convergence of vector quantizers with applications to optimal quantization suppose that a sequence of probability distribution functions f n converges weakly to a distribution function f does the sequence of optimal quantizers for the f n s converge to an optimal quantizer for f if so do the respective distortions converge to the optimal distortion for f sufficient conditions are given to guarantee the convergence for both scalar and vector quantizers with a general class of distortion measures these results are used to prove the existence of minimum rth power distortion vector quantizers and the convergence of a proposed algorithm for constructing optimal quantizers

on weierstrass semigroups and the redundancy of improved geometric goppa codes improved geometric goppa codes have a smaller redundancy and the same bound on the minimum distance as ordinary algebraic geometry codes for an asymptotically good sequence of function fields we give a formula for the redundancy

robust linear least squares regression we consider the problem of robustly predicting as well as the best linear combination of d given functions in least squares regression and variants of this problem including constraints on the parameters of the linear combination for the ridge estimator and the ordinary least squares estimator and their variants we provide new risk bounds of order d n without logarithmic factor unlike some standard results where n is the size of the training data we also provide a new estimator with better deviations in presence of heavy tailed noise it is based on truncating differences of losses in a min max framework and satisfies a d n risk bound both in expectation and in deviations the key common surprising factor of these results is the absence of exponential moment condition on the output distribution while achieving exponential deviations all risk bounds are obtained through a pac bayesian analysis on truncated differences of losses experimental results strongly back up our truncated min max estimator

the order bound on the minimum distance of the one point codes associated to the garcia stichtenoth tower garcia and stichtenoth discovered a tower of function fields that meets the drinfeld vladut bound on the ratio of the number of points to the genus for this tower pellikaan stichtenoth and torres derived a recursive description of the weierstrass semigroups associated to a tower of points on the associated curves in this correspondence a nonrecursive description of the semigroups is given and from this the enumeration of each of the semigroups is derived as well as its inverse this enables us to find an explicit formula for the order feng rao bound on the minimum distance of the associated one point codes

the multi armed bandit problem with covariates we consider a multi armed bandit problem in a setting where each arm produces a noisy reward realization which depends on an observable random covariate as opposed to the traditional static multi armed bandit problem this setting allows for dynamically changing rewards that better describe applications where side information is available we adopt a nonparametric model where the expected rewards are smooth functions of the covariate and where the hardness of the problem is captured by a margin parameter to maximize the expected cumulative reward we introduce a policy called adaptively binned successive elimination abse that adaptively decomposes the global problem into suitably localized static bandit problems this policy constructs an adaptive partition using a variant of the successive elimination se policy our results include sharper regret bounds for the se policy in a static bandit problem and minimax optimal regret bounds for the abse policy in the dynamic problem

on the generalized feng rao numbers of numerical semigroups generated by intervals we give some general results concerning the computation of the generalized feng rao numbers of numerical semigroups in the case of a numerical semigroup generated by an interval a formula for the rth feng rao number is obtained

robust forecast combinations abstract forecast outliers commonly occur in economic financial and other areas of forecasting applications in the literature of forecast combinations there have been only a few studies exploring how to deal with outliers in this work we propose two robust combining methods based on the after algorithm yang 2004a our approach utilizes robust loss functions in order to reduce the influence of outliers oracle inequalities for certain versions of these methods are obtained which show that the combined forecasts automatically perform as well as the best individual among the pool of original forecasts systematic simulations and data examples show that the robust methods outperform the after algorithm when outliers are likely to occur and perform on par with after when there are no outliers comparison of the robust afters with some commonly used combining methods also shows their potential advantages

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

quantum query complexity of entropy estimation estimation of shannon and renyi entropies of unknown discrete distributions is a fundamental problem in statistical property testing in this paper we give the first quantum algorithms for estimating alpha renyi entropies shannon entropy being 1 renyi entropy in particular we demonstrate a quadratic quantum speedup for shannon entropy estimation and a generic quantum speedup for alpha renyi entropy estimation for all alpha geq 0 values including tight bounds for the shannon entropy the hartley entropy alpha 0 and the collision entropy alpha 2 we also provide quantum upper bounds for estimating min entropy alpha infty as well as the kullback leibler divergence we complement our results with quantum lower bounds on alpha renyi entropy estimation for all alpha geq 0 values our approach is inspired by the pioneering work of bravyi harrow and hassidim bhh however with many new technical ingredients 1 we improve the error dependence of the bhh framework by a fine tuned error analysis together with montanaro s approach to estimating the expected output of quantum subroutines for alpha 0 1 2 we develop a procedure similar to cooling schedules in simulated annealing for general alpha geq 0 and 3 in the cases of integer alpha geq 2 and alpha infty we reduce the entropy estimation problem to the alpha distinctness and lceil log n rceil distinctness problems respectively

comment on tests of scaling and universality of the distributions of trade size and share volume evidence from three distinct markets in a recent publication plerou and stanley phys rev e 76 046109 2007 use the meerschaert scheffler estimator to verify the inverse half cubic law of trade size distributions we show that this procedure systematically underestimates these tail exponents

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

partially finite programming in l_1 and the existence of maximum entropy estimates best entropy estimation is a technique that has been widely applied in many areas of science it consists of estimating an unknown density from some of its moments by maximizing some measure of the entropy of the estimate this problem can be modelled as a partially finite convex program with an integrable function as the variable a complete duality and existence theory is developed for this problem and for an associated extended problem which allows singular measure theoretic solutions this theory explains the appearance of singular components observed in the literature when the burg entropy is used it also provides a unified treatment of existence conditions when the burg boltzmann shannon or some other entropy is used as the objective some examples are discussed

a note on numerical semigroups this correspondence is a short extension to the previous article bras amoroacutes 2004 in that work some results were given on one point codes related to numerical semigroups one of the crucial concepts in the discussion was the so called nu sequence of a semigroup this sequence has been used in the literature to derive bounds on the minimum distance as well as for defining improvements on the dimension of existing codes it was proven in that work that the nu sequence of a semigroup uniquely determines it here this result is extended to another object related to a semigroup the oplus operation this operation has also been important in the literature for defining other classes of improved codes it is also proven here that although the infinite set of values in the nu sequence resp the oplus values uniquely determines the associated semigroup no finite part of it can determine it because it is shared by infinitely many semigroups in that reference the proof of the fact that the nu sequence of a numerical semigroup uniquely determines it is constructive the result here presented shows that however that construction can not be performed as an algorithm with finite input

asymptotics for kernel estimate of sliced inverse regression to explore nonlinear structures hidden in high dimensional data and to estimate the effective dimension reduction directions in multivariate nonparametric regression li and duan proposed the sliced inverse regression sir method which is simple to use in this paper the asymptotic properties of the kernel estimate of sliced inverse regression are investigated it turns out that regardless of the kernel function the asymptotic distribution remains the same for a wide range of smoothing parameters

network generalized hamming weight in this work we extend the notion of generalized hamming weight for classical linear block codes to linear network codes by introducing the network generalized hamming weight nghw for a given network with respect to a fixed linear network code the basic properties of nghw are studied we further show that nghw can be used as a tool to characterize the security performance of a linear network code on a wiretap network we also introduce the notion of network maximum distance separable code nmds code by extending the notion of maximum distance separable code in classical algebraic coding theory we prove that nmds codes play an important role in minimizing the information that an eavesdropper can obtain from the network

near optimal mean estimators with respect to general norms we study the problem of estimating the mean of a random vector in mathbb r d based on an i i d sample when the accuracy of the estimator is measured by a general norm on mathbb r d we construct an estimator that depends on the norm that achieves an essentially optimal accuracy confidence tradeoff under the only assumption that the random vector has a well defined covariance matrix at the heart of the argument is the construction of a uniform median of means estimator in a class of real valued functions

the weight hierarchy of some reducible cyclic codes the generalized hamming weights ghws of linear codes are fundamental parameters the knowledge of which is of great interest in many applications however to determine the ghws of linear codes is difficult in general in this paper we study the ghws for a family of reducible cyclic codes and obtain the complete weight hierarchy in several cases this is achieved by extending the idea of yang et al into higher dimension and by employing some interesting combinatorial arguments it shall be noted that these cyclic codes may have arbitrary number of nonzeros

non asymptotic analysis of a new bandit algorithm for semi bounded rewards in this paper we consider a stochastic multiarmed bandit problem it is known in this problem that deterministic minimum empirical divergence dmed policy achieves the asymptotic theoretical bound for the model where each reward distribution is supported in a known bounded interval say 0 1 however the regret bound of dmed is described in an asymptotic form and the performance in finite time has been unknown we modify this policy and derive a finite time regret bound for the new policy indexed minimum empirical divergence imed by refining large deviation probabilities to a simple nonasymptotic form further the refined analysis reveals that the finite time regret bound is valid even in the case that the reward is not bounded from below therefore our finite time result applies to the case that the minimum reward that is the maximum loss is unknown or unbounded we also present some simulation results which shows that imed much improves dmed and performs competitively to other state of the art policies

the minimum distance of codes in an array coming from telescopic semigroups the concept of an error correcting array gives a new bound on the minimum distance of linear codes and a decoding algorithm which decodes up to half this bound this gives a unified point of view which explains several improvements on the minimum distance of algebraic geometric codes moreover it is explained in terms of linear algebra and the theory of semigroups only

reply to comment on tests of scaling and universality of the distributions of trade size and share volume evidence from three distinct markets analyzing trade by trade data for three distinct markets we showed that the cumulative distributions of trade size display power law tails pqxxq with exponents q in the levy stable domain qq 2 moreover we reported that the exponent values are consistent for all stocks irrespective of stock specific variables such as market capitalization industry sector or the specific market where the stock is traded our conclusions were based on using two distinct estimation methods racz et al now propose that one of the estimators we used has slow convergence for a pure power law particularly as tail exponents approach the boundary q 2 we examine the robustness of our results to specific estimation method by additional analysis using five distinct techniques to estimate q we find results that are fully consistent with those we had reported providing compelling evidence that our conclusions hold regardless of estimation procedure

sliced inverse regression for dimension reduction abstract modern advances in computing power have greatly widened scientists scope in gathering and investigating information from many variables information which might have been ignored in the past yet to effectively scan a large pool of variables is not an easy task although our ability to interact with data has been much enhanced by recent innovations in dynamic graphics in this article we propose a novel data analytic tool sliced inverse regression sir for reducing the dimension of the input variable x without going through any parametric or nonparametric model fitting process this method explores the simplicity of the inverse view of regression that is instead of regressing the univariate output variable y against the multivariate x we regress x against y forward regression and inverse regression are connected by a theorem that motivates this method the theoretical properties of sir are investigated under a model of the form y f 1 x k x e where the k s are the unknown

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

new multi parametric analytical approximations of exponential distribution with power law tails for new cars sells and other applications abstract a multi parametric family of exponential distributions with various power law tails is introduced and is shown to describe adequately the known distributions of incomes and wealth as well as the recently measured distributions of new car sales the three or four parametric families are characterized by effective temperature in the exponential part the power exponent in the power law asymptotic part the coefficient for the transition between the above two parts and the starting value if it is not equal to zero since the new car sales distributions are found to correspond to known distributions of incomes the latter may be inferred from the former

exploration exploitation tradeoff using variance estimates in multi armed bandits algorithms based on upper confidence bounds for balancing exploration and exploitation are gaining popularity since they are easy to implement efficient and effective this paper considers a variant of the basic algorithm for the stochastic multi armed bandit problem that takes into account the empirical variance of the different arms in earlier experimental works such algorithms were found to outperform the competing algorithms we provide the first analysis of the expected regret for such algorithms as expected our results show that the algorithm that uses the variance estimates has a major advantage over its alternatives that do not use such estimates provided that the variances of the payoffs of the suboptimal arms are low we also prove that the regret concentrates only at a polynomial rate this holds for all the upper confidence bound based algorithms and for all bandit problems except those special ones where with probability one the payoff obtained by pulling the optimal arm is larger than the expected payoff for the second best arm hence although upper confidence bound bandit algorithms achieve logarithmic expected regret rates they might not be suitable for a risk averse decision maker we illustrate some of the results by computer simulations

learning without concentration we obtain sharp bounds on the estimation error of the empirical risk minimization procedure performed in a convex class and with respect to the squared loss without assuming that class members and the target are bounded functions or have rapidly decaying tails rather than resorting to a concentration based argument the method used here relies on a small ball assumption and thus holds for classes consisting of heavy tailed functions and for heavy tailed targets the resulting estimates scale correctly with the noise level of the problem and when applied to the classical bounded scenario always improve the known bounds

individual convergence rates in empirical vector quantizer design we consider the rate of convergence of the expected distortion redundancy of empirically optimal vector quantizers earlier results show that the mean squared distortion of an empirically optimal quantizer designed from n independent and identically distributed i i d source samples converges uniformly to the optimum at a rate of o 1 spl radic n and that this rate is sharp in the minimax sense we prove that for any fixed distribution supported on a given finite set the convergence rate is o 1 n faster than the minimax lower bound where the corresponding constant depends on the source distribution for more general source distributions we provide conditions implying a little bit worse o logn n rate of convergence although these conditions in general are hard to verify we show that sources with continuous densities satisfying certain regularity properties similar to the ones of pollard that were used to prove a central limit theorem for the code points of the empirically optimal quantizers are included in the scope of this result in particular scalar distributions with strictly log concave densities with bounded support such as the truncated gaussian distribution satisfy these conditions

model space of economic events abstract a method for constructing the model or virtual space of economic events when economic objects can be considered as material ones is suggested we describe change of share rates in time at stock markets as the potential difference of attracted bodies in time in this virtual space each share of each enterprise is displayed by a single particle with a unit charge it is shown that the random value of potential difference at the origin of coordinates measured at a definite time interval has the probability density coinciding with the known distribution of levy flights or levy walks a distribution of alteration in time of the standard and poor index value obtained by mantegna and stanley they shown that it is the levy walks distribution too mantegna and stanley nature 376 1995 46 is used for determination of the introduced potential dependence on coordinates in the model space a simple phenomenological model of interaction potential is introduced the potential law of each particle turns out to be closed to r 2 14 in the minimum possible three dimensional model space this model permits calculation of time of random potential correlations at a certain point of the model space these correlations could characterize the time period of making a decision by an investor at stock exchange it is shown that this time is notably shorter in unstable periods 1987 a microscopical model of interaction in the virtual space is also discussed

algebraic geometry codes the theory of error correcting codes derived from curves in an algebraic geometry was initiated by the work of goppa as generalizations of bose chaudhuri hocquenghem bch reed solomon rs and goppa codes the development of the theory has received intense consideration since that time and the purpose of the paper is to review this work elements of the theory of algebraic curves at a level sufficient to understand the code constructions and decoding algorithms are introduced code constructions from particular classes of curves including the klein quartic elliptic and hyperelliptic curves and hermitian curves are presented decoding algorithms for these classes of codes and others are considered the construction of classes of asymptotically good codes using modular curves is also discussed

sub gaussian mean estimators we discuss the possibilities and limitations of estimating the mean of a real valued random variable from independent and identically distributed observations from a non asymptotic point of view in particular we define estimators with a sub gaussian behavior even for certain heavy tailed distributions we also prove various impossibility results for mean estimators

list decoding from erasures bounds and code constructions we consider the problem of list decoding from erasures we establish lower and upper bounds on the rate of a binary linear code that can be list decoded with list size l when up to a fraction p of its symbols are adversarially erased such bounds already exist in the literature albeit under the label of generalized hamming weights and we make their connection to list decoding from erasures explicit our bounds show that in the limit of large l the rate of such a code approaches the capacity 1 p of the erasure channel such nicely list decodable codes are then used as inner codes in a suitable concatenation scheme to give a uniformly constructive family of asymptotically good binary linear codes of rate spl omega spl epsiv sup 2 log 1 spl epsiv that can be efficiently list decoded using lists of size o 1 spl epsiv when an adversarially chosen 1 spl epsiv fraction of symbols are erased for arbitrary spl epsiv 0 this improves previous results in this vein which achieved a rate of spl omega spl epsiv sup 3 log 1 spl epsiv

adaptive minimax regression estimation over sparse l q hulls given a dictionary of mn predictors in a random design regression setting with n observations we construct estimators that target the best performance among all the linear combinations of the predictors under a sparse lq norm 0 q 1 constraint on the linear coefficients besides identifying the optimal rates of convergence our universal aggregation strategies by model mixing achieve the optimal rates simultaneously over the full range of 0 q 1 for any mn and without knowledge of the lq norm of the best linear coefficients to represent the regression function to allow model misspecification our upper bound results are obtained in a framework of aggregation of estimates a striking feature is that no specific relationship among the predictors is needed to achieve the upper rates of convergence hence permitting basically arbitrary correlations between the predictors therefore whatever the true regression function assumed to be uniformly bounded our estimators automatically exploit any sparse representation of the regression function if any to the best extent possible within the lq constrained linear combinations for any 0 q 1 a sparse approximation result in the lq hulls turns out to be crucial to adaptively achieve minimax rate optimal aggregation it precisely characterizes the number of terms needed to achieve a prescribed accuracy of approximation to the best linear combination in an lq hull for 0 q 1 it offers the insight that the minimax rate of lq aggregation is basically determined by an effective model size which is a sparsity index that depends on q mn n and the lq norm bound in an easily interpretable way based on a classical model selection theory that deals with a large number of models

analytical representation of stock and stock indexes returns non gaussian random walks with various jump laws abstract empirical data for stock and stock indexes returns that is available for international markets as well as for the russian stock market is introduced and discussed random walk process with a specific law of an elementary independent increment jump in some random walk space is proposed for a proper basic description an analytical representation of a random walk distribution law is derived from a general conception of stochastic elementary jumps this representation covers classic s chandrasekhar s results for gauss random walks with limited jumps and jumps having distributions with all moments as well as p levy flights with jump distributions without moments starting from the second one finally it describes truncated levy random walks for jumps having distributions with at least the second moment the natural generalization of truncated levy walks is done the random walk process is compared with known empirical data for the above discussed return fluctuations of international assets and stock indexes there is a very good agreement of the elaborated theory with empirical data

acute semigroups the order bound on the minimum distance and the feng rao improvements we introduce a new class of numerical semigroups which we call the class of acute semigroups and we prove that they generalize symmetric and pseudosymmetric numerical semigroups arf numerical semigroups and the semigroups generated by an interval for a numerical semigroup spl lambda spl lambda sub 0 i spl nu sub i 1 for all i spl ges m we prove that the only numerical semigroups for which the sequence spl nu sub i is always nondecreasing are ordinary numerical semigroups furthermore we show that a semigroup can be uniquely determined by its sequence spl nu sub i

exponential distribution of financial returns at mesoscopic time lags a new stylized fact we study the probability distribution of stock returns at mesoscopic time lags return horizons ranging from about an hour to about a month while at shorter microscopic time lags the distribution has power law tails for mesoscopic times the bulk of the distribution more than 99 of the probability follows an exponential law the slope of the exponential function is determined by the variance of returns which increases proportionally to the time lag at longer times the exponential law continuously evolves into gaussian distribution the exponential to gaussian crossover is well described by the analytical solution of the heston model with stochastic volatility r 2004 elsevier b v all rights reserved

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

strong consistency of k means clustering a random sample is divided into the k clusters that minimise the within cluster sum of squares conditions are found that ensure the almost sure convergence as the sample size increases of the set of means of the k clusters the result is proved for a more general clustering criterion

truncated levy distribution of sp500 stock index fluctuations distribution of one share fluctuations in a model space abstract truncated levy distributions of sp500 stock index fluctuations mantegna stanley nature 376 1995 46 are obtained in the formerly introduced model romanovsky physica a 265 1999 264 for stock market a one body random kinematics in this space corresponds to the one shares fluctuations on stock exchange the statistics of such kinematics is investigated and the asymptotic distribution of one share fluctuations was obtained all results are compared with the known experimental data mantegna stanley nature 376 1995 46 plerou et al phys rev e 60 1999 6519 and 5306 gopikrishnan et al eur j phys b 1998 138

levy flights from a continuous time process l evy flight dynamics can stem from simple random walks in a system whose operational time number of steps n typically grows superlinearly with physical time t thus this process is a kind of continuous time random walk ctrw dual to the typical scher montroll model in which n grows sublinearly with t models in which l evy flights emerge due to a temporal subordination allow one easily to discuss the response of a random walker to a weak outer force which is shown to be nonlinear on the other hand the relaxation of an ensemble of such walkers in a harmonic potential follows a simple exponential pattern and leads to a normal boltzmann distribution mixed models describing normal ctrw s in superlinear operational time and l evy flights under the operational time of subdiffusive ctrw s lead to a paradoxical diffusive behavior similar to the one found in transport on polymer chains the relaxation to the boltzmann distribution in such models is slow and asymptotically follows a power law

a central limit theorem for k means clustering a set of n points in euclidean space is partitioned into the k groups that minimize the within groups sum of squares under the assumption that the n points come from independent sampling on a fixed distribution conditions are found to assure asymptotic normality of the vector of means of the k groups the method of proof makes novel application of a functional central limit theorem for empirical processes a generalization of donsker s theorem due to dudley

stochastic volatility of financial markets as the fluctuating rate of trading an empirical study we present an empirical study of the subordination hypothesis for a stochastic time series of a stock price the fluctuating rate of trading is identified with the stochastic variance of the stock price as in the continuous time random walk ctrw framework the probability distribution of the stock price changes log returns for a given number of trades n is found to be approximately gaussian the probability distribution of n for a given time interval dt is nonpoissonian and has an exponential tail for large n and a sharp cutoff for small n combining these two distributions produces a non trivial distribution of log returns for a given time interval dt which has exponential tails and a gaussian central part in agreement with empirical observations r 2007 elsevier b v all rights reserved

kullback leibler upper confidence bounds for optimal sequential allocation we consider optimal sequential allocation in the context of the so called stochastic multi armed bandit model we describe a generic index policy in the sense of gittins 1979 based on upper confidence bounds of the arm payoffs computed using the kullback leibler divergence we consider two classes of distributions for which instances of this general idea are analyzed the kl ucb algorithm is designed for one parameter exponential families and the empirical kl ucb algorithm for bounded and finitely supported distributions our main contribution is a unified finite time analysis of the regret of these algorithms that asymptotically matches the lower bounds of lai and robbins 1985 and burnetas and katehakis 1996 respectively we also investigate the behavior of these algorithms when used with general bounded rewards showing in particular that they provide significant improvements over the state of the art

improved minimax bounds on the test and training distortion of empirically designed vector quantizers it has been shown by earlier results that the minimax expected test distortion redundancy of empirical vector quantizers with three or more levels designed from n independent and identically distributed i i d data points is at least omega 1 radicn for the class of distributions on a bounded set in this correspondence a much simpler construction and proof for this are given with much better constants there are similar bounds for the training distortion of the empirically optimal vector quantizer with three or more levels these rates however do not hold for a one level quantizer here the two level quantizer case is clarified showing that it already shares the behavior of the general case given that the minimax bounds are proved using a construction that involves discrete distributions one suspects that for the class of distributions with uniformly bounded continuous densities the expected distortion redundancy might decrease as o 1 radicn uniformly it is shown as well that this is not so proving that the lower bound for the expected test distortion remains true for these subclasses

scaling and correlation in financial time series we discuss the results of three recent phenomenological studies focussed on understanding the distinctive statistical properties of nancial time series i the probability distribution of stock price uctuations stock price uctuations occur in all magnitudes in analogy to earthquakes from tiny uctuations to very drastic events such as the crash of 19 october 1987 sometimes referred to as black monday the distribution of price uctuations decays with a power law tail well outside the levy stable regime and describes uctuations that dier by as much as 8 orders of magnitude in addition this distribution preserves its functional form for uctuations on time scales that dier by 3 orders of magnitude from 1 min up to approximately 10 days ii correlations in nancial time series while price uctuations themselves have rapidly decaying correlations the magnitude of uctuations measured by either the absolute value or the square of the price uctuations has correlations that decay as a power law persisting for several months iii volatility and trading activity we quantify the relation between trading activity measured by the number of transactions nt and the price change gt for a given stock over a time interval t t t we nd that nt displays long range power law correlations in time which leads to the interpretation that the long range correlations previously found for jgtj are connected to those of nt c 2000 elsevier science b v all rights reserved

empirical risk minimization for heavy tailed losses the purpose of this paper is to discuss empirical risk minimization when the losses are not necessarily bounded and may have a distribution with heavy tails in such situations usual empirical averages may fail to provide reliable estimates and empirical risk minimization may provide large excess risk however some robust mean estimators proposed in the literature may be used to replace empirical means in this paper we investigate empirical risk minimization based on a robust estimate proposed by catoni we develop performance bounds based on chaining arguments tailored to catoni s mean estimator

instance optimal geometric algorithms we prove the existence of an algorithm a for computing 2d or 3d convex hulls that is optimal for every point set in the following sense for every sequence of n points and for every algorithm a in a certain class a the running time of a on input is at most a constant factor times the running time of a on the worst possible permutation of for a in fact we can establish a stronger property for every sequence of points and every algorithm a the running time of a on is at most a constant factor times the average running time of a over all permutations of we call algorithms satisfying these properties instance optimal in the order oblivious and random order setting such instance optimal algorithms simultaneously subsume output sensitive algorithms and distribution dependent average case algorithms and all algorithms that do not take advantage of the order of the input or that assume the input are given in a random order the class a under consideration consists of all algorithms in a decision tree model where the tests involve only multilinear functions with a constant number of arguments to establish an instance specific lower bound we deviate from traditional ben or style proofs and adopt a new adversary argument for 2d convex hulls we prove that a version of the well known algorithm by kirkpatrick and seidel 1986 or chan snoeyink and yap 1995 already attains this lower bound for 3d convex hulls we propose a new algorithm we further obtain instance optimal results for a few other standard problems in computational geometry such as maxima in 2d and 3d orthogonal line segment intersection in 2d finding bichromatic l close pairs in 2d offline orthogonal range searching in 2d offline dominance reporting in 2d and 3d offline half space range reporting in 2d and 3d and offline point location in 2d our framework also reveals a connection to distribution sensitive data structures and yields new results as a byproduct for example on online orthogonal range searching in 2d and online half space range reporting in 2d and 3d

contextual bandits with similarity information in a multi armed bandit mab problem an online algorithm makes a sequence of choices in each round it chooses from a time invariant set of alternatives and receives the payoff associated with this alternative while the case of small strategy sets is by now well understood a lot of recent work has focused on mab problems with exponentially or infinitely large strategy sets where one needs to assume extra structure in order to make the problem tractable in particular recent literature considered information on similarity between arms we consider similarity information in the setting of contextual bandits a natural extension of the basic mab problem where before each round an algorithm is given the context a hint about the payoffs in this round contextual bandits are directly motivated by placing advertisements on web pages one of the crucial problems in sponsored search a particularly simple way to represent similarity information in the contextual bandit setting is via a similarity distance between the context arm pairs which bounds from above the difference between the respective expected payoffs prior work on contextual bandits with similarity uses uniform partitions of the similarity space so that each context arm pair is approximated by the closest pair in the partition algorithms based on uniform partitions disregard the structure of the payoffs and the context arrivals which is potentially wasteful we present algorithms that are based on adaptive partitions and take advantage of benign payoffs and context arrivals without sacrificing the worst case performance the central idea is to maintain a finer partition in high payoff regions of the similarity space and in popular regions of the context space our results apply to several other settings e g mab with constrained temporal change slivkins and upfal 2008 and sleeping bandits kleinberg et al 2008a

quantization and the method of k means asymptotic results from the statistical theory of k means clustering are applied to problems of vector quantization the behavior of quantizers constructed from long training sequences of data is analyzed by relating it to the consistency problem for k means

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 second feng rao number for codes coming from inductive semigroups the second feng rao number of every inductive numerical semigroup is explicitly computed this number determines the asymptotical behavior of the order bound for the second hamming weight of one point algebraic geometry codes in particular this result is applied for the codes defined by asymptotically good towers of function fields whose weierstrass semigroups are inductive in addition some properties of inductive numerical semigroups are studied the involved apery sets are computed in a recursive way and some tests to check whether given numerical semigroups are inductive or not are provided

comment on a tuning free robust and efficient approach to high dimensional regression consider the linear model 1 y x 0 e where y y 1 y n r n is a response vector x x 1 x n r n p is a design matrix 0 r p is an unknown coefficient vec

k dimensional coding schemes in hilbert spaces this paper presents a general coding method where data in a hilbert space are represented by finite dimensional coding vectors the method is based on empirical risk minimization within a certain class of linear operators which map the set of coding vectors to the hilbert space two results bounding the expected reconstruction error of the method are derived which highlight the role played by the codebook and the class of linear operators the results are specialized to some cases of practical importance including k means clustering nonnegative matrix factorization and other sparse coding methods

random generation of combinatorial structures from a uniform distribution abstract the class of problems involving the random generation of combinatorial structures from a uniform distribution is considered uniform generation problems are in computational difficulty intermediate between classical existence and counting problems it is shown that exactly uniform generation of efficiently verifiable combinatorial structures is reducible to approximate counting and hence is within the third level of the polynomial hierarchy natural combinatorial problems are presented which exhibit complexity gaps between their existence and generation and between their generation and counting versions it is further shown that for self reducible problems almost uniform generation and randomized approximate counting are inter reducible and hence of similar complexity

bandits with heavy tail the stochastic multiarmed bandit problem is well understood when the reward distributions are sub gaussian in this paper we examine the bandit problem under the weaker assumption that the distributions have moments of order 1 e for some e 0 1 surprisingly moments of order 2 i e finite variance are sufficient to obtain regret bounds of the same order as under sub gaussian reward distributions in order to achieve such regret we define sampling strategies based on refined estimators of the mean such as the truncated empirical mean catoni s m estimator and the median of means estimator we also derive matching lower bounds that also show that the best achievable regret deteriorates when e 1

quantum speedup of monte carlo methods monte carlo methods use random sampling to estimate numerical quantities which are hard to compute deterministically one important example is the use in statistical physics of rapidly mixing markov chains to approximately compute partition functions in this work we describe a quantum algorithm which can accelerate monte carlo methods in a very general setting the algorithm estimates the expected output value of an arbitrary randomized or quantum subroutine with bounded variance achieving a near quadratic speedup over the best possible classical algorithm combining the algorithm with the use of quantum walks gives a quantum speedup of the fastest known classical algorithms with rigorous performance bounds for computing partition functions which use multiple stage markov chain monte carlo techniques the quantum algorithm can also be used to estimate the total variation distance between probability distributions efficiently

on the weight hierarchy of geometric goppa codes the weight hierarchy of a linear code is the set of generalized hamming weights of the code in the paper the authors consider geometric goppa codes and provide a lower bound on their generalized hamming weights similar to goppa s lower bound on their minimum distance in the particular case of hermitian codes exact results on the second and third generalized hamming weights are given for any m except a few cases where m is a parameter that governs the dimension of these codes in many instances the authors are able to provide considerably more information on their generalized hamming weights an upper bound relating the generalized hamming weights of hermitian codes to the pole numbers at a special point on the curve is also provided similar results are given in the case of codes from some subfields of the hermitian function fields which are also maximal finally a nontrivial family of codes is also presented whose weight hierarchy is completely determined

on the performance of clustering in hilbert spaces based on randomly drawn vectors in a separable hilbert space one may construct a k means clustering scheme by minimizing an empirical squared error we investigate the risk of such a clustering scheme defined as the expected squared distance of a random vector x from the set of cluster centers our main result states that for an almost surely bounded the expected excess clustering risk is o 1 n since clustering in high or even infinite dimensional spaces may lead to severe computational problems we examine the properties of a dimension reduction strategy for clustering based on johnson lindenstrauss type random projections our results reflect a tradeoff between accuracy and computational complexity when one uses k means clustering after random projection of the data to a low dimensional space we argue that random projections work better than other simplistic dimension reduction schemes

scaling in the norwegian stock market the main objective of this paper is to investigate the validity of the much used assumptions that stock market returns follow a random walk and are normally distributed for this purpose the concepts of chaos theory and fractals are applied two independent models are used to examine price variations in the norwegian and us stock markets the first model used is the range over standard deviation or r s statistic which tests for persistence or antipersistence in the time series both the norwegian and us stock markets show significant persistence caused by long run memory components in the series in addition an average non periodic cycle of four years is found for the us stock market these results are not consistent with the random walk assumption the second model investigates the distributional scaling behaviour of the high frequency price variations in the norwegian stock market the results show a remarkable constant scaling behaviour between different time intervals this means that there is no intrinsic time scale for the dynamics of stock price variations the relationship can be expressed through a scaling exponent describing the development of the distributions as the time scale changes this description may be important when constructing or improving pricing models such that they coincide more closely with the observed market behaviour the empirical distributions of high frequency price variations for the norwegian stock market is then compared to the levy stable distribution with the relevant scaling exponent found by using the r s and distributional scaling analysis good agreement is found between the levy profile and the empirical distribution for price variations less than 6 standard deviations covering almost three orders of magnitude in the data for probabilities larger than 6 standard deviations there seem to be an exponential fall off from the levy profile in the tails which indicates that the second moment may be finite

an optimal algorithm for monte carlo estimation a typical approach to estimate an unknown quantity mu is to design an experiment that produces a random variable z distributed in 0 1 with e z mu run this experiment independently a number of times and use the average of the outcomes as the estimate in this paper we consider the case when no a priori information about z is known except that is distributed in 0 1 we describe an approximation algorithm cal a cal a which given epsilon and delta when running independent experiments with respect to any z produces an estimate that is within a factor 1 epsilon of mu with probability at least 1 delta we prove that the expected number of experiments run by cal a cal a which depends on z is optimal to within a constant factor for every z

robust estimation of high dimensional covariance and precision matrices high dimensional data are often most plausibly generated from distributions with complex structure and leptokurtosis in some or all components covariance and precision matrices provide a useful summary of such structure yet the performance of popular matrix estimators typically hinges upon a sub gaussianity assumption this paper presents robust matrix estimators whose performance is guaranteed for a much richer class of distributions the proposed estimators under a bounded fourth moment assumption achieve the same minimax convergence rates as do existing methods under a sub gaussianity assumption consistency of the proposed estimators is also established under the weak assumption of bounded 2 e moments for e 0 2 the associated convergence rates depend on e

generalized hamming weights for linear codes motivated by cryptographical applications the algebraic structure of linear codes from a new perspective is studied by viewing the minimum hamming weight as a certain minimum property of one dimensional subcodes a generalized notion of higher dimensional hamming weights is obtained these weights characterize the code performance on the wire tap channel of type ii basic properties of generalized weights are derived the values of these weights for well known classes of codes are determined and lower bounds on code parameters are obtained several open problems are also listed

fractional diffusion equation for a power law truncated levy process abstract truncated levy flights are stochastic processes which display a crossover from a heavy tailed levy behavior to a faster decaying probability distribution function pdf putting less weight on long flights overcomes the divergence of the levy distribution second moment we introduce a fractional generalization of the diffusion equation whose solution defines a process in which a levy flight of exponent is truncated by a power law of exponent 5 a closed form for the characteristic function of the process is derived the pdf of the displacement slowly converges to a gaussian in its central part showing however a power law far tail possible applications are discussed

scaling of the distribution of fluctuations of financial market indices up to more than one month we find that the distributions for dt 4d 1560 min are consistent with a power law asymptotic behavior characterized by an exponent a 3 well outside the stable levy regime 0 a 2 to test the robustness of the s p result we perform a parallel analysis on two other financial market indices database iv contains 3560 daily records of the nikkei index for the 14 year period 1984 1997 and database v contains 4649 daily records of the hang seng index for the 18 year period 1980 1997 we find estimates ofa consistent with those describing the distribution of s p 500 daily returns one possible reason for the scaling of these distributions is the long persistence of the autocorrelation function of the volatility for time scales longer than dt 3 4 d our results are consistent with a slow convergence to gaussian behavior s1063 651x 99 11211 x pacs number s 05 40 fb 05 45 tp 89 90 1n

on the parameters of algebraic geometry codes related to arf semigroups we compute the order or feng rao 1994 bound on the minimum distance of one point algebraic geometry codes c sub spl omega p spl rho sub t q when the weierstrass semigroup at the point q is an arf 91949 semigroup the results developed to that purpose also provide the dimension of the improved geometric goppa codes related to these c sub spl omega p spl rho sub t q

new lower bounds on the generalized hamming weights of ag codes a sharp upper bound for the maximum integer not belonging to an ideal of a numerical semigroup is given and the ideals attaining this bound are characterized then the result is used through the so called feng rao numbers to bound the generalized hamming weights of algebraic geometry codes this is further developed for hermitian codes and the codes on one of the garcia stichtenoth towers as well as for some more general families

randomized allocation with nonparametric estimation for a multi armed bandit problem with covariates we study a multi armed bandit problem in a setting where covariates are available we take a nonparametric approach to estimate the functional relationship between the response reward and the covariates the estimated relationships and appropriate randomization are used to select a good arm to play for a greater expected reward randomization helps balance the tendency to trust the currently most promising arm with further exploration of other arms it is shown that with some familiar nonparametric methods e g histogram the proposed strategy is strongly consistent in the sense that the accumulated reward is asymptotically equivalent to that based on the best arm which depends on the covariates almost surely 1 introduction multi armed bandit problems have been extensively studied in probability and statistics in the past few decades and still receive considerable interest in these and related fields readers are referred to berry and fristedt 1985 and gittins 1989 and the references cited therein for the history many elegant results and applications in clinical trials scheduling and other industrial problems some recent developments have been reported in for example lai and yakowitz 1995 who considered the situation in which observations are dependent berry chen zame heath and shepp 1997 who considered infinitely many arms and auer cesa bianchi freund and schapire 1995 who considered worst case performances in classic bandit problems each arm sequentially generates rewards based on a distribution with some unknown parameters and one needs to sequentially select one arm to play for the maximum expected reward in a majority of the earlier work no auxiliary information beyond the observed rewards was considered when selecting an arm to play exceptions include woodroofe 1979 sarkar 1991 and clayton 1989 who considered one armed bandit problems with covariates the first two papers studied bayesian sequential allocation in non bernoulli bandit models with parametric frameworks and showed that the myopic rule is asymptotically optimal the third studied bernoulli bandit problems

linearly parameterized bandits we consider bandit problems involving a large possibly infinite collection of arms in which the expected reward of each arm is a linear function of an r dimensional random vector z rr where r 2 the objective is to minimize the cumulative regret and bayes risk when the set of arms corresponds to the unit sphere we prove that the regret and bayes risk is of order r t by establishing a lower bound for an arbitrary policy and showing that a matching upper bound is obtained through a policy that alternates between exploration and exploitation phases the phase based policy is also shown to be effective if the set of arms satisfies a strong convexity condition for the case of a general set of arms we describe a near optimal policy whose regret and bayes risk admit upper bounds of the form o r t log3 2t

using confidence bounds for exploitation exploration trade offs we show how a standard tool from statistics namely confidence bounds can be used to elegantly deal with situations which exhibit an exploitation exploration trade off our technique for designing and analyzing algorithms for such situations is general and can be applied when an algorithm has to make exploitation versus exploration decisions based on uncertain information provided by a random process we apply our technique to two models with such an exploitation exploration trade off for the adversarial bandit problem with shifting our new algorithm suffers only o st 1 2 regret with high probability over t trials with s shifts such a regret bound was previously known only in expectation the second model we consider is associative reinforcement learning with linear value functions for this model our technique improves the regret from o t3 4 to o t1 2

on the weight hierarchy of codes coming from semigroups with two generators the weight hierarchy of one point algebraic geometry codes can be estimated by means of the generalized order bounds which are described in terms of a certain weierstrass semigroup the asymptotical behavior of such bounds for r 2 differs from that of the classical feng rao distance r 1 by the so called feng rao numbers this paper is addressed to compute the feng rao numbers for numerical semigroups of embedding dimension two with two generators obtaining a closed simple formula for the general case by using numerical semigroup techniques these involve the computation of the apery set with respect to an integer of the semigroups under consideration the formula obtained is applied to lower bounding the generalized hamming weights improving the bound given by kirfel and pellikaan in terms of the classical feng rao distance we also compare our bound with a modification of the griesmer bound improving this one in many cases

fast learning rates in statistical inference through aggregation we develop minimax optimal risk bounds for the general learning task consisting in predicting as well as the best function in a reference set mathcal g up to the smallest possible additive term called the convergence rate when the reference set is finite and when n denotes the size of the training data we provide minimax convergence rates of the form c frac log mathcal g n v with tight evaluation of the positive constant c and with exact 0 v 1 the latter value depending on the convexity of the loss function and on the level of noise in the output distribution the risk upper bounds are based on a sequential randomized algorithm which at each step concentrates on functions having both low risk and low variance with respect to the previous step prediction function our analysis puts forward the links between the probabilistic and worst case viewpoints and allows to obtain risk bounds unachievable with the standard statistical learning approach one of the key ideas of this work is to use probabilistic inequalities with respect to appropriate gibbs distributions on the prediction function space instead of using them with respect to the distribution generating the data the risk lower bounds are based on refinements of the assouad lemma taking particularly into account the properties of the loss function our key example to illustrate the upper and lower bounds is to consider the lq regression setting for which an exhaustive analysis of the convergence rates is given while q ranges in 1

on semigroups generated by two consecutive integers and improved hermitian codes analysis of the berlekamp massey sakata algorithm for decoding one point codes leads to two methods for improving code rate one method due to feng and rao removes parity checks that may be recovered by their majority voting algorithm the second method is to design the code to correct only those error vectors of a given weight that are also geometrically generic in this work formulae are given for the redundancies of hermitian codes optimized with respect to these criteria as well as the formula for the order bound on the minimum distance the results proceed from an analysis of numerical semigroups generated by two consecutive integers

robust statistical learning with lipschitz and convex loss functions we obtain estimation and excess risk bounds for empirical risk minimizers erm and minmax median of means mom estimators based on loss functions that are both lipschitz and convex results for the erm are derived under weak assumptions on the outputs and subgaussian assumptions on the design as in alquier et al estimation bounds and sharp oracle inequalities of regularized procedures with lipschitz loss functions arxiv 1702 01402 2017 the difference with alquier et al 2017 is that the global bernstein condition of this paper is relaxed here into a local assumption we also obtain estimation and excess risk bounds for minmax mom estimators under similar assumptions on the output and only moment assumptions on the design moreover the dataset may also contains outliers in both inputs and outputs variables without deteriorating the performance of the minmax mom estimators unlike alternatives based on mom s principle lecue and lerasle in ann stat 2017 lugosi and mendelson in jems 2016 the analysis of minmax mom estimators is not based on the small ball assumption sba of koltchinskii and mendelson int math res not imrn 23 12991 13008 2015 in particular the basic example of non parametric statistics where the learning class is the linear span of localized bases that does not satisfy sba saumard in bernoulli 24 3 2176 2203 2018 can now be handled finally minmax mom estimators are analysed in a setting where the local bernstein condition is also dropped out it is shown to achieve excess risk bounds with exponentially large probability under minimal assumptions insuring only the existence of all objects

coordinate independent sparse sufficient dimension reduction and variable selection sufficient dimension reduction sdr in regression which reduces the dimension by replacing original predictors with a minimal set of their linear combinations without loss of information is very helpful when the number of predictors is large the standard sdr methods suffer because the estimated linear combinations usually consist of all original predictors making it difficult to interpret in this paper we propose a unified method coordinate independent sparse estimation cise that can simultaneously achieve sparse sufficient dimension reduction and screen out irrelevant and redundant variables efficiently cise is subspace oriented in the sense that it incorporates a coordinate independent penalty term with a broad series of model based and model free sdr approaches this results in a grassmann manifold optimization problem and a fast algorithm is suggested under mild conditions based on manifold theories and techniques it can be shown that cise would perform asymptotically as well as if the true irrelevant predictors were known which is referred to as the oracle property simulation studies and a real data example demonstrate the effectiveness and efficiency of the proposed approach

a note on hermitian codes over gf q sup 2 properties of codes defined by goppa s algebraic geometric method using hermitian curves are discussed generator and parity check matrices for such codes are derived explicitly results concerning the minimum distance and the weight distribution of the codes are provided

statistical properties of share volume traded in financial markets we quantitatively investigate the ideas behind the often expressed adage it takes volume to move stock prices and study the statistical properties of the number of shares traded q deltat for a given stock in a fixed time interval deltat we analyze transaction data for the largest 1000 stocks for the two year period 1994 95 using a database that records every transaction for all securities in three major us stock markets we find that the distribution p q deltat displays a power law decay and that the time correlations in q deltat display long range persistence further we investigate the relation between q deltat and the number of transactions n deltat in a time interval deltat and find that the long range correlations in q deltat are largely due to those of n deltat our results are consistent with the interpretation that the large equal time correlation previously found between q deltat and the absolute value of price change g deltat related to volatility are largely due to n deltat