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We consider the problem of testing the fit of a discrete sample of items from many categories to the uniform distribution over the categories. As a class of alternative hypotheses, we consider the removal of an $\ell_p$ ball of radius $\epsilon$ around the uniform rate sequence for $p \leq 2$. We deliver a sharp characterization of the asymptotic minimax risk when $\epsilon \to 0$ as the number of samples and number of dimensions go to infinity, for testing based on the occurrences' histogram (number of absent categories, singletons, collisions, ...). For example, for $p=1$ and in the limit of a small expected number of samples $n$ compared to the number of categories $N$ (aka "sub-linear" regime), the minimax risk $R^*_\epsilon$ asymptotes to $2 \bar{\Phi}\left(n \epsilon^2/\sqrt{8N}\right) $, with $\bar{\Phi}(x)$ the normal survival function. Empirical studies over a range of problem parameters show that this estimate is accurate in finite samples, and that our test is significantly better than the chisquared test or a test that only uses collisions. Our analysis is based on the asymptotic normality of histogram ordinates, the equivalence between the minimax setting to a Bayesian one, and the reduction of a multi-dimensional optimization problem to a one-dimensional problem.

Minwise hashing (MinHash) is a standard algorithm widely used in the industry, for large-scale search and learning applications with the binary (0/1) Jaccard similarity. One common use of MinHash is for processing massive n-gram text representations so that practitioners do not have to materialize the original data (which would be prohibitive). Another popular use of MinHash is for building hash tables to enable sub-linear time approximate near neighbor (ANN) search. MinHash has also been used as a tool for building large-scale machine learning systems. The standard implementation of MinHash requires applying $K$ random permutations. In comparison, the method of one permutation hashing (OPH), is an efficient alternative of MinHash which splits the data vectors into $K$ bins and generates hash values within each bin. OPH is substantially more efficient and also more convenient to use. In this paper, we combine the differential privacy (DP) with OPH (as well as MinHash), to propose the DP-OPH framework with three variants: DP-OPH-fix, DP-OPH-re and DP-OPH-rand, depending on which densification strategy is adopted to deal with empty bins in OPH. A detailed roadmap to the algorithm design is presented along with the privacy analysis. An analytical comparison of our proposed DP-OPH methods with the DP minwise hashing (DP-MH) is provided to justify the advantage of DP-OPH. Experiments on similarity search confirm the merits of DP-OPH, and guide the choice of the proper variant in different practical scenarios. Our technique is also extended to bin-wise consistent weighted sampling (BCWS) to develop a new DP algorithm called DP-BCWS for non-binary data. Experiments on classification tasks demonstrate that DP-BCWS is able to achieve excellent utility at around $\epsilon = 5\sim 10$, where $\epsilon$ is the standard parameter in the language of $(\epsilon, \delta)$-DP.

We consider leader election in clique networks, where $n$ nodes are connected by point-to-point communication links. For the synchronous clique under simultaneous wake-up, i.e., where all nodes start executing the algorithm in round $1$, we show a tradeoff between the number of messages and the amount of time. More specifically, we show that any deterministic algorithm with a message complexity of $n f(n)$ requires $\Omega\left(\frac{\log n}{\log f(n)+1}\right)$ rounds, for $f(n) = \Omega(\log n)$. Our result holds even if the node IDs are chosen from a relatively small set of size $\Theta(n\log n)$, as we are able to avoid using Ramsey's theorem. We also give an upper bound that improves over the previously-best tradeoff. Our second contribution for the synchronous clique under simultaneous wake-up is to show that $\Omega(n\log n)$ is in fact a lower bound on the message complexity that holds for any deterministic algorithm with a termination time $T(n)$. We complement this result by giving a simple deterministic algorithm that achieves leader election in sublinear time while sending only $o(n\log n)$ messages, if the ID space is of at most linear size. We also show that Las Vegas algorithms (that never fail) require $\Theta(n)$ messages. For the synchronous clique under adversarial wake-up, we show that $\Omega(n^{3/2})$ is a tight lower bound for randomized $2$-round algorithms. Finally, we turn our attention to the asynchronous clique: Assuming adversarial wake-up, we give a randomized algorithm that achieves a message complexity of $O(n^{1 + 1/k})$ and an asynchronous time complexity of $k+8$. For simultaneous wake-up, we translate the deterministic tradeoff algorithm of Afek and Gafni to the asynchronous model, thus partially answering an open problem they pose.

This paper presents a personalized graph federated learning (PGFL) framework in which distributedly connected servers and their respective edge devices collaboratively learn device or cluster-specific models while maintaining the privacy of every individual device. The proposed approach exploits similarities among different models to provide a more relevant experience for each device, even in situations with diverse data distributions and disproportionate datasets. Furthermore, to ensure a secure and efficient approach to collaborative personalized learning, we study a variant of the PGFL implementation that utilizes differential privacy, specifically zero-concentrated differential privacy, where a noise sequence perturbs model exchanges. Our mathematical analysis shows that the proposed privacy-preserving PGFL algorithm converges to the optimal cluster-specific solution for each cluster in linear time. It also shows that exploiting similarities among clusters leads to an alternative output whose distance to the original solution is bounded, and that this bound can be adjusted by modifying the algorithm's hyperparameters. Further, our analysis shows that the algorithm ensures local differential privacy for all clients in terms of zero-concentrated differential privacy. Finally, the performance of the proposed PGFL algorithm is examined by performing numerical experiments in the context of regression and classification using synthetic data and the MNIST dataset.

We consider the issue of answering unions of conjunctive queries (UCQs) with disjunctive existential rules and mappings. While this issue has already been well studied from a chase perspective, query rewriting within UCQs has hardly been addressed yet. We first propose a sound and complete query rewriting operator, which has the advantage of establishing a tight relationship between a chase step and a rewriting step. The associated breadth-first query rewriting algorithm outputs a minimal UCQ-rewriting when one exists. Second, we show that for any ``truly disjunctive'' nonrecursive rule, there exists a conjunctive query that has no UCQ-rewriting. It follows that the notion of finite unification sets (fus), which denotes sets of existential rules such that any UCQ admits a UCQ-rewriting, seems to have little relevance in this setting. Finally, turning our attention to mappings, we show that the problem of determining whether a UCQ admits a UCQ-rewriting through a disjunctive mapping is undecidable. We conclude with a number of open problems.

The Tucker tensor decomposition is a natural extension of the singular value decomposition (SVD) to multiway data. We propose to accelerate Tucker tensor decomposition algorithms by using randomization and parallelization. We present two algorithms that scale to large data and many processors, significantly reduce both computation and communication cost compared to previous deterministic and randomized approaches, and obtain nearly the same approximation errors. The key idea in our algorithms is to perform randomized sketches with Kronecker-structured random matrices, which reduces computation compared to unstructured matrices and can be implemented using a fundamental tensor computational kernel. We provide probabilistic error analysis of our algorithms and implement a new parallel algorithm for the structured randomized sketch. Our experimental results demonstrate that our combination of randomization and parallelization achieves accurate Tucker decompositions much faster than alternative approaches. We observe up to a 16X speedup over the fastest deterministic parallel implementation on 3D simulation data.

The conventional understanding of adversarial training in generative adversarial networks (GANs) is that the discriminator is trained to estimate a divergence, and the generator learns to minimize this divergence. We argue that despite the fact that many variants of GANs were developed following this paradigm, the current theoretical understanding of GANs and their practical algorithms are inconsistent. In this paper, we leverage Wasserstein gradient flows which characterize the evolution of particles in the sample space, to gain theoretical insights and algorithmic inspiration of GANs. We introduce a unified generative modeling framework - MonoFlow: the particle evolution is rescaled via a monotonically increasing mapping of the log density ratio. Under our framework, adversarial training can be viewed as a procedure first obtaining MonoFlow's vector field via training the discriminator and the generator learns to draw the particle flow defined by the corresponding vector field. We also reveal the fundamental difference between variational divergence minimization and adversarial training. This analysis helps us to identify what types of generator loss functions can lead to the successful training of GANs and suggest that GANs may have more loss designs beyond the literature (e.g., non-saturated loss), as long as they realize MonoFlow. Consistent empirical studies are included to validate the effectiveness of our framework.

In a recent article the authors showed that the radiative Transfer equations with multiple frequencies and scattering can be formulated as a nonlinear integral system. In the present article, the formulation is extended to handle reflective boundary conditions. The fixed point method to solve the system is shown to be monotone. The discretization is done with a $P^1$ Finite Element Method. The convolution integrals are precomputed at every vertices of the mesh and stored in compressed hierarchical matrices, using Partially Pivoted Adaptive Cross-Approximation. Then the fixed point iterations involve only matrix vector products. The method is $O(N\sqrt[3]{N}\ln N)$, with respect to the number of vertices, when everything is smooth. A numerical implementation is proposed and tested on two examples. As there are some analogies with ray tracing the programming is complex.

We describe a measure quantization procedure i.e., an algorithm which finds the best approximation of a target probability law (and more generally signed finite variation measure) by a sum of $Q$ Dirac masses ($Q$ being the quantization parameter). The procedure is implemented by minimizing the statistical distance between the original measure and its quantized version; the distance is built from a negative definite kernel and, if necessary, can be computed on the fly and feed to a stochastic optimization algorithm (such as SGD, Adam, ...). We investigate theoretically the fundamental questions of existence of the optimal measure quantizer and identify what are the required kernel properties that guarantee suitable behavior. We propose two best linear unbiased (BLUE) estimators for the squared statistical distance and use them in an unbiased procedure, called HEMQ, to find the optimal quantization. We test HEMQ on several databases: multi-dimensional Gaussian mixtures, Wiener space cubature, Italian wine cultivars and the MNIST image database. The results indicate that the HEMQ algorithm is robust and versatile and, for the class of Huber-energy kernels, matches the expected intuitive behavior.

The practicality of reinforcement learning algorithms has been limited due to poor scaling with respect to the problem size, as the sample complexity of learning an $\epsilon$-optimal policy is $\tilde{\Omega}\left(|S||A|H^3 / \epsilon^2\right)$ over worst case instances of an MDP with state space $S$, action space $A$, and horizon $H$. We consider a class of MDPs for which the associated optimal $Q^*$ function is low rank, where the latent features are unknown. While one would hope to achieve linear sample complexity in $|S|$ and $|A|$ due to the low rank structure, we show that without imposing further assumptions beyond low rank of $Q^*$, if one is constrained to estimate the $Q$ function using only observations from a subset of entries, there is a worst case instance in which one must incur a sample complexity exponential in the horizon $H$ to learn a near optimal policy. We subsequently show that under stronger low rank structural assumptions, given access to a generative model, Low Rank Monte Carlo Policy Iteration (LR-MCPI) and Low Rank Empirical Value Iteration (LR-EVI) achieve the desired sample complexity of $\tilde{O}\left((|S|+|A|)\mathrm{poly}(d,H)/\epsilon^2\right)$ for a rank $d$ setting, which is minimax optimal with respect to the scaling of $|S|, |A|$, and $\epsilon$. In contrast to literature on linear and low-rank MDPs, we do not require a known feature mapping, our algorithm is computationally simple, and our results hold for long time horizons. Our results provide insights on the minimal low-rank structural assumptions required on the MDP with respect to the transition kernel versus the optimal action-value function.