Certified Robustness

Randomized Smoothing (RS) is a prominent technique for certifying the robustness of neural networks against adversarial perturbations. With RS, achieving high accuracy at small radii requires a small noise variance, while achieving high accuracy at large radii requires a large noise variance. However, the global noise variance used in the standard RS formulation leads to a fundamental limitation: there exists no global noise variance that simultaneously achieves strong performance at both small and large radii. To break through the global variance limitation, we propose a dual RS framework which enables input-dependent noise variances. To achieve that, we first prove that RS remains valid with input-dependent noise variances, provided the variance is locally constant around each input. Building on this result, we introduce two components: (i) a variance estimator predicts an optimal noise variance for each input, (ii) this estimated variance is then used by a standard RS classifier. The variance estimator is independently smoothed via RS to ensure local constancy, enabling flexible design. We also introduce training strategies to iteratively optimize the two components. Experiments on CIFAR-10 demonstrate that our dual RS method provides strong performance for both small and large radii-unattainable with global noise variance-while incurring only a 60% computational overhead at inference. Moreover, it outperforms prior input-dependent noise approaches across most radii, with gains at radii 0.5, 0.75, and 1.0 of 15.6%, 20.0%, and 15.7%. On ImageNet, dual RS remains effective across all radii, with advantages of 8.6%, 17.1%, and 9.1% at radii 0.5, 1.0, and 1.5. Additionally, the dual RS framework provides a routing perspective for certified robustness, improving the accuracy-robustness trade-off with off-the-shelf expert RS models.

Reinforcement learning policies parametrized by deep neural networks have achieved strong performance for continuous control, yet even small input perturbations may lead to unpredictable behavior. This sensitivity limits their use in safety-critical domains, where robustness guarantees are required. Our work addresses this gap between state-of-the-art adversarial training methods and formal verification to train verifiably robust agents. Previous works train networks with individual adversarial perturbations, making them only robust against the specific adversarial attacks used. In contrast, our approach propagates entire perturbed input sets, enclosing all possible adversarial attacks within a single network pass. We leverage this to explicitly penalize the size of the output set (minimizing closed-loop uncertainty) and thereby make the actor robust against all possible attacks. This is realized by the use of set-based policy gradients, where each output within the set has a different gradient, thereby balancing the accuracy and robustness of the network. Doing so, we achieve formal verifiability across different verification frameworks for up to 9 times larger input perturbations compared to standard reinforcement learning and improve certified worst-case performance.

As robustness verification methods are becoming more precise, training certifiably robust neural networks is becoming ever more relevant. To this end, certified training methods compute and then optimize an upper bound on the worst-case loss over a robustness specification. Curiously, training methods based on the imprecise interval bound propagation (IBP) consistently outperform those leveraging more precise bounding methods. Still, we lack an understanding of the mechanisms making IBP so successful. In this work, we thoroughly investigate these mechanisms by leveraging a novel metric measuring the tightness of IBP bounds. We first show theoretically that, for deep linear models, tightness decreases with width and depth at initialization, but improves with IBP training, given sufficient network width. We, then, derive sufficient and necessary conditions on weight matrices for IBP bounds to become exact and demonstrate that these impose strong regularization, explaining the empirically observed trade-off between robustness and accuracy in certified training. Our extensive experimental evaluation validates our theoretical predictions for ReLU networks, including that wider networks improve performance, yielding state-of-the-art results. Interestingly, we observe that while all IBP-based training methods lead to high tightness, this is neither sufficient nor necessary to achieve high certifiable robustness. This hints at the existence of new training methods that do not induce the strong regularization required for tight IBP bounds, leading to improved robustness and standard accuracy.