Mobile Context-Aware Recommender Systems can be naturally modelled as an exploration exploitation trade-off (exr exp) problem, where the system has to choose between maximizing its expected rewards dealing with its current knowledge (exploitation) and learning more about the unknown user's preferences to improve its knowledge (exploration). This problem has been addressed by the reinforcement learning community but they do not consider the risk level of the current user's situation, where it may be dangerous to recommend items the user may not desire in her current situation if the risk level is high. We introduce in this paper an algorithm named R-UCB that considers the risk level of the user's situation to adaptively balance between exr and exp. The detailed analysis of the experimental results reveals several important discoveries in the exr exp behaviour.
The first variant of the @math -greedy strategy is what @cite_1 @cite_11 refer to as the @math -beginning strategy. This strategy makes exploration all at once at the beginning. For a given number I of iterations, documents are randomly selected during the @math I first iterations; during the rest (1−- @math )I iterations, the document of highest estimated mean is selected. Another variant of the @math -greedy strategy is what @cite_11 calls the @math -decreasing. In this strategy, the document with the highest estimated mean is always selected except when a random document is selected instead with @math frequency, where @math = @math i , @math @math ]0,1] and @math is the index of the current round. Besides @math -decreasing, four other strategies was presented in @cite_14 .
@cite_14 @cite_1 @cite_11
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The goal of search result diversification is to select a subset of documents from the candidate set to satisfy as many different subtopics as possible. In general, it is a problem of subset selection and selecting an optimal subset of documents is NP-hard. Existing methods usually formalize the problem as ranking the documents with greedy sequential document selection. At each of the ranking position the document that can provide the largest amount of additional information is selected. It is obvious that the greedy selections inevitably produce suboptimal rankings. In this paper we propose to partially alleviate the problem with a Monte Carlo tree search (MCTS) enhanced Markov decision process (MDP), referred to as M @math Div. In M @math Div, the construction of diverse ranking is formalized as an MDP process where each action corresponds to selecting a document for one ranking position. Given an MDP state which consists of the query, selected documents, and candidates, a recurrent neural network is utilized to produce the policy function for guiding the document selection and the value function for predicting the whole ranking quality. The produced raw policy and value are then strengthened with MCTS through exploring the possible rankings at the subsequent positions, achieving a better search policy for decision-making. Experimental results based on the TREC benchmarks showed that M @math Div can significantly outperform the state-of-the-art baselines based on greedy sequential document selection, indicating the effectiveness of the exploratory decision-making mechanism in M @math Div. We investigate the minimization of Newton's functional for the problem of the body of minimal resistance with maximal height @math cite butt in the class of convex developable functions defined in a disc. This class is a natural candidate to find a (non-radial) minimizer in accordance with the results of cite lrp2. We prove that the minimizer in this class has a minimal set in the form of a regular polygon with @math sides centered in the disc, and numerical experiments indicate that the natural number @math is a non-decreasing function of @math . The corresponding functions all achieve a lower value of the functional than the optimal radially symmetric function with the same height @math . We study the performance of a family of randomized parallel coordinate descent methods for minimizing the sum of a nonsmooth and separable convex functions. The problem class includes as a special case L1-regularized L1 regression and the minimization of the exponential loss ("AdaBoost problem"). We assume the input data defining the loss function is contained in a sparse @math matrix @math with at most @math nonzeros in each row. Our methods need @math iterations to find an approximate solution with high probability, where @math is the number of processors and @math for the fastest variant. The notation hides dependence on quantities such as the required accuracy and confidence levels and the distance of the starting iterate from an optimal point. Since @math is a decreasing function of @math , the method needs fewer iterations when more processors are used. Certain variants of our algorithms perform on average only @math arithmetic operations during a single iteration per processor and, because @math decreases when @math does, fewer iterations are needed for sparser problems. We consider document listing on string collections, that is, finding in which strings a given pattern appears. In particular, we focus on repetitive collections: a collection of size @math over alphabet @math is composed of @math copies of a string of size @math , and @math edits are applied on ranges of copies. We introduce the first document listing index with size @math , precisely @math bits, and with useful worst-case time guarantees: Given a pattern of length @math , the index reports the @math strings where it appears in time @math , for any constant @math (and tells in time @math if @math ). Our technique is to augment a range data structure that is commonly used on grammar-based indexes, so that instead of retrieving all the pattern occurrences, it computes useful summaries on them. We show that the idea has independent interest: we introduce the first grammar-based index that, on a text @math with a grammar of size @math , uses @math bits and counts the number of occurrences of a pattern @math in time @math , for any constant @math . We also give the first index using @math bits, where @math is parsed by Lempel-Ziv into @math phrases, counting occurrences in time @math . We study the problem of approximating the commuting-operator value of a two-player non-local game. It is well-known that it is @math -complete to decide whether the classical value of a non-local game is 1 or @math . Furthermore, as long as @math is small enough, this result does not depend on the gap @math . In contrast, a recent result of Fitzsimons, Ji, Vidick, and Yuen shows that the complexity of computing the quantum value grows without bound as the gap @math decreases. In this paper, we show that this also holds for the commuting-operator value of a game. Specifically, in the language of multi-prover interactive proofs, we show that the power of @math (proofs with two provers, one round, completeness probability @math , soundness probability @math , and commuting-operator strategies) can increase without bound as the gap @math gets arbitrarily small. Our results also extend naturally in two ways, to perfect zero-knowledge protocols, and to lower bounds on the complexity of computing the approximately-commuting value of a game. Thus we get lower bounds on the complexity class @math - @math of perfect zero-knowledge multi-prover proofs with approximately-commuting operator strategies, as the gap @math gets arbitrarily small. While we do not know any computable time upper bound on the class @math , a result of the first author and Vidick shows that for @math and @math , the class @math , with constant communication from the provers, is contained in @math . We give a lower bound of @math (ignoring constants inside the function) for this class, which is tight up to polynomial factors assuming the exponential time hypothesis. The problem of finding large average submatrices of a real-valued matrix arises in the exploratory analysis of data from a variety of disciplines, ranging from genomics to social sciences. In this paper we provide a detailed asymptotic analysis of large average submatrices of an @math Gaussian random matrix. The first part of the paper addresses global maxima. For fixed @math we identify the average and the joint distribution of the @math submatrix having largest average value. As a dual result, we establish that the size of the largest square sub-matrix with average bigger than a fixed positive constant is, with high probability, equal to one of two consecutive integers that depend on the threshold and the matrix dimension @math . The second part of the paper addresses local maxima. Specifically we consider submatrices with dominant row and column sums that arise as the local optima of iterative search procedures for large average submatrices. For fixed @math , we identify the limiting average value and joint distribution of a @math submatrix conditioned to be a local maxima. In order to understand the density of such local optima and explain the quick convergence of such iterative procedures, we analyze the number @math of local maxima, beginning with exact asymptotic expressions for the mean and fluctuation behavior of @math . For fixed @math , the mean of @math is @math while the standard deviation is @math . Our principal result is a Gaussian central limit theorem for @math that is based on a new variant of Stein's method. We consider the following generalization of the binary search problem. A search strategy is required to locate an unknown target node @math in a given tree @math . Upon querying a node @math of the tree, the strategy receives as a reply an indication of the connected component of @math containing the target @math . The cost of querying each node is given by a known non-negative weight function, and the considered objective is to minimize the total query cost for a worst-case choice of the target. Designing an optimal strategy for a weighted tree search instance is known to be strongly NP-hard, in contrast to the unweighted variant of the problem which can be solved optimally in linear time. Here, we show that weighted tree search admits a quasi-polynomial time approximation scheme: for any @math , there exists a @math -approximation strategy with a computation time of @math . Thus, the problem is not APX-hard, unless @math . By applying a generic reduction, we obtain as a corollary that the studied problem admits a polynomial-time @math -approximation. This improves previous @math -approximation approaches, where the @math -notation disregards @math -factors. Consider the problem of choosing a string of actions to optimize an objective function that is string submodular. It was shown in previous papers that the greedy strategy, consisting of a string of actions that only locally maximizes the step-wise gain in the objective function, achieves at least a @math -approximation to the optimal strategy. This paper improves this approximation by introducing additional constraints on curvature, namely, total backward curvature, total forward curvature, and elemental forward curvature. We show that if the objective function has total backward curvature @math , then the greedy strategy achieves at least a @math -approximation of the optimal strategy. If the objective function has total forward curvature @math , then the greedy strategy achieves at least a @math -approximation of the optimal strategy. Moreover, we consider a generalization of the diminishing-return property by defining the elemental forward curvature. We also introduce the notion of string-matroid and consider the problem of maximizing the objective function subject to a string-matroid constraint. We investigate two applications of string submodular functions with curvature constraints: 1) choosing a string of actions to maximize the expected fraction of accomplished tasks; and 2) designing a string of measurement matrices such that the information gain is maximized. Let @math @math be a given set of @math string documents of total length @math , our task is to index @math , such that the @math most relevant documents for an online query pattern @math of length @math can be retrieved efficiently. We propose an index of size @math bits and @math query time for the basic relevance metric , where @math is the size (in bits) of a compressed full text index of @math , with @math time for searching a pattern of length @math . We further reduce the space to @math bits, however the query time will be @math , where @math is the alphabet size and @math is any constant. In the generalized Russian cards problem, we have a card deck @math of @math cards and three participants, Alice, Bob, and Cathy, dealt @math , @math , and @math cards, respectively. Once the cards are dealt, Alice and Bob wish to privately communicate their hands to each other via public announcements, without the advantage of a shared secret or public key infrastructure. Cathy should remain ignorant of all but her own cards after Alice and Bob have made their announcements. Notions for Cathy's ignorance in the literature range from Cathy not learning the fate of any individual card with certainty (weak @math -security) to not gaining any probabilistic advantage in guessing the fate of some set of @math cards (perfect @math -security). As we demonstrate, the generalized Russian cards problem has close ties to the field of combinatorial designs, on which we rely heavily, particularly for perfect security notions. Our main result establishes an equivalence between perfectly @math -secure strategies and @math -designs on @math points with block size @math , when announcements are chosen uniformly at random from the set of possible announcements. We also provide construction methods and example solutions, including a construction that yields perfect @math -security against Cathy when @math . We leverage a known combinatorial design to construct a strategy with @math , @math , and @math that is perfectly @math -secure. Finally, we consider a variant of the problem that yields solutions that are easy to construct and optimal with respect to both the number of announcements and level of security achieved. Moreover, this is the first method obtaining weak @math -security that allows Alice to hold an arbitrary number of cards and Cathy to hold a set of @math cards. Alternatively, the construction yields solutions for arbitrary @math , @math and any @math . We consider solving the @math -regularized least-squares ( @math -LS) problem in the context of sparse recovery for applications such as compressed sensing. The standard proximal gradient method, also known as iterative soft-thresholding when applied to this problem, has low computational cost per iteration but a rather slow convergence rate. Nevertheless, when the solution is sparse, it often exhibits fast linear convergence in the final stage. We exploit the local linear convergence using a homotopy continuation strategy, i.e., we solve the @math -LS problem for a sequence of decreasing values of the regularization parameter, and use an approximate solution at the end of each stage to warm start the next stage. Although similar strategies have been studied in the literature, there have been no theoretical analysis of their global iteration complexity. This paper shows that under suitable assumptions for sparse recovery, the proposed homotopy strategy ensures that all iterates along the homotopy sol... In this paper we study general @math l p regularized unconstrained minimization problems. In particular, we derive lower bounds for nonzero entries of the first- and second-order stationary points and hence also of local minimizers of the @math l p minimization problems. We extend some existing iterative reweighted @math l 1 ( @math IRL 1 ) and @math l 2 ( @math IRL 2 ) minimization methods to solve these problems and propose new variants for them in which each subproblem has a closed-form solution. Also, we provide a unified convergence analysis for these methods. In addition, we propose a novel Lipschitz continuous @math ∈ -approximation to @math ? x ? p p . Using this result, we develop new @math IRL 1 methods for the @math l p minimization problems and show that any accumulation point of the sequence generated by these methods is a first-order stationary point, provided that the approximation parameter @math ∈ is below a computable threshold value. This is a remarkable result since all existing iterative reweighted minimization methods require that @math ∈ be dynamically updated and approach zero. Our computational results demonstrate that the new @math IRL 1 method and the new variants generally outperform the existing @math IRL 1 methods (Chen and Zhou in 2012; Foucart and Lai in Appl Comput Harmon Anal 26:395---407, 2009). When graph Ramsey theory is viewed as a game, "Painter" 2-colors the edges of a graph presented by "Builder". Builder wins if every coloring has a monochromatic copy of a fixed graph @math . In the on-line version, iteratively, Builder presents one edge and Painter must color it. Builder must keep the presented graph in a class @math . Builder wins the game @math if a monochromatic copy of @math can be forced. The on-line degree Ramsey number @math is the least @math such that Builder wins @math when @math is the class of graphs with maximum degree at most @math . Our results include: 1) @math if and only if @math is a linear forest or each component lies inside @math . 2) @math , where @math . 3) @math for a tree @math , where @math and @math are two largest vertex degrees. 4) @math , with @math except for finitely many odd values of @math . 5) @math when @math . The lower bounds come from strategies for Painter that color edges red whenever the red graph remains in a specified class. The upper bounds use a result showing that Builder may assume that Painter plays "consistently". This paper presents, to our knowledge, the first study on analyzing mathematical expressions to detect academic plagiarism. We make the following contributions. First, we investigate confirmed cases of plagiarism to categorize the similarities of mathematical content commonly found in plagiarized publications. From this investigation, we derive possible feature selection and feature comparison strategies for developing math-based detection approaches and a ground truth for our experiments. Second, we create a test collection by embedding confirmed cases of plagiarism into the NTCIR-11 MathIR Task dataset, which contains approx. 60 million mathematical expressions in 105,120 documents from arXiv.org. Third, we develop a first math-based detection approach by implementing and evaluating different feature comparison approaches using an open source parallel data processing pipeline built using the Apache Flink framework. The best performing approach identifies all but two of our real-world test cases at the top rank and achieves a mean reciprocal rank of 0.86. The results show that mathematical expressions are promising text-independent features to identify academic plagiarism in large collections. To facilitate future research on math-based plagiarism detection, we make our source code and data available. We consider the problem of selecting the best subset of exactly @math columns from an @math matrix @math . We present and analyze a novel two-stage algorithm that runs in @math time and returns as output an @math matrix @math consisting of exactly @math columns of @math . In the first (randomized) stage, the algorithm randomly selects @math columns according to a judiciously-chosen probability distribution that depends on information in the top- @math right singular subspace of @math . In the second (deterministic) stage, the algorithm applies a deterministic column-selection procedure to select and return exactly @math columns from the set of columns selected in the first stage. Let @math be the @math matrix containing those @math columns, let @math denote the projection matrix onto the span of those columns, and let @math denote the best rank- @math approximation to the matrix @math . Then, we prove that, with probability at least 0.8, @math This Frobenius norm bound is only a factor of @math worse than the best previously existing existential result and is roughly @math better than the best previous algorithmic result for the Frobenius norm version of this Column Subset Selection Problem (CSSP). We also prove that, with probability at least 0.8, @math This spectral norm bound is not directly comparable to the best previously existing bounds for the spectral norm version of this CSSP. Our bound depends on @math , whereas previous results depend on @math ; if these two quantities are comparable, then our bound is asymptotically worse by a @math factor. We consider a system of @math identical server pools and a single dispatcher where tasks arrive as a Poisson process of rate @math . Arriving tasks cannot be queued, and must immediately be assigned to one of the server pools to start execution, or discarded. The execution times are assumed to be exponentially distributed with unit mean, and do not depend on the number of other tasks receiving service. However, the experienced performance (e.g. in terms of received throughput) does degrade with an increasing number of concurrent tasks at the same server pool. The dispatcher therefore aims to evenly distribute the tasks across the various server pools. Specifically, when a task arrives, the dispatcher assigns it to the server pool with the minimum number of tasks among @math randomly selected server pools. This assignment strategy is called the JSQ @math scheme, as it resembles the power-of- @math version of the Join-the-Shortest-Queue (JSQ) policy, and will also be referred to as such in the special case @math . We construct a stochastic coupling to bound the difference in the system occupancy processes between the JSQ policy and a scheme with an arbitrary value of @math . We use the coupling to derive the fluid limit in case @math and @math as @math , along with the associated fixed point. The fluid limit turns out to be insensitive to the exact growth rate of @math , and coincides with that for the JSQ policy. We further leverage the coupling to establish that the diffusion limit corresponds to that for the JSQ policy as well, as long as @math , and characterize the common limiting diffusion process. These results indicate that the JSQ optimality can be preserved at the fluid-level and diffusion-level while reducing the overhead by nearly a factor O( @math ) and O( @math ), respectively. Randomized coordinate descent (RCD) methods are state-of-the-art algorithms for training linear predictors via minimizing regularized empirical risk. When the number of examples ( @math ) is much larger than the number of features ( @math ), a common strategy is to apply RCD to the dual problem. On the other hand, when the number of features is much larger than the number of examples, it makes sense to apply RCD directly to the primal problem. In this paper we provide the first joint study of these two approaches when applied to L2-regularized ERM. First, we show through a rigorous analysis that for dense data, the above intuition is precisely correct. However, we find that for sparse and structured data, primal RCD can significantly outperform dual RCD even if @math , and vice versa, dual RCD can be much faster than primal RCD even if @math . Moreover, we show that, surprisingly, a single sampling strategy minimizes both the (bound on the) number of iterations and the overall expected complexity of RCD. Note that the latter complexity measure also takes into account the average cost of the iterations, which depends on the structure and sparsity of the data, and on the sampling strategy employed. We confirm our theoretical predictions using extensive experiments with both synthetic and real data sets. We analyse a randomized pursuit-evasion game played by two players on a graph, a hunter and a rabbit. Let @math be any connected, undirected graph with @math nodes. The game is played in rounds and in each round both the hunter and the rabbit are located at a node of the graph. Between rounds both the hunter and the rabbit can stay at the current node or move to another node. The hunter is assumed to be restricted to the graph @math : in every round, the hunter can move using at most one edge. For the rabbit we investigate two models: in one model the rabbit is restricted to the same graph as the hunter, and in the other model the rabbit is unrestricted, i.e., it can jump to an arbitrary node in every round.We say that the rabbit is caught as soon as hunter and rabbit are located at the same node in a round. The goal of the hunter is to catch the rabbit in as few rounds as possible, whereas the rabbit aims to maximize the number of rounds until it is caught. Given a randomized hunter strategy for @math , the escape length for that strategy is the worst case expected number of rounds it takes the hunter to catch the rabbit, where the worst case is with regard to all (possibly randomized) rabbit strategies. Our main result is a hunter strategy for general graphs with an escape length of only @math against restricted as well as unrestricted rabbits. This bound is close to optimal since @math is a trivial lower bound on the escape length in both models. Furthermore, we prove that our upper bound is optimal up to constant factors against unrestricted rabbits. We derive the mean squared error convergence rates of kernel density-based plug-in estimators of mutual information measures between two multidimensional random variables @math and @math for two cases: 1) @math and @math are both continuous; 2) @math is continuous and @math is discrete. Using the derived rates, we propose an ensemble estimator of these information measures for the second case by taking a weighted sum of the plug-in estimators with varied bandwidths. The resulting ensemble estimator achieves the @math parametric convergence rate when the conditional densities of the continuous variables are sufficiently smooth. To the best of our knowledge, this is the first nonparametric mutual information estimator known to achieve the parametric convergence rate for this case, which frequently arises in applications (e.g. variable selection in classification). The estimator is simple to implement as it uses the solution to an offline convex optimization problem and simple plug-in estimators. A central limit theorem is also derived for the ensemble estimator. Ensemble estimators that achieve the parametric rate are also derived for the first case ( @math and @math are both continuous) and another case 3) @math and @math may have any mixture of discrete and continuous components. We shall show that if the restricted isometry constant (RIC) @math of the measurement matrix @math satisfies @math then the greedy algorithm Orthogonal Matching Pursuit(OMP) will succeed. That is, OMP can recover every @math -sparse signal @math in @math iterations from @math . Moreover, we shall show the upper bound of RIC is sharp in the following sense. For any given @math , we shall construct a matrix @math with the RIC @math such that OMP may not recover some @math -sparse signal @math in @math iterations.