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Sharp Dichotomies for Regret Minimization in Metric Spaces
, 2010
"... The Lipschitz multiarmed bandit (MAB) problem generalizes the classical multiarmed bandit problem by assuming one is given side information consisting of a priori upper bounds on the difference in expected payoff between certain pairs of strategies. Classical results of LaiRobbins [28] and Auer e ..."
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The Lipschitz multiarmed bandit (MAB) problem generalizes the classical multiarmed bandit problem by assuming one is given side information consisting of a priori upper bounds on the difference in expected payoff between certain pairs of strategies. Classical results of LaiRobbins [28] and Auer et al. [3] imply a logarithmic regret bound for the Lipschitz MAB problem on finite metric spaces. Recent results on continuumarmed bandit problems and their generalizations imply lower bounds of √ t, or stronger, for many infinite metric spaces such as the unit interval. Is this dichotomy universal? We prove that the answer is yes: for every metric space, the optimal regret of a Lipschitz MAB algorithm is either bounded above by any f ∈ ω(log t), or bounded below by any g ∈ o ( √ t). Perhaps surprisingly, this dichotomy does not coincide with the distinction between finite and infinite metric spaces; instead it depends on whether the completion of the metric space is compact and countable. Our proof connects upper and lower bound techniques in online learning with classical topological notions such as perfect sets and the CantorBendixson theorem. We also consider the fullfeedback (a.k.a., bestexpert)
Paradoxes in Göttingen
"... In 1903 Russell’s paradox came over the mathematical world with a double stroke. Bertrand Russell himself published it under the heading “The Contradiction” in chapter 10 of his Principles of Mathematics (Russell 1903). Almost at the same time Gottlob Frege (1848–1925) referred to Russell’s ..."
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In 1903 Russell’s paradox came over the mathematical world with a double stroke. Bertrand Russell himself published it under the heading “The Contradiction” in chapter 10 of his Principles of Mathematics (Russell 1903). Almost at the same time Gottlob Frege (1848–1925) referred to Russell’s
MSc in Logic
, 2011
"... Degrees of nondeterminacy and game logics on cardinals under the axiom of determinacy ..."
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Degrees of nondeterminacy and game logics on cardinals under the axiom of determinacy
Manifolds at and beyond the limit of metrisability
"... In this paper we give a brief introduction to criteria for metrisability of a manifold and to some aspects of nonmetrisable manifolds. Bias towards work currently being done by the author and his colleagues at the University of Auckland will be very evident. ..."
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In this paper we give a brief introduction to criteria for metrisability of a manifold and to some aspects of nonmetrisable manifolds. Bias towards work currently being done by the author and his colleagues at the University of Auckland will be very evident.