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Functional quadratic regression models postulate a polynomial relationship between a scalar response rather than a linear one. As in functional linear regression, vertical and specially high-leverage outliers may affect the classical estimators. For that reason, the proposal of robust procedures providing reliable estimators in such situations is an important issue. Taking into account that the functional polynomial model is equivalent to a regression model that is a polynomial of the same order in the functional principal component scores of the predictor processes, our proposal combines robust estimators of the principal directions with robust regression estimators based on a bounded loss function and a preliminary residual scale estimator. Fisher-consistency of the proposed method is derived under mild assumptions. The results of a numerical study show, for finite samples, the benefits of the robust proposal over the one based on sample principal directions and least squares. The usefulness of the proposed approach is also illustrated through the analysis of a real data set which reveals that when the potential outliers are removed the classical and robust methods behave very similarly.

Datasets with sheer volume have been generated from fields including computer vision, medical imageology, and astronomy whose large-scale and high-dimensional properties hamper the implementation of classical statistical models. To tackle the computational challenges, one of the efficient approaches is subsampling which draws subsamples from the original large datasets according to a carefully-design task-specific probability distribution to form an informative sketch. The computation cost is reduced by applying the original algorithm to the substantially smaller sketch. Previous studies associated with subsampling focused on non-regularized regression from the computational efficiency and theoretical guarantee perspectives, such as ordinary least square regression and logistic regression. In this article, we introduce a randomized algorithm under the subsampling scheme for the Elastic-net regression which gives novel insights into L1-norm regularized regression problem. To effectively conduct consistency analysis, a smooth approximation technique based on alpha absolute function is firstly employed and theoretically verified. The concentration bounds and asymptotic normality for the proposed randomized algorithm are then established under mild conditions. Moreover, an optimal subsampling probability is constructed according to A-optimality. The effectiveness of the proposed algorithm is demonstrated upon synthetic and real data datasets.

There is a rapid increase in the cooperative learning paradigm in online learning settings, i.e., federated learning (FL). Unlike most FL settings, there are many situations where the agents are competitive. Each agent would like to learn from others, but the part of the information it shares for others to learn from could be sensitive; thus, it desires its privacy. This work investigates a group of agents working concurrently to solve similar combinatorial bandit problems while maintaining quality constraints. Can these agents collectively learn while keeping their sensitive information confidential by employing differential privacy? We observe that communicating can reduce the regret. However, differential privacy techniques for protecting sensitive information makes the data noisy and may deteriorate than help to improve regret. Hence, we note that it is essential to decide when to communicate and what shared data to learn to strike a functional balance between regret and privacy. For such a federated combinatorial MAB setting, we propose a Privacy-preserving Federated Combinatorial Bandit algorithm, P-FCB. We illustrate the efficacy of P-FCB through simulations. We further show that our algorithm provides an improvement in terms of regret while upholding quality threshold and meaningful privacy guarantees.

Many convex optimization problems with important applications in machine learning are formulated as empirical risk minimization (ERM). There are several examples: linear and logistic regression, LASSO, kernel regression, quantile regression, $p$-norm regression, support vector machines (SVM), and mean-field variational inference. To improve data privacy, federated learning is proposed in machine learning as a framework for training deep learning models on the network edge without sharing data between participating nodes. In this work, we present an interior point method (IPM) to solve a general ERM problem under the federated learning setting. We show that the communication complexity of each iteration of our IPM is $\tilde{O}(d^{3/2})$, where $d$ is the dimension (i.e., number of features) of the dataset.

Wasserstein distributionally robust estimators have emerged as powerful models for prediction and decision-making under uncertainty. These estimators provide attractive generalization guarantees: the robust objective obtained from the training distribution is an exact upper bound on the true risk with high probability. However, existing guarantees either suffer from the curse of dimensionality, are restricted to specific settings, or lead to spurious error terms. In this paper, we show that these generalization guarantees actually hold on general classes of models, do not suffer from the curse of dimensionality, and can even cover distribution shifts at testing. We also prove that these results carry over to the newly-introduced regularized versions of Wasserstein distributionally robust problems.

Federated Learning, as a popular paradigm for collaborative training, is vulnerable against privacy attacks. Different privacy levels regarding users' attitudes need to be satisfied locally, while a strict privacy guarantee for the global model is also required centrally. Personalized Local Differential Privacy (PLDP) is suitable for preserving users' varying local privacy, yet only provides a central privacy guarantee equivalent to the worst-case local privacy level. Thus, achieving strong central privacy as well as personalized local privacy with a utility-promising model is a challenging problem. In this work, a general framework (APES) is built up to strengthen model privacy under personalized local privacy by leveraging the privacy amplification effect of the shuffle model. To tighten the privacy bound, we quantify the heterogeneous contributions to the central privacy user by user. The contributions are characterized by the ability of generating "echos" from the perturbation of each user, which is carefully measured by proposed methods Neighbor Divergence and Clip-Laplace Mechanism. Furthermore, we propose a refined framework (S-APES) with the post-sparsification technique to reduce privacy loss in high-dimension scenarios. To the best of our knowledge, the impact of shuffling on personalized local privacy is considered for the first time. We provide a strong privacy amplification effect, and the bound is tighter than the baseline result based on existing methods for uniform local privacy. Experiments demonstrate that our frameworks ensure comparable or higher accuracy for the global model.

Consensus algorithms play a critical role in blockchains and directly impact their performance. During consensus processing, nodes need to validate and order the pending transactions into a new block, which requires verifying the application-specific data encapsulated within a transaction. This exposes the underlying data to the consensus nodes, presenting privacy concerns. Existing consensus algorithms focus on realizing application security and performance goals, but lack privacy-by-design properties or are resource-heavy and intended for securing permissionless blockchain networks. In this paper, we propose P-CFT, a zero-knowledge and crash fault tolerant consensus algorithm for permissioned blockchains. The proposed consensus algorithm provides inherent data privacy directly to the consensus layer, while still providing guarantees of crash fault tolerance. We conduct experiments using the Hyperledger Ursa cryptographic library, and the results show promise for integrating P-CFT into existing permissioned blockchain systems requiring privacy-preserving and crash fault tolerant features.

This paper studies robust nonparametric regression, in which an adversarial attacker can modify the values of up to $q$ samples from a training dataset of size $N$. Our initial solution is an M-estimator based on Huber loss minimization. Compared with simple kernel regression, i.e. the Nadaraya-Watson estimator, this method can significantly weaken the impact of malicious samples on the regression performance. We provide the convergence rate as well as the corresponding minimax lower bound. The result shows that, with proper bandwidth selection, $\ell_\infty$ error is minimax optimal. The $\ell_2$ error is optimal if $q\lesssim \sqrt{N/\ln^2 N}$, but is suboptimal with larger $q$. The reason is that this estimator is vulnerable if there are many attacked samples concentrating in a small region. To address this issue, we propose a correction method by projecting the initial estimate to the space of Lipschitz functions. The final estimate is nearly minimax optimal for arbitrary $q$, up to a $\ln N$ factor.

We propose the first theoretical and methodological framework for Gaussian process regression subject to privacy constraints. The proposed method can be used when a data owner is unwilling to share a high-fidelity supervised learning model built from their data with the public due to privacy concerns. The key idea of the proposed method is to add synthetic noise to the data until the predictive variance of the Gaussian process model reaches a prespecified privacy level. The optimal covariance matrix of the synthetic noise is formulated in terms of semi-definite programming. We also introduce the formulation of privacy-aware solutions under continuous privacy constraints using kernel-based approaches, and study their theoretical properties. The proposed method is illustrated by considering a model that tracks the trajectories of satellites.

Existing private synthetic data generation algorithms are agnostic to downstream tasks. However, end users may have specific requirements that the synthetic data must satisfy. Failure to meet these requirements could significantly reduce the utility of the data for downstream use. We introduce a post-processing technique that improves the utility of the synthetic data with respect to measures selected by the end user, while preserving strong privacy guarantees and dataset quality. Our technique involves resampling from the synthetic data to filter out samples that do not meet the selected utility measures, using an efficient stochastic first-order algorithm to find optimal resampling weights. Through comprehensive numerical experiments, we demonstrate that our approach consistently improves the utility of synthetic data across multiple benchmark datasets and state-of-the-art synthetic data generation algorithms.