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In Statistical Relational Artificial Intelligence, a branch of AI and machine learning which combines the logical and statistical schools of AI, one uses the concept {\em para\-metrized probabilistic graphical model (PPGM)} to model (conditional) dependencies between random variables and to make probabilistic inferences about events on a space of "possible worlds". The set of possible worlds with underlying domain $D$ (a set of objects) can be represented by the set $\mathbf{W}_D$ of all first-order structures (for a suitable signature) with domain $D$. Using a formal logic we can describe events on $\mathbf{W}_D$. By combining a logic and a PPGM we can also define a probability distribution $\mathbb{P}_D$ on $\mathbf{W}_D$ and use it to compute the probability of an event. We consider a logic, denoted $PLA$, with truth values in the unit interval, which uses aggregation functions, such as arithmetic mean, geometric mean, maximum and minimum instead of quantifiers. However we face the problem of computational efficiency and this problem is an obstacle to the wider use of methods from Statistical Relational AI in practical applications. We address this problem by proving that the described probability will, under certain assumptions on the PPGM and the sentence $\varphi$, converge as the size of $D$ tends to infinity. The convergence result is obtained by showing that every formula $\varphi(x_1, \ldots, x_k)$ which contains only "admissible" aggregation functions (e.g. arithmetic and geometric mean, max and min) is asymptotically equivalent to a formula $\psi(x_1, \ldots, x_k)$ without aggregation functions.

Anomaly detection among a large number of processes arises in many applications ranging from dynamic spectrum access to cybersecurity. In such problems one can often obtain noisy observations aggregated from a chosen subset of processes that conforms to a tree structure. The distribution of these observations, based on which the presence of anomalies is detected, may be only partially known. This gives rise to the need for a search strategy designed to account for both the sample complexity and the detection accuracy, as well as cope with statistical models that are known only up to some missing parameters. In this work we propose a sequential search strategy using two variations of the Generalized Local Likelihood Ratio statistic. Our proposed Hierarchical Dynamic Search (HDS) strategy is shown to be order-optimal with respect to the size of the search space and asymptotically optimal with respect to the detection accuracy. An explicit upper bound on the error probability of HDS is established for the finite sample regime. Extensive experiments are conducted, demonstrating the performance gains of HDS over existing methods.

Although robust learning and local differential privacy are both widely studied fields of research, combining the two settings is just starting to be explored. We consider the problem of estimating a discrete distribution in total variation from $n$ contaminated data batches under a local differential privacy constraint. A fraction $1-\epsilon$ of the batches contain $k$ i.i.d. samples drawn from a discrete distribution $p$ over $d$ elements. To protect the users' privacy, each of the samples is privatized using an $\alpha$-locally differentially private mechanism. The remaining $\epsilon n $ batches are an adversarial contamination. The minimax rate of estimation under contamination alone, with no privacy, is known to be $\epsilon/\sqrt{k}+\sqrt{d/kn}$, up to a $\sqrt{\log(1/\epsilon)}$ factor. Under the privacy constraint alone, the minimax rate of estimation is $\sqrt{d^2/\alpha^2 kn}$. We show that combining the two constraints leads to a minimax estimation rate of $\epsilon\sqrt{d/\alpha^2 k}+\sqrt{d^2/\alpha^2 kn}$ up to a $\sqrt{\log(1/\epsilon)}$ factor, larger than the sum of the two separate rates. We provide a polynomial-time algorithm achieving this bound, as well as a matching information theoretic lower bound.

Many existing algorithms for streaming geometric data analysis have been plagued by exponential dependencies in the space complexity, which are undesirable for processing high-dimensional data sets. In particular, once $d\geq\log n$, there are no known non-trivial streaming algorithms for problems such as maintaining convex hulls and L\"owner-John ellipsoids of $n$ points, despite a long line of work in streaming computational geometry since [AHV04]. We simultaneously improve these results to $\mathrm{poly}(d,\log n)$ bits of space by trading off with a $\mathrm{poly}(d,\log n)$ factor distortion. We achieve these results in a unified manner, by designing the first streaming algorithm for maintaining a coreset for $\ell_\infty$ subspace embeddings with $\mathrm{poly}(d,\log n)$ space and $\mathrm{poly}(d,\log n)$ distortion. Our algorithm also gives similar guarantees in the \emph{online coreset} model. Along the way, we sharpen results for online numerical linear algebra by replacing a log condition number dependence with a $\log n$ dependence, answering a question of [BDM+20]. Our techniques provide a novel connection between leverage scores, a fundamental object in numerical linear algebra, and computational geometry. For $\ell_p$ subspace embeddings, we give nearly optimal trade-offs between space and distortion for one-pass streaming algorithms. For instance, we give a deterministic coreset using $O(d^2\log n)$ space and $O((d\log n)^{1/2-1/p})$ distortion for $p>2$, whereas previous deterministic algorithms incurred a $\mathrm{poly}(n)$ factor in the space or the distortion [CDW18]. Our techniques have implications in the offline setting, where we give optimal trade-offs between the space complexity and distortion of subspace sketch data structures. To do this, we give an elementary proof of a "change of density" theorem of [LT80] and make it algorithmic.

Privacy protection is an essential issue in personalized news recommendation, and federated learning can potentially mitigate the privacy concern by training personalized news recommendation models over decentralized user data.For a theoretical privacy guarantee, differential privacy is necessary. However, applying differential privacy to federated recommendation training and serving conventionally suffers from the unsatisfactory trade-off between privacy and utility due to the high-dimensional characteristics of model gradients and hidden representations. In addition, there is no formal privacy guarantee for both training and serving in federated recommendation. In this paper, we propose a unified federated news recommendation method for effective and privacy-preserving model training and online serving with differential privacy guarantees. We first clarify the notion of differential privacy over users' behavior data for both model training and online serving in the federated recommendation scenario. Next, we propose a privacy-preserving online serving mechanism under this definition with differentially private user interest decomposition. More specifically, it decomposes the high-dimensional and privacy-sensitive user embedding into a combination of public basic vectors and adds noise to the combination coefficients. In this way, it can avoid the dimension curse and improve the utility by reducing the required noise intensity for differential privacy. Besides, we design a federated recommendation model training method with differential privacy, which can avoid the dimension-dependent noise for large models via label permutation and differentially private attention modules. Experiments on real-world news recommendation datasets validate the effectiveness of our method in achieving a good trade-off between privacy protection and utility for federated news recommendations.

We study the acceleration of the Local Polynomial Interpolation-based Gradient Descent method (LPI-GD) recently proposed for the approximate solution of empirical risk minimization problems (ERM). We focus on loss functions that are strongly convex and smooth with condition number $\sigma$. We additionally assume the loss function is $\eta$-H\"older continuous with respect to the data. The oracle complexity of LPI-GD is $\tilde{O}\left(\sigma m^d \log(1/\varepsilon)\right)$ for a desired accuracy $\varepsilon$, where $d$ is the dimension of the parameter space, and $m$ is the cardinality of an approximation grid. The factor $m^d$ can be shown to scale as $O((1/\varepsilon)^{d/2\eta})$. LPI-GD has been shown to have better oracle complexity than gradient descent (GD) and stochastic gradient descent (SGD) for certain parameter regimes. We propose two accelerated methods for the ERM problem based on LPI-GD and show an oracle complexity of $\tilde{O}\left(\sqrt{\sigma} m^d \log(1/\varepsilon)\right)$. Moreover, we provide the first empirical study on local polynomial interpolation-based gradient methods and corroborate that LPI-GD has better performance than GD and SGD in some scenarios, and the proposed methods achieve acceleration.

Vector Perturbation Precoding (VPP) can speed up downlink data transmissions in Large and Massive Multi-User MIMO systems but is known to be NP-hard. While there are several algorithms in the literature for VPP under total power constraint, they are not applicable for VPP under per-antenna power constraint. This paper proposes a novel, parallel tree search algorithm for VPP under per-antenna power constraint, called \emph{\textbf{TreeStep}}, to find good quality solutions to the VPP problem with practical computational complexity. We show that our method can provide huge performance gain over simple linear precoding like Regularised Zero Forcing. We evaluate TreeStep for several large MIMO~($16\times16$ and $24\times24$) and massive MIMO~($16\times32$ and $24\times 48$) and demonstrate that TreeStep outperforms the popular polynomial-time VPP algorithm, the Fixed Complexity Sphere Encoder, by achieving the extremely low BER of $10^{-6}$ at a much lower SNR.

Federated learning with differential privacy, or private federated learning, provides a strategy to train machine learning models while respecting users' privacy. However, differential privacy can disproportionately degrade the performance of the models on under-represented groups, as these parts of the distribution are difficult to learn in the presence of noise. Existing approaches for enforcing fairness in machine learning models have considered the centralized setting, in which the algorithm has access to the users' data. This paper introduces an algorithm to enforce group fairness in private federated learning, where users' data does not leave their devices. First, the paper extends the modified method of differential multipliers to empirical risk minimization with fairness constraints, thus providing an algorithm to enforce fairness in the central setting. Then, this algorithm is extended to the private federated learning setting. The proposed algorithm, \texttt{FPFL}, is tested on a federated version of the Adult dataset and an "unfair" version of the FEMNIST dataset. The experiments on these datasets show how private federated learning accentuates unfairness in the trained models, and how FPFL is able to mitigate such unfairness.

It is shown, with two sets of indicators that separately load on two distinct factors, independent of one another conditional on the past, that if it is the case that at least one of the factors causally affects the other, then, in many settings, the process will converge to a factor model in which a single factor will suffice to capture the covariance structure among the indicators. Factor analysis with one wave of data can then not distinguish between factor models with a single factor versus those with two factors that are causally related. Therefore, unless causal relations between factors can be ruled out a priori, alleged empirical evidence from one-wave factor analysis for a single factor still leaves open the possibilities of a single factor or of two factors that causally affect one another. The implications for interpreting the factor structure of psychological scales, such as self-report scales for anxiety and depression, or for happiness and purpose, are discussed. The results are further illustrated through simulations to gain insight into the practical implications of the results in more realistic settings prior to the convergence of the processes. Some further generalizations to an arbitrary number of underlying factors are noted.

We provide a new analysis of local SGD, removing unnecessary assumptions and elaborating on the difference between two data regimes: identical and heterogeneous. In both cases, we improve the existing theory and provide values of the optimal stepsize and optimal number of local iterations. Our bounds are based on a new notion of variance that is specific to local SGD methods with different data. The tightness of our results is guaranteed by recovering known statements when we plug $H=1$, where $H$ is the number of local steps. The empirical evidence further validates the severe impact of data heterogeneity on the performance of local SGD.