded number of recursive calls; and programs with one linear recursive clause of constant locality. These results immediately imply the non-learnability of any more general class of programs. We also show that learning a constant-depth determinate program with either two linear recursive clauses or one linear recursive clause and one non-recursive clause is as hard as learning boolean DNF. Together with positive results from the companion paper, these negative results establish a boundary of efficient learnability for recursive function-free clauses.

Theory revision integrates inductive learning and background knowledge by combining training examples with a coarse domain theory to produce a more accurate theory. There are two challenges that theory revision and other theory-guided systems face. First, a representation language appropriate for the initial theory may be inappropriate for an improved theory. While the original representation may concisely express the initial theory, a more accurate theory forced to use that same representation may be bulky, cumbersome, and difficult to reach. Second, a theory structure suitable for a coarse domain theory may be insufficient for a fine-tuned theory. Systems that produce only small, local changes to a theory have limited value for accomplishing complex structural alterations that may be required. Consequently, advanced theory-guided learning systems require flexible representation and flexible structure. An analysis of various theory revision systems and theory-guided learning systems reveals specific strengths and weaknesses in terms of these two desired properties. Designed to capture the underlying qualities of each system, a new system uses theory-guided constructive induction. Experiments in three domains show improvement over previous theory-guided systems. This leads to a study of the behavior, limitations, and potential of theory-guided constructive induction.