Federated learning (FL) can achieve privacy-safe and reliable collaborative training without collecting users' private data. Its excellent privacy security potential promotes a wide range of FL applications in Internet-of-Things (IoT), wireless networks, mobile devices, autonomous vehicles, and cloud medical treatment. However, the FL method suffers from poor model performance on non-i.i.d. data and excessive traffic volume. To this end, we propose a personalized FL algorithm using a hierarchical proximal mapping based on the moreau envelop, named sparse federated learning with hierarchical personalized models (sFedHP), which significantly improves the global model performance facing diverse data. A continuously differentiable approximated L1-norm is also used as the sparse constraint to reduce the communication cost. Convergence analysis shows that sFedHP's convergence rate is state-of-the-art with linear speedup and the sparse constraint only reduces the convergence rate to a small extent while significantly reducing the communication cost. Experimentally, we demonstrate the benefits of sFedHP compared with the FedAvg, HierFAVG (hierarchical FedAvg), and personalized FL methods based on local customization, including FedAMP, FedProx, Per-FedAvg, pFedMe, and pFedGP.

Consider a set of points sampled independently near a smooth compact submanifold of Euclidean space. We provide mathematically rigorous bounds on the number of sample points required to estimate both the dimension and the tangent spaces of that manifold with high confidence. The algorithm for this estimation is Local PCA, a local version of principal component analysis. Our results accommodate for noisy non-uniform data distribution with the noise that may vary across the manifold, and allow simultaneous estimation at multiple points. Crucially, all of the constants appearing in our bound are explicitly described. The proof uses a matrix concentration inequality to estimate covariance matrices and a Wasserstein distance bound for quantifying nonlinearity of the underlying manifold and non-uniformity of the probability measure.

We propose a cluster-based method to detect and locate eavesdropping events in optical line systems characterized by small power losses. Our findings indicate that detecting such subtle losses from eavesdropping can be accomplished solely through optical performance monitoring (OPM) data collected at the receiver. On the other hand, the localization of such events can be effectively achieved by leveraging in-line OPM data.

In this work we consider the problem of estimating the principal subspace (span of the top r singular vectors) of a symmetric matrix in a federated setting, when each node has access to estimates of this matrix. We study how to make this problem Byzantine resilient. We introduce a novel provably Byzantine-resilient, communication-efficient, and private algorithm, called Subspace-Median, to solve it. We also study the most natural solution for this problem, a geometric median based modification of the federated power method, and explain why it is not useful. We consider two special cases of the resilient subspace estimation meta-problem - federated principal components analysis (PCA) and the spectral initialization step of horizontally federated low rank column-wise sensing (LRCCS) in this work. For both these problems we show how Subspace Median provides a resilient solution that is also communication-efficient. Median of Means extensions are developed for both problems. Extensive simulation experiments are used to corroborate our theoretical guarantees. Our second contribution is a complete AltGDmin based algorithm for Byzantine-resilient horizontally federated LRCCS and guarantees for it. We do this by developing a geometric median of means estimator for aggregating the partial gradients computed at each node, and using Subspace Median for initialization.

We present Zeroth-order Riemannian Averaging Stochastic Approximation (\texttt{Zo-RASA}) algorithms for stochastic optimization on Riemannian manifolds. We show that \texttt{Zo-RASA} achieves optimal sample complexities for generating $\epsilon$-approximation first-order stationary solutions using only one-sample or constant-order batches in each iteration. Our approach employs Riemannian moving-average stochastic gradient estimators, and a novel Riemannian-Lyapunov analysis technique for convergence analysis. We improve the algorithm's practicality by using retractions and vector transport, instead of exponential mappings and parallel transports, thereby reducing per-iteration complexity. Additionally, we introduce a novel geometric condition, satisfied by manifolds with bounded second fundamental form, which enables new error bounds for approximating parallel transport with vector transport.

One of the most natural connections between quantum and classical machine learning has been established in the context of kernel methods. Kernel methods rely on kernels, which are inner products of feature vectors living in large feature spaces. Quantum kernels are typically evaluated by explicitly constructing quantum feature states and then taking their inner product, here called embedding quantum kernels. Since classical kernels are usually evaluated without using the feature vectors explicitly, we wonder how expressive embedding quantum kernels are. In this work, we raise the fundamental question: can all quantum kernels be expressed as the inner product of quantum feature states? Our first result is positive: Invoking computational universality, we find that for any kernel function there always exists a corresponding quantum feature map and an embedding quantum kernel. The more operational reading of the question is concerned with efficient constructions, however. In a second part, we formalize the question of universality of efficient embedding quantum kernels. For shift-invariant kernels, we use the technique of random Fourier features to show that they are universal within the broad class of all kernels which allow a variant of efficient Fourier sampling. We then extend this result to a new class of so-called composition kernels, which we show also contains projected quantum kernels introduced in recent works. After proving the universality of embedding quantum kernels for both shift-invariant and composition kernels, we identify the directions towards new, more exotic, and unexplored quantum kernel families, for which it still remains open whether they correspond to efficient embedding quantum kernels.

A crucial task for a randomized controlled trial (RCT) is to specify a statistical method that can yield an efficient estimator and powerful test for the treatment effect. A novel and effective strategy to obtain efficient and powerful treatment effect inferences is to incorporate predictions from generative artificial intelligence (AI) algorithms into covariate adjustment for the regression analysis of a RCT. Training a generative AI algorithm on historical control data enables one to construct a digital twin generator (DTG) for RCT participants, which utilizes a participant's baseline covariates to generate a probability distribution for their potential control outcome. Summaries of the probability distribution from the DTG are highly predictive of the trial outcome, and adjusting for these features via regression can thus improve the quality of treatment effect inferences, while satisfying regulatory guidelines on statistical analyses, for a RCT. However, a critical assumption in this strategy is homoskedasticity, or constant variance of the outcome conditional on the covariates. In the case of heteroskedasticity, existing covariate adjustment methods yield inefficient estimators and underpowered tests. We propose to address heteroskedasticity via a weighted prognostic covariate adjustment methodology (Weighted PROCOVA) that adjusts for both the mean and variance of the regression model using information obtained from the DTG. We prove that our method yields unbiased treatment effect estimators, and demonstrate via comprehensive simulation studies and case studies from Alzheimer's disease that it can reduce the variance of the treatment effect estimator, maintain the Type I error rate, and increase the power of the test for the treatment effect from 80% to 85%~90% when the variances from the DTG can explain 5%~10% of the variation in the RCT participants' outcomes.

Federated Learning (FL) is a decentralized machine learning framework that enables collaborative model training while respecting data privacy. In various applications, non-uniform availability or participation of users is unavoidable due to an adverse or stochastic environment, the latter often being uncontrollable during learning. Here, we posit a generic user selection mechanism implementing a possibly randomized, stationary selection policy, suggestively termed as a Random Access Model (RAM). We propose a new formulation of the FL problem which effectively captures and mitigates limited participation of data originating from infrequent, or restricted users, at the presence of a RAM. By employing the Conditional Value-at-Risk (CVaR) over the (unknown) RAM distribution, we extend the expected loss FL objective to a risk-aware objective, enabling the design of an efficient training algorithm that is completely oblivious to the RAM, and with essentially identical complexity as FedAvg. Our experiments on synthetic and benchmark datasets show that the proposed approach achieves significantly improved performance as compared with standard FL, under a variety of setups.

We address a classical problem in statistics: adding two-way interaction terms to a regression model. As the covariate dimension increases quadratically, we develop an estimator that adapts well to this increase, while providing accurate estimates and appropriate inference. Existing strategies overcome the dimensionality problem by only allowing interactions between relevant main effects. Building on this philosophy, we implement a softer link between the two types of effects using a local shrinkage model. We empirically show that borrowing strength between the amount of shrinkage for main effects and their interactions can strongly improve estimation of the regression coefficients. Moreover, we evaluate the potential of the model for inference, which is notoriously hard for selection strategies. Large-scale cohort data are used to provide realistic illustrations and evaluations. Comparisons with other methods are provided. The evaluation of variable importance is not trivial in regression models with many interaction terms. Therefore, we derive a new analytical formula for the Shapley value, which enables rapid assessment of individual-specific variable importance scores and their uncertainties. Finally, while not targeting for prediction, we do show that our models can be very competitive to a more advanced machine learner, like random forest, even for fairly large sample sizes. The implementation of our method in RStan is fairly straightforward, allowing for adjustments to specific needs.

We consider the problem of identifying, from statistics, a distribution of discrete random variables $X_1,\ldots,X_n$ that is a mixture of $k$ product distributions. The best previous sample complexity for $n \in O(k)$ was $(1/\zeta)^{O(k^2 \log k)}$ (under a mild separation assumption parameterized by $\zeta$). The best known lower bound was $\exp(\Omega(k))$. It is known that $n\geq 2k-1$ is necessary and sufficient for identification. We show, for any $n\geq 2k-1$, how to achieve sample complexity and run-time complexity $(1/\zeta)^{O(k)}$. We also extend the known lower bound of $e^{\Omega(k)}$ to match our upper bound across a broad range of $\zeta$. Our results are obtained by combining (a) a classic method for robust tensor decomposition, (b) a novel way of bounding the condition number of key matrices called Hadamard extensions, by studying their action only on flattened rank-1 tensors.