PAC-Bayesian Generalization Bounds

We present a distributionally robust PAC-Bayesian framework for certifying the performance of learning-based finite-horizon controllers. While existing PAC-Bayes control literature typically assumes bounded losses and matching training and deployment distributions, we explicitly address unbounded losses and environmental distribution shifts (the sim-to-real gap). We achieve this by drawing on two modern lines of research, namely the PAC-Bayes generalization theory and distributionally robust optimization via the type-1 Wasserstein distance. By leveraging the System Level Synthesis (SLS) reparametrization, we derive a sub-Gaussian loss proxy and a bound on the performance loss due to distribution shift. Both are tied directly to the operator norm of the closed-loop map. For linear time-invariant systems, this yields a computationally tractable optimization-based framework together with high-probability safety certificates for deployment in real-world environments that differ from those used in training.

Graph neural networks have demonstrated excellent applicability to a wide range of domains, including social networks, biological systems, recommendation systems, and wireless communications. Yet a principled theoretical understanding of their generalization behavior remains limited, particularly for graph classification tasks where complex interactions between model parameters and graph structure play a crucial role. Among existing theoretical tools, PAC-Bayesian norm-based generalization bounds provide a flexible and data-dependent framework; however, current results for GNNs often restrict the exploitation of graph structures. In this work, we propose a topology-aware PAC-Bayesian norm-based generalization framework for graph convolutional networks (GCNs) that extends a previously developed framework to graph-structured models. Our approach reformulates the derivation of generalization bounds as a stochastic optimization problem and introduces sensitivity matrices that measure the response of classification outputs with respect to structured weight perturbations. By imposing different structures on sensitivity matrices from both spatial and spectral perspectives, we derive a family of generalization error bounds with graph structures explicitly embedded. Such bounds could recover existing results as special cases, while yielding bounds that are tighter than state-of-the-art PAC-Bayesian bounds for GNNs. Notably, the proposed framework explicitly integrates graph structural properties into the generalization analysis, enabling a unified inspection of GNN generalization behavior from both spatial aggregation and spectral filtering viewpoints.

Generalization is a central concept in machine learning theory, yet for quantum models, it is predominantly analyzed through uniform bounds that depend on a model's overall capacity rather than the specific function learned. These capacity-based uniform bounds are often too loose and entirely insensitive to the actual training and learning process. Previous theoretical guarantees have failed to provide non-uniform, data-dependent bounds that reflect the specific properties of the learned solution rather than the worst-case behavior of the entire hypothesis class. To address this limitation, we derive the first PAC-Bayesian generalization bounds for a broad class of quantum models by analyzing layered circuits composed of general quantum channels, which include dissipative operations such as mid-circuit measurements and feedforward. Through a channel perturbation analysis, we establish non-uniform bounds that depend on the norms of learned parameter matrices; we extend these results to symmetry-constrained equivariant quantum models; and we validate our theoretical framework with numerical experiments. This work provides actionable model design insights and establishes a foundational tool for a more nuanced understanding of generalization in quantum machine learning.

We consider a PAC-Bayes type learning rule for binary classification, balancing the training error of a randomized ''posterior'' predictor with its KL divergence to a pre-specified ''prior''. This can be seen as an extension of a modified two-part-code Minimum Description Length (MDL) learning rule, to continuous priors and randomized predictions. With a balancing parameter of $λ=1$ this learning rule recovers an (empirical) Bayes posterior and a modified variant recovers the profile posterior, linking with standard Bayesian prediction (up to the treatment of the single-parameter noise level). However, from a risk-minimization prediction perspective, this Bayesian predictor overfits and can lead to non-vanishing excess loss in the agnostic case. Instead a choice of $λ\gg 1$, which can be seen as using a sample-size-dependent-prior, ensures uniformly vanishing excess loss even in the agnostic case. We precisely characterize the effect of under-regularizing (and over-regularizing) as a function of the balance parameter $λ$, understanding the regimes in which this under-regularization is tempered or catastrophic. This work extends previous work by Zhu and Srebro [2025] that considered only discrete priors to PAC Bayes type learning rules and, through their rigorous Bayesian interpretation, to Bayesian prediction more generally.

Despite being widely adopted as a canonical framework for learning robust models, adversarial training suffers from robust overfitting. Existing empirical and theoretical explorations fail to provide a satisfactory mechanistic interpretation of the phenomenon. By modeling adversarial training with momentum SGD as a discrete-time dynamical system, we propose a PAC-Bayesian analytical framework that proves time-resolved robust generalization bounds. Specifically, our framework tracks the closed-form evolution of the posterior mean and covariance under both stationary and non-stationary transient regimes, connecting the model's robust generalization performance to learning rate, local loss geometry, and mini-batch stochastic gradients. By estimating the key quantities associated with the bound, we illustrate the underlying mechanism of robust overfitting. Our framework also shows how adversarial weight perturbation reduces robust generalization gaps by suppressing dominant loss-curvature modes, while suggesting that excessive penalization can be sub-optimal for optimization.