PAC-Bayesian Generalization Bounds

Identifying linear time-invariant (LTI) dynamical systems is challenging when trajectories are short, noisy, or high-dimensional. Traditional system identification typically treats each system independently and cannot exploit shared structure across related systems. We propose PBML-LTI, a PAC-Bayesian meta-learning framework for few-shot LTI system identification that learns a transferable prior over task-specific dynamics while preserving task heterogeneity. Each task corresponds to an unknown LTI system, and the meta-learner uses training trajectories to learn a data-dependent prior over transition matrices. For a new system with limited data, PBML-LTI performs Bayesian adaptation under this prior to obtain a task-specific posterior, providing accurate estimates and principled uncertainty quantification. A key challenge is temporal dependence, since LTI trajectories violate the i.i.d. assumptions underlying most PAC-Bayes meta-learning analyses. We address this with a martingale PAC-Bayes analysis for dependent trajectory losses and derive a support-query predictive-risk bound that motivates a fit-KL meta-training objective. The bound clarifies the roles of empirical fit, posterior complexity, and prior quality in few-shot adaptation under sequential dependence. We further derive corollaries for transition-matrix recovery and multi-step trajectory prediction, connecting uncertainty-aware meta-identification with finite-sample guarantees for dependent dynamical data.

Weighted majority votes are central to many successful ensemble methods. PAC-Bayesian theory provides tight generalization guarantees for such models by analyzing the expected risk of stochastic classifiers, while analyzing the risk of deterministic majority votes relies on surrogate bounds. To avoid these surrogates, Zantedeschi et al. ( 2021) introduced guarantees for stochastic majority votes, but the resulting models remain randomized. In this paper, we propose a derandomization framework for stochastic majority votes. To do so, we apply recent advances in disintegrated PAC-Bayesian theory directly to the space of majority vote weight vectors, transforming stochastic guarantees into certificates for a single deterministic majority vote. We derive two families of high-probability generalization bounds, covering both data-independent and data-dependent constructions of the ensemble, which naturally lead to a self-bounding learning algorithm optimizing deterministic majority vote guarantees.

In this paper we derive a Probably Approximately Correct (PAC)-Bayesian error bound for partially observed linear time-invariant (LTI) stochastic dynamical systems in state-space form with inputs and sub-Gaussian noise. Such bounds are widespread in machine learning, and they are useful for characterizing the predictive power of models learned from finitely many data points. The bound derived in this paper relates the expectation of prediction errors with the prediction error generated by the model on the data used for learning. In addition, we show that it can also be used to derive bounds for the parameter estimation error. In turn, this allows us to provide finite-sample error bounds for the prediction error and parameter estimation error for a wide class of system identification algorithms. Furthermore, as LTI systems are a sub-class of recurrent neural networks (RNNs), these error bounds could be a first step towards PAC-Bayesian bounds for RNNs.

PAC-Bayes theory provides generalization guarantees by controlling the Kullback--Leibler (KL) divergence between posterior and prior distributions over a chosen hypothesis representation. However, predictive risk depends only on the predictive behavior induced by a hypothesis, not on the particular internal realization that implements that behavior. In over-parameterized systems, many distinct configurations induce identical predictive behavior, yet the classical PAC-Bayes KL divergence does not distinguish uncertainty over predictive behavior from variation among behaviorally equivalent realizations. We show that this distinction induces an exact structural decomposition of classical PAC-Bayes complexity. We formalize behavioral equivalence through a measurable behavior map and use measure disintegration to decompose probability measures on the configuration space into a distribution over predictive behaviors and conditional distributions over behavioral fibers. This yields an exact decomposition of the classical PAC-Bayes KL divergence into a behavior-selection term and a realization-level term given by an expected conditional KL within fibers. We define Z-information as the negative of this realization-level contribution: the exact gap between the KL divergence and the complexity of uncertainty over predictive behavior alone. We further show that the behavior-selection term admits an exact variational characterization: it is the minimum KL divergence among all posteriors inducing the same distribution over predictive behaviors, attained by a canonical fiber-symmetrized representative. Finally, we show that symmetry, behavior-preserving directions, fiber geometry, and invariance under fiber-preserving perturbations arise naturally from the same behavior-map structure. Together, these results identify predictive behavior as the natural object of PAC-Bayes complexity.

Diffusion Policies (DPs) are able to perform complex manipulation tasks. However, DPs are typically trained by minimizing a denoising objective, which provides limited control over generalization in the finite-data regimes common in robotics. In this letter, we propose PAC-DP, an approach that increases the performance of DPs in robotic manipulation tasks. By modeling the DP as a Bayesian neural network, and defining a PAC-Bayes generalization bound, we derive a novel training objective that augments the standard denoising loss with a Kullback-Leibler divergence regularizer between the posterior and prior parameter distributions. From the theoretical perspective, our approach provides a principled approach to regularize the training of DPs without significantly increasing the training time. From the practical point of view, experimental results demonstrate improved denoising performance, lower variational negative log-likelihood, and higher success rates across multiple robotic manipulation benchmarks. Crucially, the largest improvements are observed in low-data training regimes and complex tasks, establishing PAC-DP as a theoretically grounded framework for robot policy learning.

Overparameterized models often have continuous parameter symmetries, so different parameters define the same predictor. We show that PAC--Bayesian analysis should be performed on the quotient predictor space: pushing a prior and posterior to the quotient preserves the empirical and population Gibbs risks while removing the nonnegative KL contribution caused solely by how the two distributions differ among parameterizations of the same predictor. Quotienting alone does not determine which prior to use. We construct a canonical choice of one parameterization for each predictor and account for the geometric volume of its equivalent parameterizations. This transforms a neutral reference prior into a data-independent prior that reflects the model's implicit bias. It approximates the ideal but inadmissible posterior-matched prior, which would minimize the KL term by depending on the training data. The resulting certificate is tighter exactly when this geometry-induced prior has smaller KL divergence from the learned quotient posterior than the neutral prior. We test this prediction in Fourier regression with a Hadamard parameterization and in Query-Key attention, using ordinary SGD without an explicit regularizer. The implicit-bias prior reduces the mean quotient-space KL by $40.69\%$ and the mean PAC--Bayes certificate by $21.40\%$ in the Fourier-Hadamard experiment. The smaller, prior-scale-dependent improvement in Query-Key attention confirms the predicted conditional nature of the effect.

While reinforcement learning with verifiable rewards (RLVR) is widely used to improve the reasoning capabilities of large language models (LLMs), the generalizability of the resulting models remains poorly understood. In this work, we establish the first non-vacuous generalization bounds for parameter-efficient RLVR fine-tuning at the billion-parameter scale. Our approach adapts PAC-Bayes compression bounds to this setting, and addresses the inherent stochasticity of token generation by applying the Gumbel-max reparameterization trick. To operationalize these bounds, we propose the Progressive RLVR framework, which integrates RLVR with on-policy distillation, TinyLoRA, and model quantization. Progressive RLVR empirically retains 84-97% performance of standard LoRA fine-tuning while producing models that are 14,796x more compressible. We show that this framework yields non-vacuous generalization bounds in four domains: mathematical problem-solving, programming, general-knowledge reasoning, and Text-to-SQL. Our bounds exceed the accuracy of the base model by 9-51% and lie within 6-11% of the accuracy of the fine-tuned models.

Parameterized quantum circuits (PQCs) are increasingly used as policies and value functions in quantum reinforcement learning, yet it remains unclear when and why quantum policies generalize. We give a PAC-Bayesian account in which generalization is governed not by the raw number of circuit parameters, but by the effective dimension of the Fisher geometry induced by the circuit. This quantity is inflated by entanglement, making entangling connectivity an independent axis of complexity.In controlled experiments that fix the number of trainable rotations and vary only entanglement, we find that circuits with larger Fisher effective dimension exhibit larger train-test gaps, while parameter count is a weak predictor. The resulting bound acts primarily as a ranking certificate: it correctly orders circuits with identical parameter count, which parameter-counting bounds cannot do. We validate this mechanism across supervised classification, quantum contextual bandits, and value-function generalization, where entangled circuits consistently generalize worse than non-entangled circuits of equal parameter count, with gaps shrinking as sample size increases.Our strongest evidence comes from low-variance decision models, including single-observable classifiers, value heads, and one-step policies. In end-to-end multi-step policy learning, entanglement effects remain statistically significant but high return variance leaves the full ordering only partially resolved. Partial-correlation analysis shows that Fisher effective dimension screens off entangling pattern, and controls for training accuracy, readout, and optimizer rule out major optimization confounders. The effect also persists on an IBM Heron quantum processor under real noise. Overall, our results reframe quantum policy design around an entanglement--generalization trade-off rather than expressivity alone.

PAC-Bayesian bounds provide finite-sample guarantees for data-dependent randomized predictors, but applying them to learning-based control is difficult because the natural objective is a quadratic trajectory cost. Such losses are unbounded, non-Lipschitz , and lead to response-dependent Chernoff terms. We employ System Level Synthesis parameterization, which exposes the closed-loop trajectory map of a linear system directly and makes the quadratic control loss amenable to explicit certification. Moreover, we provide a set of PAC-Bayes-Chernoff certificates for posterior distributions over feasible closed-loop responses. For Gaussian disturbance trajectories with arbitrary covariance, we derive an exact one-sided Gaussian transform and a tractable quadratic upper bound expressed through closed-loop sensitivity quantities. We also derive a posterior-localized surrogate for settings where pointwise closed-loop response certificates are unavailable or have support related admissibility issues. Although PAC-Bayes certifies a non-degenerate posterior, the convex quadratic form of the SLS loss transfers the certificate to the posterior mean response. We present a deterministic mean response deployment result that is particularly suitable for control while retaining the stochastic posterior in the bound. Additionally, we provide a data-driven bound for this deployment, transitioning away from an oracle bound. Minimizing this bound naturally results in a learning algorithm for control selection from data. Numerical experiments on a double integrator show that the algorithm acts as a sensitivity-aware finite-sample regularizer, improving held-out cost and reducing closed-loop sensitivity in the low-data regime

Process Reward Models (PRMs) provide step-level verification for Large Language Model (LLM) reasoning, yet their training data acquisition remains a bottleneck: human annotation is costly and Monte Carlo roll-out estimates are noisy. A recent approach, FOVER, trains PRMs on step-level error labels automatically annotated by formal verification tools such as Z3 and Isabelle, and empirically observes cross-task generalization from symbolic tasks to diverse reasoning benchmarks. However, this generalization phenomenon lacks any theoretical explanation, and no formal bounds exist on the generalization error, sample complexity, convergence rate, or downstream Best-of-K performance of such PRMs. We propose VeriBound, a theoretical framework that provides PAC-Bayesian generalization bounds for PRMs trained with formal verification tools. We establish four main results: (i) a PAC-Bayesian generalization bound that relates the empirical verification error on formal-verification-annotated training data to the expected error on unseen reasoning tasks, with the bound depending on the formal verification accuracy and the divergence between training and test task distributions; (ii) a sample complexity result showing that $O(d \log(d/δ) / ε^2)$ formal-verification-annotated examples suffice to achieve generalization error $ε$ with probability $1-δ$, where $d$ is the complexity of the PRM hypothesis class; (iii) a convergence analysis proving that PRM training with formal verification labels converges at a linear rate under $L$-smoothness and bounded variance conditions; and (iv) an error propagation bound that relates step-level verification error to Best-of-K performance degradation.

We study PAC-Bayes derandomization for smooth loss functions. Our goal is to obtain generalization bounds that hold with high probability for deterministic predictors by exploiting smoothness properties of both the loss and the predictor class. We show that passing from the Gibbs predictor to the deterministic predictor at the posterior mean has a precise cost, given by the generalization gap of the Jensen gap class. We control this class through its Rademacher complexity, leading to bounds for deterministic predictors that involve flatness quantities expressed in terms of parameter Jacobians and Hessians of the score map. The framework applies to both bounded and unbounded smooth loss functions, and we specialize the results to linear predictors and smooth neural networks. Finally, the Jacobian and Hessian quantities appearing in the theory motivate a practical regularizer. For BatchNorm networks, we compute this regularizer with respect to effective BatchNorm weights obtained by folding the BatchNorm transformation into the adjacent affine weights. Experiments on CIFAR-10 illustrate the behavior of this regularizer under different batch sizes.

This thesis investigates how Bayesian principles can deepen our understanding of modern deep learning systems. While neural networks achieve remarkable predictive performance, their ability to generalize and to quantify uncertainty remains only partly understood. This thesis approaches this challenge from both methodological and theoretical angles: unifying Bayesian inference, function-space modeling, and large-deviation theory under a common probabilistic perspective. On the methodological side, the thesis introduces the Deep Variational Implicit Process (DVIP), a scalable Bayesian framework that extends implicit processes to deep architectures. Complementing this, two post-hoc methods -- the Variational Linearized Laplace Approximation (VaLLA) and the Fixed-Mean Gaussian Process (FMGP) -- are proposed to equip pretrained deterministic networks with calibrated uncertainty estimates. The theoretical contributions focus on one of the central open questions in modern machine learning: why do large, over-parameterized neural networks generalize so well? To address this, the thesis develops a unified probabilistic framework that connects three key mechanisms -- diversity, smoothness, and stochasticity -- within the language of PAC-Bayesian and large-deviation theory.

Whilst the vulnerability of graph neural networks (GNNs) to adversarial attacks poses a critical threat to graph representation learning, the understanding of the robust generalization behavior remains a fundamental challenge in the adversarial setting. Recently, PAC-Bayesian margin-based generalization analysis substantially advances this line of research by providing a flexible and data-dependent analytical framework. However, existing robust analyses often rely on isotropic Gaussian posteriors and control weight perturbations in the full parameter space, which limits the ability to capture heterogeneous parameter sensitivity yet hinges on hidden-width-dependent complexity terms, resulting in not-tight-enough generalization bounds. In this paper, we extend a recently proposed sensitivity-aware PAC-Bayesian framework from deep neural networks to message passing GNNs (MPGNNs) and derive a tighter robust generalization bound in the adversarial setting. Specifically, we first quantify how sensitive the perturbations across different parameter blocks are to the network outputs by deriving the output Jacobians with respect to the weight parameters. Exploiting the fact that these Jacobian matrices have rank at most $K$ in $K$-class graph classification, we then construct Jacobian-aligned sensitivity matrices and use anisotropic Gaussian posteriors with optimized covariances to upper bound the KL divergence in a tight way. Notably, by refining the spectral-norm dependence on the learned weights and reducing the leading dimension factor from hidden-width-dependent terms to the number of classes $K$, our analysis yields much tighter robust generalization guarantees for MPGNNs, thereby guiding their designs to enhance adversarial robustness.

Critical sequential decisions are rarely single-timescale: a strategic decision causally shapes the context in which every subsequent tactical choice is made; standard bandit and reinforcement-learning theory does not capture this causal coupling between timescales. We formalise the problem class as Nested Contextual Causal Bandits (NCCBs), a hierarchical SCM where each level's action sets the next level's context distribution, and propose Nested Causal Thompson Sampling (NCTS), which draws one mechanism-factorised belief per episode and acts recursively under it. Our main theoretical result is a causal PAC-Bayesian excess-risk bound that certifies any candidate deployment policy from historic data alone, off-policy and anytime, answering the deployment question: can we trust this agent here, and at what risk? Experiments on a hierarchical SCM show that, against a matched RFF-GP joint regression on the same function class, the factorised SCM-mechanism posterior transfers significantly better zero-shot under exogenous distribution shifts, the recursive meta-to-inner commit significantly dominates the joint-commit alternative in distribution, and the certificate significantly contracts as offline data accumulates. Combining these results, we establish progressive certified handover, a safe-deployment method: each timescale flips from a legacy controller to NCTS when gains can be certified, independently of the others.

Physics-informed machine learning (PIML) integrates mechanistic knowledge, typically in the form of partial differential equations (PDE), into data-driven models. Despite strong empirical performance, its statistical generalisation properties remain poorly understood, particularly in the regression setting with unbounded losses. Existing analyses rely on approximation or stability arguments and do not fully capture how physical structure influences generalisation from finite data. In this work, we develop a PAC-Bayesian framework for PIML that provides high-probability generalisation guarantees in the presence of unbounded losses. We adopt a multi-task perspective that jointly treats data fidelity, PDE residuals, initial and boundary conditions, avoiding the looseness induced by standard union-bound approaches. Our analysis leverages the structure of physics-informed objectives to derive novel bounds where the complexity scales with input-gradient norms of the losses, revealing a direct link between physical regularity and generalisation. We instantiate this framework under Sobolev and Poincaré-type assumptions, yielding two classes of bounds that trade off statistical complexity and smoothness in different regimes. Building on these results, we propose a self-bounding-aware learning algorithm that directly optimises tractable surrogates of the derived bounds, along with a practical procedure to estimate the associated constants in realistic settings. Empirical evaluations on standard PDE benchmarks demonstrate that our bounds are non-vacuous, significantly tighter than union-bound baselines, and can be effectively minimised during training. Overall, our results provide a principled statistical foundation for the generalisation of physics-informed models.

Understanding the relationship between generalization and privacy remains a challenge in modern machine learning theory, particularly for deep networks that are trained by variants of differentially private stochastic gradient descent (DP- SGD). In this work we make progress on this persistent open problem. First, we derive explicit upper bounds on the approximate max-information of any algorithm that fulfills $(ε, δ)$-differential privacy or Rényi differential privacy, thereby going beyond the classical results for pure $ε$-differential privacy. Subsequently, we show even stronger guarantees for two common private learning algorithms, output perturbation with the Gaussian mechanism, and streaming DP-SGD, by exploiting the structure of their internal randomization. As an application of our results, we demonstrate how to obtain non-vacuous PAC-Bayes generalization bounds for deep networks, in which the prior distribution is learned by DP-SGD instead of the classical way of choosing it in a data-independent way.

Test-time adaptation (TTA) methods improve model performance under distribution shift but lack formal guarantees connecting shift magnitude to prediction reliability. We develop a PAC-Bayesian framework yielding generalization bounds explicitly parameterized by the maximum mean discrepancy (MMD) between source and target distributions. Our principal contribution is interpreting MMD-balls around the source distribution as credal sets in Walley's imprecise probability theory, yielding natural epistemic uncertainty quantification. We establish: (i) a PAC-Bayesian bound with an MMD-dependent shift penalty under an RKHS-Lipschitz loss assumption; (ii) a finite-sample version via MMD concentration; (iii) a uniform worst-case risk bound over all distributions in the credal set, with a lower-upper risk decomposition; and (iv) geodesic preservation bounds explaining why kernel-guided adaptation protects local feature geometry. The credal set interpretation separates epistemic from aleatoric uncertainty and provides a principled decision criterion for when adaptation is warranted.

We study transformers' generalization behavior on boolean domains from the perspective of the Fourier spectra of their target functions. In contrast to prior work (Edelman et al., 2022; Trauger & Tosh, 2024), which derived generalization bounds from Rademacher complexity, we investigate the feasibility of obtaining generalization bounds via PAC-Bayes theory. We show that sparse spectra concentrated on low-degree components enable low-sharpness constructions with good generalization properties. Our idea is to show the existence of flat minima implementing any boolean function of sparsity no greater than the context length, and then apply a PAC-Bayes bound to an idealized low-sharpness learner, resulting in a non-vacuous generalization bound. We use this to give a formal account of why chain-of-thought improves generalization for high-degree target functions, and show that the complexity parameters in our bound can be efficiently estimated via property testing. We evaluate predictions empirically and conduct a mechanistic interpretability study to support the realism of our theoretical construction in real transformers.

A unified framework for learning under covariate shift is presented, in which a constrained density-ratio network approximates the Radon-Nikodym derivative $r^\star = dP/dQ$ and feeds an anytime PAC-Bayes generalization certificate. A change-of-measure identity decomposes the gap between target risk and importance-weighted source risk into a ratio-bias term governed by $\|r_θ- r^\star\|_{L^2(Q)}$ and a generalization-gap term governed by the variability of the weighted loss. Normalization and moment-matching identities are enforced as hard integral constraints through an augmented-Lagrangian scheme, with a second-moment penalty controlling the effective sample size. PAC-Bayes is instantiated on the weighted risk in a fixed-time regime that yields Bernoulli-KL bounds, identifies the network-weighted Gibbs posterior as the unique KL-regularized minimizer, and quantifies stability under $L^2(Q)$ perturbations of the learned ratio, and is then strengthened by geometric peeling to an anytime certificate uniform in $t \geq t_{\min}$. A pre-registered two-campaign protocol combining a patch test against analytic ground truth with a real-data deployment validates the framework: the network produces calibrated ratios, reduces target $0/1$ loss against unweighted ERM and classical direct ratio-estimation baselines, and attains the anytime certificate. A single fixed-time coverage failure is recorded, with per-split coverage aligning one-to-one with the magnitude of the label shift, confirming that the covariate-only assumption is operationally tight rather than a defect of the certificate.

Deep neural networks generalize well despite being heavily overparameterized, in apparent contradiction with classical learning theory based on uniform convergence over fixed hypothesis spaces. Uniform bounds over the entire parameter space are vacuous in this regime, and recent work has shown that non-vacuous guarantees can be recovered by restricting attention to the part of parameter space that the algorithm actually visits. This survey paper organizes this line of work around three steps: extending PAC-Bayesian theory to random, data-dependent hypothesis sets (arXiv:2404.17442); refining the complexity term with geometric and topological descriptors of the optimization trajectory, including fractal dimensions, alpha-weighted lifetime sums, and positive magnitude (arXiv:2006.09313, arXiv:2302.02766, arXiv:2407.08723); and replacing the resulting information-theoretic terms by stability assumptions (arXiv:2507.06775). We unify these contributions around a single template inequality and a head-to-head comparison of the resulting bounds.

We derive explicit non-asymptotic PAC-Bayes generalization bounds for Gibbs posteriors, that is, data-dependent distributions over model parameters obtained by exponentially tilting a prior with the empirical risk. Unlike classical worst-case complexity bounds based on uniform laws of large numbers, which require explicit control of the model space in terms of metric entropy (integrals), our analysis yields posterior-averaged risk bounds that can be applied to overparameterized models and adapt to the data structure and the intrinsic model complexity. The bound involves a marginal-type integral over the parameter space, which we analyze using tools from singular learning theory to obtain explicit and practically meaningful characterizations of the posterior risk. Applications to low-rank matrix completion and ReLU neural network regression and classification show that the resulting bounds are analytically tractable and substantially tighter than classical complexity-based bounds. Our results highlight the potential of PAC-Bayes analysis for precise finite-sample generalization guarantees in modern overparameterized and singular models.

Early-exit neural networks enable adaptive computation by allowing confident predictions to exit at intermediate layers, achieving 2-8$\times$ inference speedup. Despite widespread deployment, their generalization properties lack theoretical understanding -- a gap explicitly identified in recent surveys. This paper establishes a unified PAC-Bayesian framework for adaptive-depth networks. (1) Novel Entropy-Based Bounds: We prove the first generalization bounds depending on exit-depth entropy $H(D)$ and expected depth $\mathbb{E}[D]$ rather than maximum depth $K$, with sample complexity $\mathcal{O}((\mathbb{E}[D] \cdot d + H(D))/ε^2)$. (2) Explicit Constructive Constants: Our analysis yields the leading coefficient $\sqrt{2\ln 2} \approx 1.177$ with complete derivation. (3) Provable Early-Exit Advantages: We establish sufficient conditions under which adaptive-depth networks strictly outperform fixed-depth counterparts. (4) Extension to Approximate Label Independence: We relax the label-independence assumption to $ε$-approximate policies, broadening applicability to learned routing. (5) Comprehensive Validation: Experiments across 6 architectures on 7 benchmarks demonstrate tightness ratios of 1.52-3.87$\times$ (all $p < 0.001$) versus $>$100$\times$ for classical bounds. Bound-guided threshold selection matches validation-tuned performance within 0.1-0.3%.

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.