Robustness of neural networks to random noise perturbations of their inputs

Learning from imperfect data is a central theme in machine learning, connecting practical questions of robustness to fundamental questions of learnability. Here we examine attribute noise: learning from corrupted inputs while keeping the labels intact, a setting that has received considerably less analytical attention than its label-noise counterpart. We consider two types of corruption models: additive noise and replacement noise. Through experiments with multi-layer perceptrons (MLPs) on corrupted classification datasets, we find that neural networks remain robust, maintaining well-above-chance accuracy even when inputs are >90% corrupted -- far beyond human recognition. To understand this robustness, we analyze infinite-width networks in the heavy-corruption regime using a mean-field-inspired approach and derive a leading-order decision rule for the classification outcome: the network implements a prototype rule, the nearest-class-mean, assigning each test point to the class whose training-set average it most closely resembles. This leading-order decision rule is universal across a broad range of MLP architectures, holding for any depth, as well as a wide class of activation functions and noise distributions. The same centroid mechanism closely matches finite-width network behavior in our experiments and provides an interpretable and analytically tractable account of why learning can succeed even when individual training examples carry almost no signal.

Neural responses in cortex exhibit substantial trial-to-trial variability in response to repeated stimuli, while peripheral sensory neurons respond far more consistently, leading many to wonder whether stochasticity may carry meaning. Existing work has argued that noise and signal correlations may be optimized for discrimination in animals, whereas artificial neural network (ANN) studies have shown similar benefits of noise in machine learning tasks, although most ANN work has neglected the effects of correlations. Here we investigate whether correlated noise improves the robustness of artificial neural networks to adversarial attacks and naturalistic image modifications. Using the covariance of activations under modified versus clean inputs, we find that structured noise may significantly improve network robustness. Robustness to naturalistic image modifications benefits most from structure, but this structure transfers poorly across modification types. In contrast, noise structure from adversarial attacks can generalize to other kinds of attacks. These results suggest that structured noise in ANN activations generally improves robustness, establishing a biologically plausible strategy for creating robust artificial neural networks that only relies on local information.

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.