Abstract: Multiarmed bandit problem is a typical example of a dilemma between exploration and exploitation in reinforcement learning. This problem is expressed as a model of a gambler playing a slot machine with multiple arms. We study stochastic bandit problem where each arm has a reward distribution supported in a known bounded interval, e.g. [0, 1]. In this model, Auer et al. (2002) proposed practical policies called UCB and derived finite-time regret of UCB policies. However, policies achieving the asymptotic bound given by Burnetas and Katehakis (1996) have been unknown for the model. We propose Deterministic Minimum Empirical Divergence (DMED) policy and prove that DMED achieves the asymptotic bound. Furthermore, the index used in DMED for choosing an arm can be computed easily by a convex optimization technique. Although we do not derive a finite-time regret, we confirm by simulations that DMED achieves a regret close to the asymptotic bound in finite time.

Abstract: We prove an optimal $\Omega(n)$ lower bound on the randomized communication complexity of the much-studied Gap-Hamming-Distance problem. As a consequence, we obtain essentially optimal multi-pass space lower bounds in the data stream model for a number of fundamental problems, including the estimation of frequency moments. The Gap-Hamming-Distance problem is a communication problem, wherein Alice and Bob receive $n$-bit strings $x$ and $y$, respectively. They are promised that the Hamming distance between $x$ and $y$ is either at least $n/2+\sqrt{n}$ or at most $n/2-\sqrt{n}$, and their goal is to decide which of these is the case. Since the formal presentation of the problem by Indyk and Woodruff (FOCS, 2003), it had been conjectured that the naive protocol, which uses $n$ bits of communication, is asymptotically optimal. The conjecture was shown to be true in several special cases, e.g., when the communication is deterministic, or when the number of rounds of communication is limited. The proof of our aforementioned result, which settles this conjecture fully, is based on a new geometric statement regarding correlations in Gaussian space, related to a result of C. Borell (1985). To prove this geometric statement, we show that random projections of not-too-small sets in Gaussian space are close to a mixture of translated normal variables.

Abstract: We present a novel distributed algorithm for the maximal independent set problem (This is an extended journal version of Schneider and Wattenhofer in Twenty-seventh annual ACM SIGACT-SIGOPS symposium on principles of distributed computing, 2008). On bounded-independence graphs our deterministic algorithm finishes in O(log* n) time, n being the number of nodes. In light of Linial’s Ω(log* n) lower bound our algorithm is asymptotically optimal. Furthermore, it solves the connected dominating set problem for unit disk graphs in O(log* n) time, exponentially faster than the state-of-the-art algorithm. With a new extension our algorithm also computes a δ + 1 coloring and a maximal matching in O(log* n) time, where δ is the maximum degree of the graph.

Abstract: In 1985, for detecting a change in distribution, Pollak introduced a specific minimax performance metric and a randomized version of the Shiryaev–Roberts procedure where the zero initial condition is replaced by a random variable sampled from the quasi-stationary distribution of the Shiryaev–Roberts statistic. Pollak proved that this procedure is third-order asymptotically optimal as the mean time to false alarm becomes large. The question of whether Pollak’s procedure is strictly minimax for any false alarm rate has been open for more than two decades, and there were several attempts to prove this strict optimality. In this paper, we provide a counterexample which shows that Pollak’s procedure is not optimal and that there is a strictly optimal procedure which is nothing but the Shiryaev–Roberts procedure that starts with a specially designed deterministic point.

Abstract: We prove asymptotically optimal bounds on the Gaussian noise sensitivity of degree-d polynomial threshold functions. These bounds translate into optimal bounds on the Gaussian surface area of such functions, and therefore imply new bounds on the running time of agnostic learning algorithms.

Abstract: Sigma-Delta modulation is a popular method for analog-to-digital conversion of bandlimited signals that employs coarse quantization coupled with oversampling. The standard mathematical model for the error analysis of the method measures the performance of a given scheme by the rate at which the associated reconstruction error decays as a function of the oversampling ratio $\lambda$. It was recently shown that exponential accuracy of the form $O(2^{-r\lambda})$ can be achieved by appropriate one-bit Sigma-Delta modulation schemes. By general information-entropy arguments $r$ must be less than 1. The current best known value for $r$ is approximately 0.088. The schemes that were designed to achieve this accuracy employ the "greedy" quantization rule coupled with feedback filters that fall into a class we call "minimally supported". In this paper, we study the minimization problem that corresponds to optimizing the error decay rate for this class of feedback filters. We solve a relaxed version of this problem exactly and provide explicit asymptotics of the solutions. From these relaxed solutions, we find asymptotically optimal solutions of the original problem, which improve the best known exponential error decay rate to $r \approx 0.102$. Our method draws from the theory of orthogonal polynomials; in particular, it relates the optimal filters to the zero sets of Chebyshev polynomials of the second kind.

Abstract: We study the consistency of parameter estimators in adaptive designs generated by a one-step ahead D-optimal algorithm. We show that when the design space is finite, under mild conditions the least-squares estimator in a nonlinear regression model is strongly consistent and the information matrix evaluated at the current estimated value of the parameters strongly converges to the D-optimal matrix for the unknown true value of the parameters. A similar property is shown to hold for maximum-likelihood estimation in Bernoulli trials (dose-response experiments). Some examples are presented.

Abstract: Path diversity works by setting up multiple parallel connections between the endpoints using the topological path redundancy of the network. In this paper, forward error correction (FEC) is applied across multiple independent paths to enhance the end-to-end reliability. We prove that the probability of irrecoverable loss (PE) decays exponentially with the number of paths. Furthermore, the rate allocation (RA) problem across independent paths is studied. Our objective is to find the optimal RA, i.e., the allocation that minimizes PE. The RA problem is solved for a large number of paths. Moreover, it is shown that in such asymptotically optimal RA, each path is assigned a positive rate iff its quality is above a certain threshold. Finally, using memoization technique, a heuristic suboptimal algorithm with polynomial runtime is proposed for RA over a finite number of paths. This algorithm converges to the asymptotically optimal RA when the number of paths is large. For a practical number of paths, the simulation results demonstrate the close-to-optimal performance of the proposed algorithm .

Abstract: We propose a new protocol solving the fundamental problem of disseminating a piece of information to all members of a group of n players. It builds upon the classical randomized rumor spreading protocol and several extensions. The main achievements are the following: Our protocol spreads the rumor to all other nodes in the asymptotically optimal time of (1 + o(1)) \log_2 n. The whole process can be implemented in a way such that only O(n f(n)) calls are made, where f(n)= \omega(1) can be arbitrary. In contrast to other protocols suggested in the literature, our algorithm only uses push operations, i.e., only informed nodes take active actions in the network. To the best of our knowledge, this is the first randomized push algorithm that achieves an asymptotically optimal running time.

Abstract: Conditional density estimation in a parametric regression setting, where the problem is to estimate a parametric density of the response given the predictor, is a classical and prominent topic in regression analysis. This article explores this problem in a nonparametric setting where no assumption about shape of an underlying conditional density is made. For the first time in the literature, it is proved that there exists a nonparametric data-driven estimator that matches performance of an oracle which: (i) knows the underlying conditional density, (ii) adapts to an unknown design of predictors, (iii) performs a dimension reduction if the response does not depend on the predictor, (iv) is minimax over a vast set of anisotropic bivariate function classes. All these results are established via an oracle inequality which is on par with ones known in the univariate density estimation literature. Further, the asymptotically optimal estimator is tested on an interesting actuarial example which explores a relationship between credit scoring and premium for basic auto-insurance for 54 undergraduate college students.

Abstract: We analyze the performance limits of data dissemination with multichannel, single radio sensors under random packet loss. We formulate the problem of minimizing the average delay of data dissemination as a stochastic shortest path problem and show that, for an arbitrary topology network, an optimal control policy can be found in a finite number of steps, using value iteration or Dijkstra's algorithm. However, the computational complexity of this solution is generally prohibitive. We thus focus on two special classes of network topologies of practical interest, namely single-hop clusters and multihop cluster chains. For these topologies, we derive the structure of policies that achieve an asymptotically optimal average delay, in networks with large number of nodes. Our analysis reveals that a single radio in each node suffices to achieve performance gain directly proportional to the total number of channels available. Through simulation, we show that the derived policies perform close to optimal even for networks with small and moderate numbers of nodes and can be implemented with limited overhead.

Abstract: Publisher Summary Random processes with second-order temporal statistical structures are classically divided into two broad classes—wide-sense stationary (WSS) processes and nonstationary processes. WSS processes are characterized by the property that the statistical correlation between any two samples thereof depends only on the time-difference between the sampling instants. Such processes exhibit many significant properties in the time and frequency domains, which can be conveniently exploited. The second-order statistics (SOS)-based separation approaches for WSS sources can roughly be divided into two categories—approaches exploiting the special structure of the correlation matrices through (approximate) joint diagonalization and approaches based on the principle of maximum likelihood (ML). ML estimation is based on more than SOS in general, but under the assumption of Gaussian sources, the ML estimate takes a relatively simple form and is indeed based on SOS alone. Using the Gaussianity assumption, it is also possible to apply optimal weighting to the joint-diagonalization based approach, thus obtaining estimates which are asymptotically optimal, and are thus asymptotically equivalent to the ML estimate.

Abstract: An optical orthogonal signature pattern code (OOSPC) finds application in transmitting 2-D images through multicore fiber in code-division multiple-access (CDMA) communication systems. Observing a one-to-one correspondence between an OOSPC and a certain combinatorial subject, called a packing design, we present a construction of optimal OOSPCs with weight 4 and maximum collision parameter 2, which generalizes a well-known Kohler construction of optimal optical orthogonal codes (OOC) with weight 4 and maximum collision parameter 2. Using this new construction enables one to obtain infinitely many optimal OOSPCs, whose existence was previously unknown. We prove that for a multiple n of 4, there exists no optimal OOSPC of size 6 n with weight 4 and maximum collision parameter 2, together with a report which shows a gap between optimal OOCs and optimal OOSPCs when 6 and n are not coprime. We also present a recursive construction of OOSPCs which are asymptotically optimal with respect to the Johnson bound. As a by-product, we obtain an asymptotically optimal (m, n, 4, 2)-OOSPC for all positive integers m and n.

Abstract: While training and estimation for Pattern Recognition (PR) have been extensively studied, the question of achieving these when the resources are both limited and constrained is relatively open. This is the focus of this paper. We consider the problem of allocating limited sampling resources in a "real-time" manner, with the explicit purpose of estimating multiple binomial proportions (the extension of these results to non-binomial proportions is, in our opinion, rather straightforward). More specifically, the user is presented with `n' training sets of data points, S 1,S 2,?,S n , where the set S i has N i points drawn from two classes {? 1,? 2}. A random sample in set S i belongs to ? 1 with probability u i and to ? 2 with probability 1?u i , with {u i }, i=1,2,?n, being the quantities to be learnt. The problem is both interesting and non-trivial because while both n and each N i are large, the number of samples that can be drawn is bounded by a constant, c. A web-related problem which is based on this model (Snaprud et al., The Accessibility for All Conference, 2003) is intriguing because the sampling resources can only be allocated optimally if the binomial proportions are already known. Further, no non-automaton solution has ever been reported if these proportions are unknown and must be sampled. Using the general LA philosophy as a paradigm to tackle this real-life problem, our scheme improves a current solution in an online manner, through a series of informed guesses which move towards the optimal solution. We solve the problem by first modelling it as a Stochastic Non-linear Fractional Knapsack Problem. We then present a completely new on-line Learning Automata (LA) system, namely, the Hierarchy of Twofold Resource Allocation Automata (H-TRAA), whose primitive component is a Twofold Resource Allocation Automaton (TRAA), both of which are asymptotically optimal. Furthermore, we demonstrate empirically that the H-TRAA provides orders of magnitude faster convergence compared to the Learning Automata Knapsack Game (LAKG) which represents the state-of-the-art. Finally, in contrast to the LAKG, the H-TRAA scales sub-linearly. Based on these results, we believe that the H-TRAA has also tremendous potential to handle demanding real-world applications, particularly those dealing with the world wide web.

Abstract: This paper is about the design and analysis of an index-assignment (IA)-based multiple-description coding scheme for the n-channel asymmetric case. We use entropy constrained lattice vector quantization and restrict attention to simple reconstruction functions, which are given by the inverse IA function when all descriptions are received or otherwise by a weighted average of the received descriptions. We consider smooth sources with finite differential entropy rate and MSE fidelity criterion. As in previous designs, our construction is based on nested lattices which are combined through a single IA function. The results are exact under high-resolution conditions and asymptotically as the nesting ratios of the lattices approach infinity. For any n, the design is asymptotically optimal within the class of IA-based schemes. Moreover, in the case of two descriptions and finite lattice vector dimensions greater than one, the performance is strictly better than that of existing designs. In the case of three descriptions, we show that in the limit of large lattice vector dimensions, points on the inner bound of Pradhan can be achieved. Furthermore, for three descriptions and finite lattice vector dimensions, we show that the IA-based approach yields, in the symmetric case, a smaller rate loss than the recently proposed source-splitting approach.

Abstract: A double transformation kernel density estimator that is suitable for heavy-tailed distributions is presented. Using a double transformation, an asymptotically optimal bandwidth parameter can be calculated when minimizing the expression of the asymptotic mean integrated squared error of the transformed variable. Simulation results are presented showing that this approach performs better than existing alternatives. An application to insurance claim cost data is included.

Abstract: In this article we present an I/O-efficient algorithm for the batched (off-line) version of the union-find problem. Given any sequence of N union and find operations, where each union operation joins two distinct sets, our algorithm uses O(SORT(N)) = O(N/B logM/BN/B) I/Os, where M is the memory size and B is the disk block size. This bound is asymptotically optimal in the worst case. If there are union operations that join a set with itself, our algorithm uses O(SORT(N) + MST(N)) I/Os, where MST(N) is the number of I/Os needed to compute the minimum spanning tree of a graph with N edges. We also describe a simple and practical O(SORT(N) log(N/M))-I/O algorithm for this problem, which we have implemented.We are interested in the union-find problem because of its applications in terrain analysis. A terrain can be abstracted as a height function defined over R2, and many problems that deal with such functions require a union-find data structure. With the emergence of modern mapping technologies, huge amount of elevation data is being generated that is too large to fit in memory, thus I/O-efficient algorithms are needed to process this data efficiently. In this article, we study two terrain-analysis problems that benefit from a union-find data structure: (i) computing topological persistence and (ii) constructing the contour tree. We give the first O(SORT(N))-I/O algorithms for these two problems, assuming that the input terrain is represented as a triangular mesh with N vertices.

Abstract: We construct unsatisfiable k-CNF formulas where every clause has k distinct literals and every variable appears in at most (2/e + o(1))2^{k}/k clauses. The Lopsided Local Lemma, applied with assignment of random values according to counterintuitive probabilities, shows that our result is asymptotically best possible. The determination of this extremal function is particularly important as it represents the value where the k-SAT problem exhibits its complexity hardness jump: from having every instance being a YES-instance it becomes NP-hard just by allowing each variable to occur in one more clause. The asymptotics of other related extremal functions are also determined. Let l(k) denote the maximum number, such that every k-CNF formula with each clause containing k distinct literals and each clause having a common variable with at most l(k) other clauses, is satisfiable. We establish that the lower bound on l(k) obtained from the Local Lemma is asymptotically optimal, i.e., l(k) = (1/e + o(1))2^{k}. The construction of our unsatisfiable CNF-formulas is based on the binary tree approach of [16] and thus the constructed formulas are in the class MU(1)of minimal unsatisfiable formulas having one more clauses than variables. To obtain the asymptotically optimal binary trees we consider a continuous approximation of the problem, set up a differential equation and estimate its solution. The trees are then obtained through a discretization of this solution. The binary trees constructed also give asymptotically precise answers for seemingly unrelated problems like the European Tenure Game introduced by Doerr [9] and the search problem with bounded number of consecutive lies, considered in a problem of the 2012 IMO contest. As yet another consequence we slightly improve two bounds related to the Neighborhood Conjecture of Beck.

Abstract: The Local Lemma is a fundamental tool of probabilistic combinatorics and theoretical computer science, yet there are hardly any natural problems known where it provides an asymptotically tight answer. The main theme of our paper is to identify several of these problems, among them a couple of widely studied extremal functions related to certain restricted versions of the k-SAT problem, where the Local Lemma does give essentially optimal answers. As our main contribution, we construct unsatisfiable k-CNF formulas where every clause has k distinct literals and every variable appears in at most (2/e + o(1))*2^k/k clauses. The Lopsided Local Lemma shows that this is asymptotically best possible. The determination of this extremal function is particularly important as it represents the value where the corresponding k-SAT problem exhibits a complexity hardness jump: from having every instance being a YES-instance it becomes NP-hard just by allowing each variable to occur in one more clause. The construction of our unsatisfiable CNF-formulas is based on the binary tree approach of [16] and thus the constructed formulas are in the class MU(1) of minimal unsatisfiable formulas having one more clauses than variables. The main novelty of our approach here comes in setting up an appropriate continuous approximation of the problem. This leads us to a differential equation, the solution of which we are able to estimate. The asymptotically optimal binary trees are then obtained through a discretization of this solution. The importance of the binary trees constructed is also underlined by their appearance in many other scenarios. In particular, they give asymptotically precise answers for seemingly unrelated problems like the European Tenure Game introduced by Doerr [9] and a search problem allowing a limited number of consecutive lies.

Abstract: We propose a class of distribution-free rank-based tests for the null hypothesis of a unit root. This class is indexed by the choice of a reference density g, which needs not coincide with the unknown actual innovation density f. The validity of these tests, in terms of exact finite sample size, is guaranteed, irrespective of the actual underlying density, by distribution-freeness. Those tests are locally and asymptotically optimal under a particular asymptotic scheme, for which we provide a complete analysis of asymptotic relative efficiencies. Rather than asymptotic optimality, however, we emphasize finite-sample performances, and show that our rank-based tests perform significantly better than the traditional Dickey-Fuller tests.

Abstract: We investigate the relation between the time complexity and the space complexity for the rendezvous problem with k agents in asynchronous tree networks. The rendezvous problem requires that all the agents in the system have to meet at a single node within finite time. First, we consider asymptotically time-optimal algorithms and investigate the minimum memory requirement per agent for asymptotically time-optimal algorithms. We show that there exists a tree with n nodes in which Ω(n) bits of memory per agent is required to solve the rendezvous problem in O(n) time (asymptotically time-optimal). Then, we present an asymptotically time-optimal rendezvous algorithm. This algorithm can be executed if each agent has O(n) bits of memory. From this lower/upper bound, this algorithm is asymptotically space-optimal on the condition that the time complexity is asymptotically optimal. Finally, we consider asymptotically space-optimal algorithms while allowing slowdown in time required to achieve rendezvous. We present an asymptotically space-optimal algorithm that each agent uses only O(logn) bits of memory. This algorithm terminates in O(Δn8) time where Δ is the maximum degree of the tree.

Abstract: We describe quantization designs which lead to asymptotically and order optimal functional quantizers for Gaussian processes in a Hilbert space setting. Regular variation of the eigenvalues of the covariance operator plays a crucial role to achieve these rates. For the development of a constructive quantization scheme we rely on the knowledge of the eigenvectors of the covariance operator in order to transform the problem into a finite dimensional quantization problem of normal distributions.

Abstract: This paper is about the design and analysis of an index-assignment (IA) based multiple-description coding scheme for the n-channel asymmetric case. We use entropy constrained lattice vector quantization and restrict attention to simple reconstruction functions, which are given by the inverse IA function when all descriptions are received or otherwise by a weighted average of the received descriptions. We consider smooth sources with finite differential entropy rate and MSE fidelity criterion. As in previous designs, our construction is based on nested lattices which are combined through a single IA function. The results are exact under high-resolution conditions and asymptotically as the nesting ratios of the lattices approach infinity. For any n, the design is asymptotically optimal within the class of IA-based schemes. Moreover, in the case of two descriptions and finite lattice vector dimensions greater than one, the performance is strictly better than that of existing designs. In the case of three descriptions, we show that in the limit of large lattice vector dimensions, points on the inner bound of Pradhan et al. can be achieved. Furthermore, for three descriptions and finite lattice vector dimensions, we show that the IA-based approach yields, in the symmetric case, a smaller rate loss than the recently proposed source-splitting approach.

Abstract: We apply large deviations theory to study asymptotic performance of running consensus distributed detection in sensor networks. Running consensus is a stochastic approximation type algorithm, recently proposed. At each time step k, the state at each sensor is updated by a local averaging of the sensor's own state and the states of its neighbors (consensus) and by accounting for the new observations (innovation).We assume Gaussian, spatially correlated observations. We allow the underlying network be time varying, provided that the graph that collects the union of links that are online at least once over a finite time window is connected. This paper shows through large deviations that, under stated assumptions on the network connectivity and sensors' observations, the running consensus detection asymptotically approaches in performance the optimal centralized detection. That is, the Bayes probability of detection error (with the running consensus detector) decays exponentially to zero as k → ∞ at the Chernoff information rate-the best achievable rate of the asymptotically optimal centralized detector

Abstract: Recent results concerning asymptotic Bayes-optimality under sparsity (ABOS) of multiple testing procedures are extended to fairly generally distributed effect sizes under the alternative. An asymptotic framework is considered where both the number of tests m and the sample size m go to infinity, while the fraction p of true alternatives converges to zero. It is shown that under mild restrictions on the loss function nontrivial asymptotic inference is possible only if n increases to infinity at least at the rate of log m. Based on this assumption precise conditions are given under which the Bonferroni correction with nominal Family Wise Error Rate (FWER) level alpha and the Benjamini- Hochberg procedure (BH) at FDR level alpha are asymptotically optimal. When n is proportional to log m then alpha can remain fixed, whereas when n increases to infinity at a quicker rate, then alpha has to converge to zero roughly like n^(-1/2). Under these conditions the Bonferroni correction is ABOS in case of extreme sparsity, while BH adapts well to the unknown level of sparsity. In the second part of this article these optimality results are carried over to model selection in the context of multiple regression with orthogonal regressors. Several modifications of Bayesian Information Criterion are considered, controlling either FWER or FDR, and conditions are provided under which these selection criteria are ABOS. Finally the performance of these criteria is examined in a brief simulation study.

Abstract: We analyse some QR decomposition algorithms, and show that the I/O complexity of the tile based algorithm is asymptotically the same as that of matrix multiplication. This algorithm, we show, performs the best when the tile size is chosen so that exactly one tile fits in the main memory. We propose a constant factor improvement, as well as a new recursive cache oblivious algorithm with the same asymptotic I/O complexity. We design Hessenberg, tridiagonal, and bidiagonal reductions that use banded intermediate forms, and perform only asymptotically optimal numbers of I/Os; these are the first I/O optimal algorithms for these problems. In particular, we show that known slab based algorithms for two sided reductions all have suboptimal asymptotic I/O performances, even though they have been reported to do better than the traditional algorithms on the basis of empirical evidence. We propose new tile based variants of multishift QR and QZ algorithms that under certain conditions on the number of shifts, have better seek and I/O complexities than all known variants. We show that techniques like rescheduling of computational steps, appropriate choosing of the blocking parameters and incorporating of more matrix-matrix operations, can be used to improve the I/O and seek complexities of matrix computations.