Higher-Order Certified Robustness for Regression

Randomized smoothing (RS) uses a smoothed classifier to provide architecture-agnostic certificates of $\ell_2$ classification robustness, but its dependence on per-input Monte Carlo (MC) sampling undermines its use in real-time systems. We argue that this cost is structural rather than fundamental, such that it can be significantly reduced by sharing information across the deployment stream. We introduce RRISE, an RS framework that compresses certification into a single forward pass through a learned surrogate. RRISE trains the surrogate against precomputed MC class-count targets via a soft-label cross-entropy loss and converts surrogate predictions into provably conservative certified radii through a one-time conformal calibration step. The resulting certificate is deployment-verifiable: whenever the calibrated radius is positive, the surrogate's prediction provably matches the smoothed classifier's and the smoothed classifier is constant on a ball of that radius around the input. Across image classification benchmarks, RRISE matches fixed-budget MC certified accuracy within $0.84$ percentage points while replacing up to $10^4$ noisy base-model evaluations per query with a single surrogate forward pass, recouping MC training cost after $\approx 10^5$ deployment queries. On CIFAR-100 and Tiny ImageNet, where the only prior offline-surrogate method collapses, RRISE achieves $1.23$ to $1.91\times$ higher certified accuracy, establishing efficient randomized smoothing as a practical path to certified robustness in repeated-deployment settings.

Randomized Smoothing (RS) offers formal $\ell_2$ guarantees for arbitrary base classifiers but faces two key practical bottlenecks: (i) it often relies on noise-augmented training to achieve nontrivial certificates, which increases training cost, can reduce clean accuracy, and weakens RS as a genuinely post-hoc defense; and (ii) certification is computationally expensive, typically requiring tens of thousands of noisy forward passes per input, which hinders deployment, especially on resource-constrained edge devices. To address both limitations, we propose Laplace-Bridged Smoothing (LBS), an analytic reformulation of RS that replaces high-dimensional input-space Monte Carlo (MC) sampling with efficient computations in a low-dimensional probability space. LBS preserves formal robustness guarantees without requiring noise-augmented training while substantially reducing certification burden. On CIFAR-10 and ImageNet, LBS attains stronger certified robustness than RS and reduces per-sample certification cost by nearly an order of magnitude. Notably, on NVIDIA Jetson Orin Nano and Raspberry Pi 4, LBS achieves speedups of up to $494\times$, enabling practical certified deployment on real-world edge devices. Finally, we provide theoretical justification for the analytic formulation and certificate validity of LBS.

Randomized Smoothing (RS) provides rigorous robustness guarantees for neural networks without architectural constraints, yet its adoption is limited by extreme computational costs. Standard RS requires tens of thousands of model evaluations per input and forces practitioners to commit to fixed sample sizes a priori. In this work, we present a novel meta-learning framework for anytime-valid certified robustness that adaptively deploys computational resources. By using a lightweight meta-learner to predict image-specific priors for a sequential E-process, we achieve a 20-fold reduction in sample complexity compared to traditional methods while maintaining rigorous statistical guarantees. Beyond raw efficiency, we demonstrate how anytime-validity enables adaptively allocating compute based upon application-specific risk thresholds, a form of resource triage impossible under classic certification frameworks. That this is achievable while also providing similar certification performance demonstrates that our approach provides a pathway for real-time, safety-critical certification deployments.