the continuum armed bandit problem in this paper we consider the multiarmed bandit problem where the arms are chosen from a subset of the real line and the mean rewards are assumed to be a continuous function of the arms the problem with an infinite number of arms is much more difficult than the usual one with a finite number of arms because the built in learning task is now infinite dimensional we devise a kernel estimator based learning scheme for the mean reward as a function of the arms using this learning scheme we construct a class of certainty equivalence control with forcing schemes and derive asymptotic upper bounds on their learning loss to the best of our knowledge these bounds are the strongest rates yet available moreover they are stronger than the o n required for optimality with respect to the average cost per unit time criterion

on conformally invariant equations on rn ii exponential invariance abstract in this paper we provide a complete characterization of fully nonlinear differential operators of any integer order on r n which exhibit conformal invariance of exponential type in this way we intend to complete the work that we undertook in li et al 2011 2 where we introduced the family of elementary conformal tensors t m u in order to describe all fully nonlinear differential operators of any integer order on r n which are conformally invariant of degree 0 examples of the differential operators that we study in this paper are those related to the q curvature equation on r 4 and to the gauss equation on r 2

kernel estimation and model combination in a bandit problem with covariates multi armed bandit problem is an important optimization game that requires an exploration exploitation tradeoff to achieve optimal total reward motivated from industrial applications such as online advertising and clinical research we consider a setting where the rewards of bandit machines are associated with covariates and the accurate estimation of the corresponding mean reward functions plays an important role in the performance of allocation rules under a exible problem setup we establish asymptotic strong consistency and perform a finite time regret analysis for a sequential randomized allocation strategy based on kernel estimation in addition since many nonparametric and parametric methods in supervised learning may be applied to estimating the mean reward functions but guidance on how to choose among them is generally unavailable we propose a model combining allocation strategy for adaptive performance simulations and a real data evaluation are conducted to illustrate the performance of the proposed allocation strategy

on the decomposition of global conformal invariants i this is the first of two papers where we address and partially confirm a conjecture of deser and schwimmer originally postulated in high energy physics the objects of study are scalar riemannian quantities constructed out of the curvature and its covariant derivatives whose integrals over compact manifolds are invariant under conformal changes of the underlying metric our main conclusion is that each such quantity that locally depends only on the curvature tensor without covariant derivatives can be written as a linear combination of the chern gauss bonnet integrand and a scalar conformal invariant

peak solutions for the fractional nirenberg problem abstract we consider the fractional nirenberg problem u k u n 2 n 2 u 0 in r n assume k x 1 e k x and k x has at least two critical points satisfying some local conditions the existence of two peak solutions is obtained when the positive number e is small enough

classification of singular radial solutions to the σk yamabe equation on annular domains the study of the kth elementary symmetric function of the weyl schouten curvature tensor of a riemannian metric the so called k curvature has produced many fruitful results in conformal geometry in recent years in these studies in conformal geometry the deforming conformal factor is considered to be a solution of a fully nonlinear elliptic pde important advances have been made in recent years in the understanding of the analytic behavior of solutions of the pde however the singular behavior of these solutions which is important in describing many important questions in conformal geometry is little understood this note classifies all possible radial solutions in particular the singular solutions of the k yamabe equation which describes conformal metrics whose k curvature equals a constant although the analysis involved is of elementary nature these results should provide useful guidance in studying the behavior of singular solutions in the general situation

kinetics of domain growth in systems with local barriers we study domain growth in spin 1 lattice models having nonconserved spin flip kinetics with local barriers our primary motivation is to model the relaxational behavior of physical systems in which molecular motion is impeded by local kinetic barriers the kinetic constraint is such that a spin from an up down state can flip to a down up state only via the zero state which has a higher energy we examine how the usual curvature driven domain growth is affected by these local barriers and whether the single spin barriers have a collective effect this paper presents comprehensive numerical results for phase ordering dynamics in this model using monte carlo simulations we demonstrate dynamical scaling for domain size distribution functions and spatial correlation functions we also present results for the time dependence of characteristic length scales and autocorrelation functions the length scale behavior is interpreted in terms of the random walk of steps on domain boundaries furthermore we present a simple stochastic model to derive an analytic expression for the autocorrelation function which exhibits a stretched exponential behavior over an extended regime in agreement with our numerical simulations

singular sets of a class of locally conformally flat manifolds we look at complete locally conformally flat lcf metrics defined on a domain sn there is a lot of information about the singular set contained in the positivity of k and in particular we obtain a bound for the hausdorff dimension of in relation to schoen and yau s work 18 for the scalar curvature on the other hand since some locally conformally flat manifolds can be embedded into sn this dimension bound implies several topological corollaries

learning to optimize via posterior sampling this paper considers the use of a simple posterior sampling algorithm to balance between exploration and exploitation when learning to optimize actions such as in multiarmed bandit problems the algorithm also known as thompson sampling and as probability matching offers significant advantages over the popular upper confidence bound ucb approach and can be applied to problems with finite or infinite action spaces and complicated relationships among action rewards we make two theoretical contributions the first establishes a connection between posterior sampling and ucb algorithms this result lets us convert regret bounds developed for ucb algorithms into bayesian regret bounds for posterior sampling our second theoretical contribution is a bayesian regret bound for posterior sampling that applies broadly and can be specialized to many model classes this bound depends on a new notion we refer to as the eluder dimension which measures the degree of dependence among action rewards compared to ucb

cover tree bayesian reinforcement learning this paper proposes an online tree based bayesian approach for reinforcement learning for inference we employ a generalised context tree model this defines a distribution on multivariate gaussian piecewise linear models which can be updated in closed form the tree structure itself is constructed using the cover tree method which remains efficient in high dimensional spaces we combine the model with thompson sampling and approximate dynamic programming to obtain effective exploration policies in unknown environments the flexibility and computational simplicity of the model render it suitable for many reinforcement learning problems in continuous state spaces we demonstrate this in an experimental comparison with a gaussian process model a linear model and simple least squares policy iteration

fractional laplacian in conformal geometry abstract in this note we study the connection between the fractional laplacian operator that appeared in the recent work of caffarelli and silvestre and a class of conformally covariant operators in conformal geometry

crossover in growth law and violation of superuniversality in the random field ising model we study the nonconserved phase ordering dynamics of the d 2 3 random field ising model quenched to below the critical temperature motivated by the puzzling results of previous work in two and three dimensions reporting a crossover from power law to logarithmic growth together with superuniversal behavior of the correlation function we have undertaken a careful investigation of both the domain growth law and the autocorrelation function our main results are as follows we confirm the crossover to asymptotic logarithmic behavior in the growth law but at variance with previous findings we find the exponent in the preasymptotic power law to be disorder dependent rather than being that of the pure system furthermore we find that the autocorrelation function does not display superuniversal behavior this restores consistency with previous results for the d 1 system and fits nicely into the unifying scaling scheme we have recently proposed in the study of the random bond ising model

the nonstochastic multiarmed bandit problem in the multiarmed bandit problem a gambler must decide which arm of k nonidentical slot machines to play in a sequence of trials so as to maximize his reward this classical problem has received much attention because of the simple model it provides of the trade off between exploration trying out each arm to find the best one and exploitation playing the arm believed to give the best payoff past solutions for the bandit problem have almost always relied on assumptions about the statistics of the slot machines in this work we make no statistical assumptions whatsoever about the nature of the process generating the payoffs of the slot machines we give a solution to the bandit problem in which an adversary rather than a well behaved stochastic process has complete control over the payoffs in a sequence of t plays we prove that the per round payoff of our algorithm approaches that of the best arm at the rate o t 1 2 we show by a matching lower bound that this is the best possible we also prove that our algorithm approaches the per round payoff of any set of strategies at a similar rate if the best strategy is chosen from a pool of n strategies then our algorithm approaches the per round payoff of the strategy at the rate o log n1 2 t 1 2 finally we apply our results to the problem of playing an unknown repeated matrix game we show that our algorithm approaches the minimax payoff of the unknown game at the rate o t 1 2

conformally invariant powers of the laplacian q curvature and tractor calculus we describe an elementary algorithm for expressing as explicit formulae in tractor calculus the conformally invariant gjms operators due to c r graham et alia these differential operators have leading part a power of the laplacian conformal tractor calculus is the natural induced bundle calculus associated to the conformal cartan connection applications discussed include standard formulae for these operators in terms of the levi civita connection and its curvature and a direct definition and formula for t branson s so called q curvature which integrates to a global conformal invariant as well as generalisations of the operators and the q curvature among examples the operators of order 4 6 and 8 and the related q curvatures are treated explicitly the algorithm exploits the ambient metric construction of fefferman and graham and includes a procedure for converting the ambient curvature and its covariant derivatives into tractor calculus expressions this is partly based on 12 where the relationship of the normal standard tractor bundle to the ambient construction is described

non algebraic domain growth in binary alloys with quenched disorder the authors present detailed numerical results from a computationally efficient cell dynamical system model of domain growth in binary alloys with quenched disorder their numerical results suggest that the domain growth law for the disordered case is compatible with r t approximately 1nt x where x has a weak dependence on the disorder amplitude however it is possible that their simulations do not access the true asymptotic regime they also find that the scaled structure factor for the disordered case is independent of the amplitude of disorder and is the same as that for the pure system

prescribing symmetric functions of the eigenvalues of the ricci tensor we study the problem of conformally deforming a metric to a prescribed symmetric function of the eigenvalues of the ricci tensor we prove an existence theorem for a wide class of symmetric functions on manifolds with positive ricci curvature provided the conformal class admits an admissible metric

non algebraic domain growth in random magnets a cell dynamical approach the authors develop a novel numerical approach based on a computationally efficient cell dynamical system cds model for studying the kinetics of ordering in systems described by a non conserved order parameter with quenched disorder evolving from unstable initial states they use this model to study the kinetics of domain growth in a coarse grained version of the random exchange ising model their numerical data strongly indicate quantitative agreement with the theoretically predicted asymptotic growth law over a limited range of disorder amplitudes they also compare their observations with those in laboratory experiments and make important predictions regarding dynamical scaling in these systems

correction to scaling in the response function of the two dimensional kinetic ising model the aging part r ag t s of the impulsive response function of the two dimensional ferromagnetic ising model quenched below the critical point is studied numerically employing an algorithm without the imposition of the external field we find that the simple scaling form r ag t s s ensuremath 1 a f t s which is usually believed to hold in the aging regime is not obeyed we analyze the data assuming the existence of a correction to scaling we find a 0 273 ifmmode pm else textpm fi 0 006 in agreement with previous numerical results obtained from the zero field cooled magnetization we investigate in detail also the scaling function f t s and we compare the results with the predictions of analytical theories we make an ansatz for the correction to scaling deriving an analytical expression for r ag t s this gives a satisfactory qualitative agreement with the numerical data for r ag t s and for the integrated response functions with the analytical model we explore the overall behavior extrapolating beyond the time regime accessible with the simulations we explain why the data for the zero field cooled susceptibility are not too sensitive to the existence of the correction to scaling in r ag t s making this quantity the most convenient for the study of the asymptotic scaling properties

existence of conformal metrics with constant q curvature given a compact four dimensional manifold we prove existence of conformal metrics with constant q curvature under generic assumptions the problem amounts to solving a fourth order nonlinear elliptic equation with variational structure since the corresponding euler functional is in general unbounded from above and from below we employ topological methods and min max schemes jointly with the compactness result of 35

regularity of a fourth order nonlinear pde with critical exponent in this paper we demonstrate the regularity of minimizers for a variational problem a special case of which arises in spectral theory and conformal geometry the associated euler lagrange equation is fourth order semilinear the leading term is the bilaplacian and lower order terms appear at critical powers

on fully nonlinear cr invariant equations on the heisenberg group abstract in this paper we provide a characterization of second order fully nonlinear cr invariant equations on the heisenberg group which is the analogue in the cr setting of the result proved in the euclidean setting by a li and the first author in li and li 2003 21 we also prove a comparison principle for solutions of second order fully nonlinear cr invariant equations defined on bounded domains of the heisenberg group and a comparison principle for solutions of a family of second order fully nonlinear equations on a punctured ball

an equation of monge ampere type in conformal geometry and four manifolds of positive ricci curvature we formulate natural conformally invariant conditions on a 4 manifold for the existence of a metric whose schouten tensor satisfies a quadratic inequality this inequality implies that the eigenvalues of the ricci tensor are positively pinched

regret minimization for reserve prices in second price auctions we show a regret minimization algorithm for setting the reserve price in a sequence of second price auctions under the assumption that all bids are independently drawn from the same unknown and arbitrary distribution our algorithm is computationally efficient and achieves a regret of o t in a sequence of t auctions this holds even when the number of bidders is stochastic with a known distribution

regret and convergence bounds for a class of continuum armed bandit problems we consider a class of multi armed bandit problems where the set of available actions can be mapped to a convex compact region of ropfd sometimes denoted as the ldquocontinuum armed banditrdquo problem the paper establishes bounds on the efficiency of any arm selection procedure under certain conditions on the class of possible underlying reward functions both finite time lower bounds on the growth rate of the regret as well as asymptotic upper bounds on the rates of convergence of the selected control values to the optimum are derived we explicitly characterize the dependence of these convergence rates on the minimal rate of variation of the mean reward function in a neighborhood of the optimal control the bounds can be used to demonstrate the asymptotic optimality of the kiefer wolfowitz method of stochastic approximation with regard to a large class of possible mean reward functions

on conformally invariant equations on rn abstract in this paper we provide a complete characterization of fully nonlinear conformally invariant differential operators of any integer order on r n which extends the result proved for operators of the second order by a li and the first author in li and li 2003 1 in particular we prove existence and uniqueness of a family of tensors suitably invariant under mobius transformations which are the basic building blocks that appear in the definition of all conformally invariant differential operators on r n we also explicitly compute the tensors that are related to operators of order up to four

x armed bandits we consider a generalization of stochastic bandits where the set of arms x is allowed to be a generic measurable space and the mean payoff function is locally lipschitz with respect to a dissimilarity function that is known to the decision maker under this condition we construct an arm selection policy called hoo hierarchical optimistic optimization with improved regret bounds compared to previous results for a large class of problems in particular our results imply that if x is the unit hypercube in a euclidean space and the mean payoff function has a finite number of global maxima around which the behavior of the function is locally continuous with a known smoothness degree then the expected regret of hoo is bounded up to a logarithmic factor by n that is the rate of growth of the regret is independent of the dimension of the space we also prove the minimax optimality of our algorithm when the dissimilarity is a metric our basic strategy has quadratic computational complexity as a function of the number of time steps and does not rely on the doubling trick we also introduce a modified strategy which relies on the doubling trick but runs in linearithmic time both results are improvements with respect to previous approaches

the yamabe problem for higher order curvatures let m be a compact riemannian manifold of dimension n 2 the k curvature for k 1 2 n is defined as the k th elementary symmetric polynomial of the eigenvalues of the schouten tenser the k yamabe problem is to prove the existence of a conformal metric whose k curvature is a constant when k 1 it reduces to the well known yamabe problem under the assumption that the metric is admissible the existence of solutions is known for the case k 2 n 4 for locally conformally flat manifolds and for the cases k n 2 in this paper we prove the solvability of the k yamabe problem in the remaining cases k n 2 under the hypothesis that the problem is variational this includes all of the cases k 2 as well as the locally conformally flat case

pure exploration in finitely armed and continuous armed bandits we consider the framework of stochastic multi armed bandit problems and study the possibilities and limitations of forecasters that perform an on line exploration of the arms these forecasters are assessed in terms of their simple regret a regret notion that captures the fact that exploration is only constrained by the number of available rounds not necessarily known in advance in contrast to the case when the cumulative regret is considered and when exploitation needs to be performed at the same time we believe that this performance criterion is suited to situations when the cost of pulling an arm is expressed in terms of resources rather than rewards we discuss the links between the simple and the cumulative regret one of the main results in the case of a finite number of arms is a general lower bound on the simple regret of a forecaster in terms of its cumulative regret the smaller the latter the larger the former keeping this result in mind we then exhibit upper bounds on the simple regret of some forecasters the paper ends with a study devoted to continuous armed bandit problems we show that the simple regret can be minimized with respect to a family of probability distributions if and only if the cumulative regret can be minimized for it based on this equivalence we are able to prove that the separable metric spaces are exactly the metric spaces on which these regrets can be minimized with respect to the family of all probability distributions with continuous mean payoff functions

conformally invariant differential operators on minkowski space and their curved analogues this article describes the construction of a natural family of conformally invariant differential operators on a four dimensional pseudo riemannian manifold included in this family are the usual massless field equations for arbitrary helicity but there are many more besides the article begins by classifying the invariant operators on flat space this is a fairly straightforward task in representation theory best solved through the theory of verma modules the method generates conformally invariant operators in the curved case by means of penrose s local twistor transport

higher symmetries of the laplacian we identify the symmetry algebra of the laplacian on euclidean space as an explicit quotient of the universal enveloping algebra of the lie algebra of conformal motions we construct analogues of these symmetries on a general conformal manifold

ranked bandits in metric spaces learning diverse rankings over large document collections most learning to rank research has assumed that the utility of different documents is independent which results in learned ranking functions that return redundant results the few approaches that avoid this have rather unsatisfyingly lacked theoretical foundations or do not scale we present a learning to rank formulation that optimizes the fraction of satisfied users with several scalable algorithms that explicitly takes document similarity and ranking context into account our formulation is a non trivial common generalization of two multi armed bandit models from the literature ranked bandits radlinski et al 2008 and lipschitz bandits kleinberg et al 2008b we present theoretical justifications for this approach as well as a near optimal algorithm our evaluation adds optimizations that improve empirical performance and shows that our algorithms learn orders of magnitude more quickly than previous approaches

geometric inequalities on locally conformally flat manifolds through the study of some elliptic and parabolic fully nonlinear pdes we establish conformal versions of the quermassintegral inequality the sobolev inequality and the moser trudinger inequality for the geometric quantities associated to the schouten tensor on locally conformally flat manifolds

off equilibrium response function in the one dimensional random field ising model a thorough numerical investigation of the slow dynamics in the d 1 random field ising model in the limit of an infinite ferromagnetic coupling is presented in this paper crossovers from the preasymptotic pure regime tothe asymptotic sinai regime are investigated for the average domain size the autocorrelation function and staggered magnetization by switching on an additional small random field at the time t w the linear off equilibrium response function is obtained which displays as well the crossover from the nontrivial behavior of the d 1 pure ising model to the asymptotic behavior where it vanishes identically

non algebraic domain growth for phase ordering dynamics in a random field discusses the three stage numerical exposition of the effects of quenched disorder on phase ordering dynamics the authors study the effects of random fields on domain growth the numerical results indicate that the domain growth law is logarithmic for both the cases with non conserve and conserved order parameter this is compatible with theoretical expectations they also study the dynamical scaling of the structure factor for both the non conserved and conserved cases