Robust Predictive Uncertainty and Double Descent in Contaminated Bayesian Random Features

Random features provide a standard approach to scaling kernel methods to large datasets by approximating the kernel with a finite-dimensional feature map. Existing works of random feature methods primarily focus on well-specified models and light-tailed noise conditions, leaving a gap in understanding their performance under more general conditions, such as misspecified models and heavy-tailed noise. In this paper, we provide a comprehensive and improved analysis of kernel ridge regression with random features (KRR-RF) under general conditions. In the light-tailed noise setting, we improve upon the worst-case uniform bounds of the feature map used in the previous works and lead to the minimax optimal learning rates with $2r+γ>α_0$, where $(r,γ,α_0)$ denote the regularity of the target function, the effective dimension of the kernel, and the embedding index of the RKHS, respectively. In the heavy-tailed noise setting, we leverage a recently developed Hilbert-valued Fuk-Nagaev inequality and extend the same optimal results under finite weak moment conditions. The additional cost to achieve robustness is a mildly stronger explicit sample-size requirement than in the light-tailed case. For both settings, our results recover the conditions random features budgets (up to log factors) required in the previous works. In the most benign case when $α_0=γ$, KRR-RF are minimax optimal for all $r \in (0,1]$. To the best of our knowledge, this is the first work to establish optimal learning rates for KRR-RF under weak moment conditions in the misspecified setting. Numerical experiments and a real data analysis validate the derived learning rates and demonstrate their practical relevance in the presence of mis-specification and heavy-tailed noise.

Reliably quantifying predictive uncertainty is difficult for complex, high-dimensional, or misspecified models. Both fully Bayesian and bootstrap resampling methods provide principled uncertainty estimates but are often too expensive for modern machine-learning models because they require posterior sampling or repeated model refitting. We introduce Ribbon, a scalable approximation to Dirichlet-reweighted bootstrap uncertainty. Ribbon replaces repeated refitting with an influence-function linearization around a single fitted model, preserving the first-order data-reweighting structure of the Bayesian bootstrap while requiring only post-hoc linear algebra. Ribbon approximates the Bayesian-bootstrap or weighted-likelihood-bootstrap refitting target. With a general concentration parameter, Ribbon gives a calibrated Dirichlet-reweighting family whose uncertainty scale can be tuned on validation data. We show that Ribbon is asymptotically equivalent to a flat-prior Laplace approximation under correct likelihood specification and recovers the robust sandwich covariance under misspecification. Across synthetic regression, MNIST classification, and California Housing benchmarks, Ribbon provides competitive predictive performance and improved calibration in several settings while avoiding repeated model retraining.

Uncertainty quantification is central to safe and efficient deployments of deep learning models, yet many computationally practical methods lack lacking rigorous theoretical motivation. Random network distillation (RND) is a lightweight technique that measures novelty via prediction errors against a fixed random target. While empirically effective, it has remained unclear what uncertainties RND measures and how its estimates relate to other approaches, e.g. Bayesian inference or deep ensembles. This paper establishes these missing theoretical connections by analyzing RND within the neural tangent kernel framework in the limit of infinite network width. Our analysis reveals two central findings in this limit: (1) The uncertainty signal from RND -- its squared self-predictive error -- is equivalent to the predictive variance of a deep ensemble. (2) By constructing a specific RND target function, we show that the RND error distribution can be made to mirror the centered posterior predictive distribution of Bayesian inference with wide neural networks. Based on this equivalence, we moreover devise a posterior sampling algorithm that generates i.i.d. samples from an exact Bayesian posterior predictive distribution using this modified Bayesian RND model. Collectively, our findings provide a unified theoretical perspective that places RND within the principled frameworks of deep ensembles and Bayesian inference, and offer new avenues for efficient yet theoretically grounded uncertainty quantification methods.