Massimiliano Datres

Ernesto Araya, Massimiliano Datres, Gitta Kutyniok

Spiking neural networks (SNNs) are a promising paradigm for energy-efficient computation, yet their theoretical foundations-especially regarding stability and robustness-remain limited compared to artificial neural networks. In this work, we study discrete-time leaky integrate-and-fire (LIF) SNNs through the lens of Boolean function analysis. We focus on noise sensitivity and stability in classification tasks, quantifying how input perturbations affect outputs. Our main result shows that wide LIF-SNN classifiers are stable on average, a property explained by the concentration of their Fourier spectrum on low-frequency components. Motivated by this, we introduce the notion of spectral simplicity, which formalizes simplicity in terms of Fourier spectrum concentration and connects our analysis to the simplicity bias observed in deep networks. Within this framework, we show that random LIF-SNNs are biased toward simple functions. Experiments on trained networks confirm that these stability properties persist in practice. Together, these results provide new insights into the stability and robustness properties of SNNs.

Lorenzo Mazza, Massimiliano Datres, Ariel Rodriguez et al.

Behavioral cloning becomes difficult when the same observation admits several valid actions. We study this problem for action-chunking policies and show that different multimodal parameterizations fail in different ways. For latent-variable policies, posterior-prior regularization makes deployment-time sampling more reliable, but excessive regularization removes the action-conditioned information needed to distinguish demonstrated modes. Reducing this regularization can preserve mode information, but then success depends on whether the prior covers the relevant latent regions. For action-space generative policies, multimodality is constrained by the smoothness of the base-to-action transport: a map with small Lipschitz constant cannot assign substantial probability to many well-separated modes. Covering many modes therefore requires either sharp transitions in base space or off-support bridge regions in action space. Experiments on synthetic multimodal tasks and robotic simulation benchmarks support these mechanisms.

Jonas von Berg, Adalbert Fono, Massimiliano Datres et al.

The relationship between overparameterization, stability, and generalization remains incompletely understood in the setting of discontinuous classifiers. We address this gap by establishing a generalization bound for finite function classes that improves inversely with class stability, defined as the expected distance to the decision boundary in the input domain (margin). Interpreting class stability as a quantifiable notion of robustness, we derive as a corollary a law of robustness for classification that extends the results of Bubeck and Sellke beyond smoothness assumptions to discontinuous functions. In particular, any interpolating model with $p \approx n$ parameters on $n$ data points must be unstable, implying that substantial overparameterization is necessary to achieve high stability. We obtain analogous results for parameterized infinite function classes by analyzing a stronger robustness measure derived from the margin in the codomain, which we refer to as the normalized co-stability. Experiments support our theory: stability increases with model size and correlates with test performance, while traditional norm-based measures remain largely uninformative.

Massimiliano Datres, Gian Paolo Leonardi, Alessio Figalli et al.

We introduce a novel capacity measure 2sED for statistical models based on the effective dimension. The new quantity provably bounds the generalization error under mild assumptions on the model. Furthermore, simulations on standard data sets and popular model architectures show that 2sED correlates well with the training error. For Markovian models, we show how to efficiently approximate 2sED from below through a layerwise iterative approach, which allows us to tackle deep learning models with a large number of parameters. Simulation results suggest that the approximation is good for different prominent models and data sets.