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603
A Model of Inductive Bias Learning
 Journal of Artificial Intelligence Research
, 2000
"... A major problem in machine learning is that of inductive bias: how to choose a learner's hypothesis space so that it is large enough to contain a solution to the problem being learnt, yet small enough to ensure reliable generalization from reasonablysized training sets. Typically such bias is ..."
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Cited by 175 (0 self)
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A major problem in machine learning is that of inductive bias: how to choose a learner's hypothesis space so that it is large enough to contain a solution to the problem being learnt, yet small enough to ensure reliable generalization from reasonablysized training sets. Typically such bias is supplied by hand through the skill and insights of experts. In this paper a model for automatically learning bias is investigated. The central assumption of the model is that the learner is embedded within an environment of related learning tasks. Within such an environment the learner can sample from multiple tasks, and hence it can search for a hypothesis space that contains good solutions to many of the problems in the environment. Under certain restrictions on the set of all hypothesis spaces available to the learner, we show that a hypothesis space that performs well on a sufficiently large number of training tasks will also perform well when learning novel tasks in the same environment. Exp...
Countable Borel Equivalence Relations
 J. MATH. LOGIC
"... This paper is a contribution to a new direction in descriptive set theory that is being extensively pursued over the last decade or so. It deals with the development of the theory of definable actions of Polish groups, the structure and classification of their orbit spaces, and the closely related s ..."
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Cited by 75 (11 self)
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This paper is a contribution to a new direction in descriptive set theory that is being extensively pursued over the last decade or so. It deals with the development of the theory of definable actions of Polish groups, the structure and classification of their orbit spaces, and the closely related study of definable equivalence relations. This study is motivated by basic foundational questions, like understanding the nature of complete classification of mathematical objects, up to some notion of equivalence, by invariants, and creating a mathematical framework for measuring the complexity of such classification problems. (For an extensive discussion of these matters, see, e.g., Hjorth [00], Kechris [99, 00a].) This theory is developed within the context of descriptive set theory, which provides the basic underlying concepts and methods. On the other hand, in view of its broad scope, there are natural interactions of it with other areas of mathematics, such as model theory, recursion theory, the theory of topological groups and their representations, topological dynamics, ergodic theory, and operator algebras
Turbulence, amalgamation, and generic automorphisms of homogeneous structures
, 2004
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Dynamic Psychological Games
, 2006
"... The motivation of decision makers who care for emotions, reciprocity, or social conformity may depend directly on beliefs (about choices, beliefs, or information). Geanakoplos, Pearce & Stacchetti (Games and Economic Behavior, 1989) point out that traditional game theory is illequipped to addre ..."
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Cited by 41 (2 self)
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The motivation of decision makers who care for emotions, reciprocity, or social conformity may depend directly on beliefs (about choices, beliefs, or information). Geanakoplos, Pearce & Stacchetti (Games and Economic Behavior, 1989) point out that traditional game theory is illequipped to address such matters, and they pioneer a new framework which does. However, their toolbox — psychological game theory — incorporates several restrictions that rule out plausible forms of beliefdependent motivation. Building on recent work on dynamic interactive epistemology, we propose a more general framework. Updated higherorder beliefs, beliefs of others, and plans of action may influence motivation, and we can capture dynamic psychological effects (such as sequential reciprocity, psychological forward induction, regret, and anxiety) that were previously ruled out. We develop solution concepts, provide examples, and explore properties.
Rigidity Theorems for Actions of Product Groups and Countable Borel Equivalence Relations
"... This paper is a contribution to the theory of countable Borel equivalence relations on standard Borel spaces. As usual, by a standard Borel space we mean a Polish (complete separable metric) space equipped with its #algebra of Borel sets. An equivalence relation E on a standard Borel space X is Bor ..."
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Cited by 38 (7 self)
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This paper is a contribution to the theory of countable Borel equivalence relations on standard Borel spaces. As usual, by a standard Borel space we mean a Polish (complete separable metric) space equipped with its #algebra of Borel sets. An equivalence relation E on a standard Borel space X is Borel if it is a Borel subset of X². Given two
Topological groups: where to from here?
, 2000
"... This is an account of one man’s view of the current perspective of theory of topological groups. We survey some recent developments which are, from our viewpoint, indicative of the future directions, concentrating on actions of topological groups on compacta, embeddings of topological groups, free t ..."
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Cited by 30 (2 self)
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This is an account of one man’s view of the current perspective of theory of topological groups. We survey some recent developments which are, from our viewpoint, indicative of the future directions, concentrating on actions of topological groups on compacta, embeddings of topological groups, free topological groups, and ‘massive’ groups (such as groups of homeomorphisms of compacta and groups of isometries of various metric spaces).
Games with secure equilibria
 In Logic in Computer Science
, 2004
"... Abstract. In 2player nonzerosum games, Nash equilibria capture the options for rational behavior if each player attempts to maximize her payoff. In contrast to classical game theory, we consider lexicographic objectives: first, each player tries to maximize her own payoff, and then, the player tr ..."
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Cited by 29 (9 self)
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Abstract. In 2player nonzerosum games, Nash equilibria capture the options for rational behavior if each player attempts to maximize her payoff. In contrast to classical game theory, we consider lexicographic objectives: first, each player tries to maximize her own payoff, and then, the player tries to minimize the opponent’s payoff. Such objectives arise naturally in the verification of systems with multiple components. There, instead of proving that each component satisfies its specification no matter how the other components behave, it often suffices to prove that each component satisfies its specification provided that the other components satisfy their specifications. We say that a Nash equilibrium is secure if it is an equilibrium with respect to the lexicographic objectives of both players. We prove that in graph games with Borel winning conditions, which include the games that arise in verification, there may be several Nash equilibria, but there is always a unique maximal payoff profile of a secure equilibrium. We show how this equilibrium can be computed in the case of ωregular winning conditions, and we characterize the memory requirements of strategies that achieve the equilibrium.
ORBIT INEQUIVALENT ACTIONS OF NONAMENABLE GROUPS
, 2008
"... Consider two free measure preserving group actions Γ � (X, µ), ∆ � (X, µ), and a measure preserving action ∆ � a (Z, ν) where (X, µ), (Z, ν) are standard probability spaces. We show how to construct free measure preserving actions Γ � c (Y, m), ∆ � d (Y, m) on a standard probability space such ..."
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Cited by 29 (3 self)
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Consider two free measure preserving group actions Γ � (X, µ), ∆ � (X, µ), and a measure preserving action ∆ � a (Z, ν) where (X, µ), (Z, ν) are standard probability spaces. We show how to construct free measure preserving actions Γ � c (Y, m), ∆ � d (Y, m) on a standard probability space such that E d ∆ ⊂ E c Γ and d has a as a factor. This generalizes the standard notion of coinduction of actions of groups from actions of subgroups. We then use this construction to show that if Γ is a countable nonamenable group, then Γ admits continuum many orbit inequivalent free, measure preserving, ergodic actions on a standard probability space.
Effective Borel measurability and reducibility of functions
 Mathematical Logic Quarterly
"... The investigation of computational properties of discontinuous functions is an important concern in computable analysis. One method to deal with this subject is to consider effective variants of Borel measurable functions. We introduce such a notion of Borel computability for singlevalued as well ..."
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Cited by 27 (7 self)
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The investigation of computational properties of discontinuous functions is an important concern in computable analysis. One method to deal with this subject is to consider effective variants of Borel measurable functions. We introduce such a notion of Borel computability for singlevalued as well as for multivalued functions by a direct effectivization of the classical definition. On Baire space the finite levels of the resulting hierarchy of functions can be characterized using a notion of reducibility for functions and corresponding complete functions. We use this classification and an effective version of a Selection Theorem of BhattacharyaSrivastava in order to prove a generalization of the Representation Theorem of KreitzWeihrauch for Borel measurable functions on computable metric spaces: such functions are Borel measurable on a certain finite level, if and only if they admit a realizer on Baire space of the same quality. This Representation Theorem enables us to introduce a realizer reducibility for functions on metric spaces and we can extend the completeness result to this reducibility. Besides being very useful by itself, this reducibility leads to a new and effective proof of the BanachHausdorffLebesgue Theorem which connects Borel measurable functions with the Baire functions. Hence, for certain metric spaces the class of Borel computable functions on a certain level is exactly the class of functions which can be expressed as a limit of a pointwise convergent and computable sequence of functions of the next lower level.