The growing prevalence of streaming data and increasing concerns over data privacy pose significant challenges for traditional nonparametric regression methods, which are often ill-suited for real-time, privacy-aware learning. In this paper, we tackle these issues by first proposing a novel one-pass online functional stochastic gradient descent algorithm that leverages the Huber loss (H-FSGD), to improve robustness against outliers and heavy-tailed errors in dynamic environments. To further accommodate privacy constraints, we introduce a locally differentially private extension, Private H-FSGD (PH-FSGD), designed to real-time, privacy-preserving estimation. Theoretically, we conduct a comprehensive non-asymptotic convergence analysis of the proposed estimators, establishing finite-sample guarantees and identifying optimal step size schedules that achieve optimal convergence rates. In particular, we provide practical insights into the impact of key hyperparameters, such as step size and privacy budget, on convergence behavior. Extensive experiments validate our theoretical findings, demonstrating that our methods achieve strong robustness and privacy protection without sacrificing efficiency.

We develop a comprehensive theory for regularized M-estimation in reproducing kernel Hilbert spaces. Under mild conditions on the loss we establish existence and measurability of the estimator, covering a wide range of convex and non-convex losses, including bounded robust losses. We further prove sharp rates of convergence with an explicit bias-variance decomposition governed by a novel complexity measure. We show that the variance is independent of misspecification, while the bias depends on a source condition parameter known in the learning literature. For tensor product Sobolev spaces we obtain new rates that connect to spaces of functions with dominating mixed smoothness, substantially extending existing results and explaining why these estimators circumvent the curse of dimensionality. Our methodology, combining elements from both functional analysis and empirical process theory, allows for an asymptotic linearisation of the objective function that avoids both closed-form solutions and global Lipschitz assumptions, and may be of independent interest. The estimators are implemented in C++ and theory is supported by numerical experiments.

In this paper, we investigate the adversarial robustness of nonparametric regression, a fundamental problem in machine learning, under the setting where an adversary can arbitrarily corrupt a subset of the input data. While the robustness of parametric regression has been extensively studied, its nonparametric counterpart remains largely unexplored. We characterize the adversarial robustness in nonparametric regression, assuming the regression function belongs to the second-order Sobolev space (i.e., it is square integrable up to its second derivative). The contribution of this paper is two-fold: (i) we establish a minimax lower bound on the estimation error, revealing a fundamental limit that no estimator can overcome, and (ii) we show that, perhaps surprisingly, the classical smoothing spline estimator, when properly regularized, exhibits robustness against adversarial corruption. These results imply that if o(n) out of n samples are corrupted, the estimation error of the smoothing spline vanishes as n . On the other hand, when a constant fraction of the data is corrupted, no estimator can guarantee vanishing estimation error, implying the optimality of the smoothing spline in terms of maximum tolerable number of corrupted samples.

In this paper, we investigate the adversarial robustness of nonparametric regression, a fundamental problem in machine learning, under the setting where an adversary can arbitrarily corrupt a subset of the input data. While the robustness of parametric regression has been extensively studied, its nonparametric counterpart remains largely unexplored. We characterize the adversarial robustness in nonparametric regression, assuming the regression function belongs to the second-order Sobolev space (i.e., it is square integrable up to its second derivative). The contribution of this paper is two-fold: (i) we establish a minimax lower bound on the estimation error, revealing a fundamental limit that no estimator can overcome, and (ii) we show that, perhaps surprisingly, the classical smoothing spline estimator, when properly regularized, exhibits robustness against adversarial corruption. These results imply that if $o(n)$ out of $n$ samples are corrupted, the estimation error of the smoothing spline vanishes as $n \to \infty$. On the other hand, when a constant fraction of the data is corrupted, no estimator can guarantee vanishing estimation error, implying the optimality of the smoothing spline in terms of maximum tolerable number of corrupted samples.

Deep learning models are widely deployed in safety-critical domains, but remain vulnerable to adversarial attacks. In this paper, we study the adversarial robustness of NTK neural networks in the context of nonparametric regression. We establish minimax optimal rates for adversarial regression in Sobolev spaces and then show that NTK neural networks, trained via gradient flow with early stopping, can achieve this optimal rate. However, in the overfitting regime, we prove that the minimum norm interpolant is vulnerable to adversarial perturbations.

Neural Information Processing Systems http://nips.cc/

Neural Information Processing Systems http://nips.cc/

As a result, the demands for interval estimation, and consequently for its validity and precision, have experienced a sustained increase over time and are reflected in a number of recent studies. For example, in proteomics, confidence intervals are employed to assess the association between post-translational modifications and intrinsically disordered regions of proteins, validating hypotheses derived from predictive models and facilitating large-scale functional analyses (Tunyasuvunakool et al., 2021; Bludau et al., 2022). In genomic research, confidence intervals are leveraged to characterize the distribution of gene expression levels, enabling robust inferences about promoter sequence effects and genetic variability (Vaishnav et al., 2022). In the realm of environmental science, interval estimation can be used to monitor deforestation rates of forests, yielding uncertainty-aware insights critical for climate policy formulation (Bullock et al., 2020). As for social sciences, confidence intervals are utilized to evaluate relationships between socioeconomic factors, bolstering the robustness of conclusions drawn from census data (Ding et al., 2021).