Abstract. This paper studies efficient learning with respect to mind changes. Our starting point is the idea that a learner that is efficient with respect to mind changes minimizes mind changes not only globally in the entire learning problem, but also locally in subproblems after receiving some evidence. Formalizing this idea leads to the notion of uniform mind change optimality. We characterize the structure of language classes that can be identified with at most α mind changes by some learner (not necessarily effective): A language class L is identifiable with α mind changes iff the accumulation order of L is at most α. Accumulation order is a classic concept from point-set topology. To aid the construction of learning algorithms, we show that the characteristic property of uniformly mind change optimal learners is that they output conjectures (languages) with maximal accumulation order. We illustrate the theory by describing mind change optimal learners for various problems such as identifying linear subspaces and one-variable patterns. 1

The nature of empirical simplicity and its relationship to scientific truth are long-standing puzzles. In this paper, empirical simplicity is explicated in terms of empirical effects, which are defined in terms of the structure of the inference problem addressed. Problem instances are classified according to the number of empirical effects they present. Simple answers are satisfied by simple worlds. An efficient solution achieves optimum worst-case cost over each complexity class with respect to such costs such as the number of retractions or errors prior to convergence and elapsed time to convergence. It is shown that always choosing the simplest theory compatible with experience and hanging onto it while it remains simplest is both necessary and sufficient for efficiency. 1 The Simplicity Puzzle Machine learning, statistics, and the philosophy of science all recommend the selection of simple theories or models on the basis of empirical data, where simplicity has something to do with minimizing independent entities, principles, causes, or equational coefficients. This intuitive preference for simplicity is called Ockham’s razor, after the fourteenth century theologian and logician William of Ockham, whose work exemplified a similar tendency. But in spite of its intuitive appeal, how could Ockham’s razor possibly help us find the true theory? For if we already know that the simplest theory is true or probably true, we don’t need Ockham’s razor to infer that it is. And if we don’t know that the simplest theory is true or probably true, how do we know that simplicity steers us in the right direction? It doesn’t help to say that simplicity is associated with other virtues such as testability (Popper 1968), unity (Friedman 1983), better explanations (Harman 1965), higher “confirmation ” (Carnap 1950, Glymour 1980), minimization of predictive risk (Akaike 1973), or minimum description length (Vitanyi and Li 2000), since if the truth weren’t simple, it wouldn’t have these nice properties either. To assume otherwise is to engage in wishful thinking (vanFraassen 1981). 1 Over-fitting arguments based upon minimization of predictive risk might seem to

In science, one faces the problem of selecting the true theory from a range of alternative theories. The typical response is to select the simplest theory compatible with available evidence, on the authority of “Ockham’s Razor”. But how can a fixed bias toward simplicity help one find possibly complex truths? A short survey of standard answers to this question reveals them to be either wishful, circular, or irrelevant. A new explanation is presented, based on minimizing the reversals of opinion prior to convergence to the truth. According to this alternative approach, Ockham’s razor does not inform one which theory is true but is, nonetheless, the uniquely most efficient strategy for arriving at the true theory, where efficiency is a matter of minimizing reversals of opinion prior to finding the true theory. 1

This paper presents a new explanation of how preferring the simplest theory compatible with experience assists one in finding the true answer to a scientific question when the answers are theories or models. Science is portrayed as an infinite game between science and nature. Simplicity is a structural invariant reflecting sequences of theory choices nature could force the scientist to produce. It is demonstrated that among the methods that converge to the truth in an empirical problem, the ones that do so with a minimum number of reversals of opinion prior to convergence are exactly the ones that prefer simple theories. The idea explains not only simplicity tastes in model selection, but aspects of theory testing and the unwillingness of natural science to break symmetries without a reason. In natural science, one typically faces a situation in which several (or even infinitely many) available theories are compatible with experience. Standard practice is to choose the simplest theory among them and to cite “Ockham’s razor ” as the excuse (figure

Both in learning and in natural science, one faces the problem of selecting among a range of theories, all of which are compatible with the available evidence. The traditional response to this problem has been to select the simplest such theory on the basis of “Ockham’s Razor”. But how can a fixed bias toward simplicity help us find possibly complex truths? I survey the current, textbook answers to this question and find them all to be wishful, circular, or irrelevant. Then I present a new approach based on minimizing the number of reversals of opinion prior to convergence to the truth. According to this alternative approach, Ockham’s razor is a good idea when it seems to be (e.g., in selecting among parametrized models) and is not a good idea when it feels dubious (e.g., in the inference of arbitrary computable functions). Hence, the proposed vindication of Ockham’s razor can be used to separate vindicated applications In science and learning, one must eventually face up to the problem of choosing among several or even infinitely many theories compatible with all available information. How ought one to choose? The traditional answer is to choose the “simplest ” and to invoke

Learning theories play a significant role to machine learning as computability and complexity theories to software engineering. Gold’s language learning paradigm is one cornerstone of modern learning theories. The aim of this thesis is to establish an inductive principle in Gold’s language learning paradigm to guide the design of machine learning algorithms. We follow the common practice of using the number of mind changes to measure complexity of Gold’s language learning problems, and study efficient learning with respect to mind changes. Our starting point is the idea that a learner that is efficient with respect to mind changes minimizes mind changes not only globally in the entire learning problem, but also locally in subproblems after receiving some evidence. Formalizing this idea leads to the notion of mind change optimality. We characterize mind change complexity of language collections with Cantor’s classic concept of accumulation order. We show that the characteristic property of mind change optimal learners is that they output conjectures (languages) with maximal accumulation order. Therefore, we obtain an inductive principle in Gold’s language learning paradigm based on the simple topological concept accumulation order. The new

Ockham’s razor is the principle that, all other things being equal, scientists ought to prefer simpler theories. In recent years, philosophers have argued that simpler theories make better predictions, possess theoretical virtues like explanatory power, and have other pragmatic virtues like computational tractability. However, such arguments fail to explain how and why a preference for simplicity can help one find true theories in scientific inquiry, unless one already assumes that the truth is simple. One new solution to that problem is the Ockham efficiency theorem (Kelly 2002, 2004, 2007a-d, Kelly and Glymour 2004), which states that scientists who heed Ockham’s razor retract their opinions less often and sooner than do their non-Ockham competitors. The theorem neglects, however, to consider competitors following random (“mixed”) strategies and in many applications random strategies are known to achieve better worst-case loss than deterministic strategies. In this paper, we describe two ways to extend the result to a very general class of random, empirical strategies. The first extension concerns expected retractions, retraction times, and errors and the second extension concerns retractions in chance, times of retractions in chance, and chances of errors.