In the last decade, significant efforts have been made to reduce the false positive rate of approximate membership checking structures. This has led to the development of new structures such as cuckoo filters and xor filters. Adaptive filters that can react to false positives as they occur to avoid them for future queries to the same elements have also been recently developed. In this paper, we propose a new type of static filters that completely avoid false positives for a given set of negative elements and show how they can be efficiently implemented using xor probing filters. Several constructions of these filters with a false positive free set are proposed that minimize the memory and speed overheads introduced by avoiding false positives. The proposed filters have been extensively evaluated to validate their functionality and show that in many cases both the memory and speed overheads are negligible. We also discuss several use cases to illustrate the potential benefits of the proposed filters in practical applications.
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Statistical depths provide a fundamental generalization of quantiles and medians to data in higher dimensions. This paper proposes a new type of globally defined statistical depth, based upon control theory and eikonal equations, which measures the smallest amount of probability density that has to be passed through in a path to points outside the support of the distribution: for example spatial infinity. This depth is easy to interpret and compute, expressively captures multi-modal behavior, and extends naturally to data that is non-Euclidean. We prove various properties of this depth, and provide discussion of computational considerations. In particular, we demonstrate that this notion of depth is robust under an aproximate isometrically constrained adversarial model, a property which is not enjoyed by the Tukey depth. Finally we give some illustrative examples in the context of two-dimensional mixture models and MNIST.
The popular LSPE($\lambda$) algorithm for policy evaluation is revisited to derive a concentration bound that gives high probability performance guarantees from some time on.
In this paper, we propose and investigate the individually fair $k$-center with outliers (IF$k$CO). In the IF$k$CO, we are given an $n$-sized vertex set in a metric space, as well as integers $k$ and $q$. At most $k$ vertices can be selected as the centers and at most $q$ vertices can be selected as the outliers. The centers are selected to serve all the not-an-outlier (i.e., served) vertices. The so-called individual fairness constraint restricts that every served vertex must have a selected center not too far way. More precisely, it is supposed that there exists at least one center among its $\lceil (n-q) / k \rceil$ closest neighbors for every served vertex. Because every center serves $(n-q) / k$ vertices on the average. The objective is to select centers and outliers, assign every served vertex to some center, so as to minimize the maximum fairness ratio over all served vertices, where the fairness ratio of a vertex is defined as the ratio between its distance with the assigned center and its distance with a $\lceil (n - q )/k \rceil_{\rm th}$ closest neighbor. As our main contribution, a 4-approximation algorithm is presented, based on which we develop an improved algorithm from a practical perspective.
The ever-continuing explosive growth of on-demand content distribution has imposed great pressure on mobile/wireless network infrastructures. To ease congestion in the network and to increase perceived user experience, caching of popular content closer to the end-users can play a significant role and as such this issue has received significant attention over the last few years. Additionally, energy efficiency is treated as a fundamental requirement in the design of next-generation mobile networks. However, there has been little attention to the overlapping area between energy efficiency and network caching especially when considering multipath routing. To this end, this paper proposes an energy-efficient caching with multipath routing support. The proposed scheme provides a joint anchoring of popular content into a set of potential caching nodes with optimized multipath support whilst ensuring a balance between transmission and caching energy cost. The proposed model also considers different content delivery modes, such as multicast and unicast. Two separated Integer-Linear Programming (ILP) models are formulated for each delivery mode. To tackle the curse of dimensionality we then provide a greedy simulated annealing algorithm, which not only reduces the time complexity but also provides a competitive performance. A wide set of numerical investigations reveal that the proposed scheme reduces the energy consumption up to 80% compared with other widely used caching approaches under the premise of network resource limitation. Sensitivity analysis to different parameters is also meticulously discussed in this paper.
$k$-means clustering is a fundamental problem in various disciplines. This problem is nonconvex, and standard algorithms are only guaranteed to find a local optimum. Leveraging the structure of local solutions characterized in [1], we propose a general algorithmic framework for escaping undesirable local solutions and recovering the global solution (or the ground truth). This framework consists of alternating between the following two steps iteratively: (i) detect mis-specified clusters in a local solution and (ii) improve the current local solution by non-local operations. We discuss implementation of these steps, and elucidate how the proposed framework unifies variants of $k$-means algorithm in literature from a geometric perspective. In addition, we introduce two natural extensions of the proposed framework, where the initial number of clusters is misspecified. We provide theoretical justification for our approach, which is corroborated with extensive experiments.
We study the practical consequences of dataset sampling strategies on the ranking performance of recommendation algorithms. Recommender systems are generally trained and evaluated on samples of larger datasets. Samples are often taken in a naive or ad-hoc fashion: e.g. by sampling a dataset randomly or by selecting users or items with many interactions. As we demonstrate, commonly-used data sampling schemes can have significant consequences on algorithm performance. Following this observation, this paper makes three main contributions: (1) characterizing the effect of sampling on algorithm performance, in terms of algorithm and dataset characteristics (e.g. sparsity characteristics, sequential dynamics, etc.); (2) designing SVP-CF, which is a data-specific sampling strategy, that aims to preserve the relative performance of models after sampling, and is especially suited to long-tailed interaction data; and (3) developing an oracle, Data-Genie, which can suggest the sampling scheme that is most likely to preserve model performance for a given dataset. The main benefit of Data-Genie is that it will allow recommender system practitioners to quickly prototype and compare various approaches, while remaining confident that algorithm performance will be preserved, once the algorithm is retrained and deployed on the complete data. Detailed experiments show that using Data-Genie, we can discard upto 5x more data than any sampling strategy with the same level of performance.
We study constrained reinforcement learning (CRL) from a novel perspective by setting constraints directly on state density functions, rather than the value functions considered by previous works. State density has a clear physical and mathematical interpretation, and is able to express a wide variety of constraints such as resource limits and safety requirements. Density constraints can also avoid the time-consuming process of designing and tuning cost functions required by value function-based constraints to encode system specifications. We leverage the duality between density functions and Q functions to develop an effective algorithm to solve the density constrained RL problem optimally and the constrains are guaranteed to be satisfied. We prove that the proposed algorithm converges to a near-optimal solution with a bounded error even when the policy update is imperfect. We use a set of comprehensive experiments to demonstrate the advantages of our approach over state-of-the-art CRL methods, with a wide range of density constrained tasks as well as standard CRL benchmarks such as Safety-Gym.
Many representative graph neural networks, $e.g.$, GPR-GNN and ChebyNet, approximate graph convolutions with graph spectral filters. However, existing work either applies predefined filter weights or learns them without necessary constraints, which may lead to oversimplified or ill-posed filters. To overcome these issues, we propose $\textit{BernNet}$, a novel graph neural network with theoretical support that provides a simple but effective scheme for designing and learning arbitrary graph spectral filters. In particular, for any filter over the normalized Laplacian spectrum of a graph, our BernNet estimates it by an order-$K$ Bernstein polynomial approximation and designs its spectral property by setting the coefficients of the Bernstein basis. Moreover, we can learn the coefficients (and the corresponding filter weights) based on observed graphs and their associated signals and thus achieve the BernNet specialized for the data. Our experiments demonstrate that BernNet can learn arbitrary spectral filters, including complicated band-rejection and comb filters, and it achieves superior performance in real-world graph modeling tasks.
Graph convolution is the core of most Graph Neural Networks (GNNs) and usually approximated by message passing between direct (one-hop) neighbors. In this work, we remove the restriction of using only the direct neighbors by introducing a powerful, yet spatially localized graph convolution: Graph diffusion convolution (GDC). GDC leverages generalized graph diffusion, examples of which are the heat kernel and personalized PageRank. It alleviates the problem of noisy and often arbitrarily defined edges in real graphs. We show that GDC is closely related to spectral-based models and thus combines the strengths of both spatial (message passing) and spectral methods. We demonstrate that replacing message passing with graph diffusion convolution consistently leads to significant performance improvements across a wide range of models on both supervised and unsupervised tasks and a variety of datasets. Furthermore, GDC is not limited to GNNs but can trivially be combined with any graph-based model or algorithm (e.g. spectral clustering) without requiring any changes to the latter or affecting its computational complexity. Our implementation is available online.
Distance metric learning based on triplet loss has been applied with success in a wide range of applications such as face recognition, image retrieval, speaker change detection and recently recommendation with the CML model. However, as we show in this article, CML requires large batches to work reasonably well because of a too simplistic uniform negative sampling strategy for selecting triplets. Due to memory limitations, this makes it difficult to scale in high-dimensional scenarios. To alleviate this problem, we propose here a 2-stage negative sampling strategy which finds triplets that are highly informative for learning. Our strategy allows CML to work effectively in terms of accuracy and popularity bias, even when the batch size is an order of magnitude smaller than what would be needed with the default uniform sampling. We demonstrate the suitability of the proposed strategy for recommendation and exhibit consistent positive results across various datasets.
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