Out-of-distribution (OOD) generalization remains a fundamental challenge in machine learning. Models trained on one data distribution often experience substantial performance degradation when evaluated on shifted or unseen domains. To address this challenge, the present paper studies the design of training data distributions that maximize average-case OOD performance. First, a theoretical analysis establishes a family of generalization bounds that quantify how the choice of training distribution influences OOD error across a predefined family of target distributions. These insights motivate the introduction of two complementary algorithmic strategies: (i) directly formulating OOD risk minimization as a bilevel optimization problem over the space of probability measures and (ii) minimizing a theoretical upper bound on OOD error. Last, the paper evaluates the two approaches across a range of function approximation and operator learning examples. The proposed methods significantly improve OOD accuracy over standard empirical risk minimization with a fixed distribution. These results highlight the potential of distribution-aware training as a principled and practical framework for robust OOD generalization.
We consider the problem of learning the evolution operator for the time-dependent Schr\"{o}dinger equation, where the Hamiltonian may vary with time. Existing neural network-based surrogates often ignore fundamental properties of the Schr\"{o}dinger equation, such as linearity and unitarity, and lack theoretical guarantees on prediction error or time generalization. To address this, we introduce a linear estimator for the evolution operator that preserves a weak form of unitarity. We establish both upper and lower bounds on the prediction error that hold uniformly over all sufficiently smooth initial wave functions. Additionally, we derive time generalization bounds that quantify how the estimator extrapolates beyond the time points seen during training. Experiments across real-world Hamiltonians -- including hydrogen atoms, ion traps for qubit design, and optical lattices -- show that our estimator achieves relative errors $10^{-2}$ to $10^{-3}$ times smaller than state-of-the-art methods such as the Fourier Neural Operator and DeepONet.
Graph Neural Networks (GNNs) are powerful tools for learning on structured data, yet the relationship between their expressivity and predictive performance remains unclear. We introduce a family of premetrics that capture different degrees of structural similarity between graphs and relate these similarities to generalization, and consequently, the performance of expressive GNNs. By considering a setting where graph labels are correlated with structural features, we derive generalization bounds that depend on the distance between training and test graphs, model complexity, and training set size. These bounds reveal that more expressive GNNs may generalize worse unless their increased complexity is balanced by a sufficiently large training set or reduced distance between training and test graphs. Our findings relate expressivity and generalization, offering theoretical insights supported by empirical results.
This paper formalizes an emerging learning paradigm that uses a trained model as a reference to guide and enhance the training of a target model through strategic data selection or weighting, named $\textbf{model steering}$. While ad-hoc methods have been used in various contexts, including the training of large foundation models, its underlying principles remain insufficiently understood, leading to sub-optimal performance. In this work, we propose a theory-driven framework for model steering called $\textbf{DRRho risk minimization}$, which is rooted in Distributionally Robust Optimization (DRO). Through a generalization analysis, we provide theoretical insights into why this approach improves generalization and data efficiency compared to training without a reference model. To the best of our knowledge, this is the first time such theoretical insights are provided for the new learning paradigm, which significantly enhance our understanding and practice of model steering. Building on these insights and the connection between contrastive learning and DRO, we introduce a novel method for Contrastive Language-Image Pretraining (CLIP) with a reference model, termed DRRho-CLIP. Extensive experiments validate the theoretical insights, reveal a superior scaling law compared to CLIP without a reference model, and demonstrate its strength over existing heuristic approaches.
Client-level fairness metrics for federated learning are used to ensure that all clients in a federation either: a) have similar final performance on their local data distributions (i.e., client parity), or b) obtain final performance on their local data distributions relative to their contribution to the federated learning process (i.e., contribution fairness). While a handful of works that propose either client-parity or contribution-based fairness metrics ground their definitions and decisions in social theories of equality -- such as distributive justice -- most works arbitrarily choose what notion of fairness to align with which makes it difficult for practitioners to choose which fairness metric aligns best with their fairness ethics. In this work, we propose UDJ-FL (Uncertainty-based Distributive Justice for Federated Learning), a flexible federated learning framework that can achieve multiple distributive justice-based client-level fairness metrics. Namely, by utilizing techniques inspired by fair resource allocation, in conjunction with performing aleatoric uncertainty-based client weighing, our UDJ-FL framework is able to achieve egalitarian, utilitarian, Rawls' difference principle, or desert-based client-level fairness. We empirically show the ability of UDJ-FL to achieve all four defined distributive justice-based client-level fairness metrics in addition to providing fairness equivalent to (or surpassing) other popular fair federated learning works. Further, we provide justification for why aleatoric uncertainty weighing is necessary to the construction of our UDJ-FL framework as well as derive theoretical guarantees for the generalization bounds of UDJ-FL. Our code is publicly available at https://github.com/alycia-noel/UDJ-FL.
The particle filter (PF) and the ensemble Kalman filter (EnKF) are widely used for approximate inference in state-space models. From a Bayesian perspective, these algorithms represent the prior by an ensemble of particles and update it to the posterior with new observations over time. However, the PF often suffers from weight degeneracy in high-dimensional settings, whereas the EnKF relies on linear Gaussian assumptions that can introduce significant approximation errors. In this paper, we propose the Adversarial Transform Particle Filter (ATPF), a novel filtering framework that combines the strengths of the PF and the EnKF through adversarial learning. Specifically, importance sampling is used to ensure statistical consistency as in the PF, while adversarially learned transformations, such as neural networks, allow accurate posterior matching for nonlinear and non-Gaussian systems. In addition, we incorporate kernel methods to ease optimization and leverage regularization techniques based on optimal transport for better statistical properties and numerical stability. We provide theoretical guarantees, including generalization bounds for both the analysis and forecast steps of ATPF. Extensive experiments across various nonlinear and non-Gaussian scenarios demonstrate the effectiveness and practical advantages of our method.
The integration of explicit Chain-of-Thought (CoT) reasoning into training large language models (LLMs) has advanced their reasoning capabilities, yet the mechanisms by which CoT enhances generalization remain poorly understood. This work investigates (1) \textit{how} CoT training reshapes internal model representations and (2) \textit{why} it improves both in-distribution (ID) and out-of-distribution (OOD) reasoning generalization. Through controlled experiments and theoretical analysis, we derive the following key insights. \textbf{1)} Structural Advantage: CoT training internalizes reasoning into a two-stage generalizing circuit, where the number of stages corresponds to the explicit reasoning steps during training. Notably, CoT-trained models resolve intermediate results at shallower layers compared to non-CoT counterparts, freeing up deeper layers to specialize in subsequent reasoning steps. \textbf{2)} Theoretical Analysis: the information-theoretic generalization bounds via distributional divergence can be decomposed into ID and OOD components. While ID error diminishes with sufficient training regardless of CoT, OOD error critically depends on CoT: Non-CoT training fails to generalize to OOD samples due to unseen reasoning patterns, whereas CoT training achieves near-perfect OOD generalization by mastering subtasks and reasoning compositions during training. The identified mechanisms explain our experimental results: CoT training accelerates convergence and enhances generalization from ID to both ID and OOD scenarios while maintaining robust performance even with tolerable noise. These findings are further validated on complex real-world datasets. This paper offers valuable insights for designing CoT strategies to enhance LLM reasoning robustness.
Deep learning methods are known to generalize well from training to future data, even in an overparametrized regime, where they could easily overfit. One explanation for this phenomenon is that even when their *ambient dimensionality*, (i.e. the number of parameters) is large, the models' *intrinsic dimensionality* is small; specifically, their learning takes place in a small subspace of all possible weight configurations. In this work, we confirm this phenomenon in the setting of *deep multi-task learning*. We introduce a method to parametrize multi-task network directly in the low-dimensional space, facilitated by the use of *random expansions* techniques. We then show that high-accuracy multi-task solutions can be found with much smaller intrinsic dimensionality (fewer free parameters) than what single-task learning requires. Subsequently, we show that the low-dimensional representations in combination with *weight compression* and *PAC-Bayesian* reasoning lead to the *first non-vacuous generalization bounds* for deep multi-task networks.
Hypergraph neural networks have been promising tools for handling learning tasks involving higher-order data, with notable applications in web graphs, such as modeling multi-way hyperlink structures and complex user interactions. Yet, their generalization abilities in theory are less clear to us. In this paper, we seek to develop margin-based generalization bounds for four representative classes of hypergraph neural networks, including convolutional-based methods (UniGCN), set-based aggregation (AllDeepSets), invariant and equivariant transformations (M-IGN), and tensor-based approaches (T-MPHN). Through the PAC-Bayes framework, our results reveal the manner in which hypergraph structure and spectral norms of the learned weights can affect the generalization bounds, where the key technical challenge lies in developing new perturbation analysis for hypergraph neural networks, which offers a rigorous understanding of how variations in the model's weights and hypergraph structure impact its generalization behavior. Our empirical study examines the relationship between the practical performance and theoretical bounds of the models over synthetic and real-world datasets. One of our primary observations is the strong correlation between the theoretical bounds and empirical loss, with statistically significant consistency in most cases.
We investigate the phenomenon of generalization through the lens of compression. In particular, we study the complexity dynamics of neural networks to explain grokking, where networks suddenly transition from memorizing to generalizing solutions long after over-fitting the training data. To this end we introduce a new measure of intrinsic complexity for neural networks based on the theory of Kolmogorov complexity. Tracking this metric throughout network training, we find a consistent pattern in training dynamics, consisting of a rise and fall in complexity. We demonstrate that this corresponds to memorization followed by generalization. Based on insights from rate--distortion theory and the minimum description length principle, we lay out a principled approach to lossy compression of neural networks, and connect our complexity measure to explicit generalization bounds. Based on a careful analysis of information capacity in neural networks, we propose a new regularization method which encourages networks towards low-rank representations by penalizing their spectral entropy, and find that our regularizer outperforms baselines in total compression of the dataset.
In this work, we introduce novel information-theoretic generalization bounds using the conditional $f$-information framework, an extension of the traditional conditional mutual information (MI) framework. We provide a generic approach to derive generalization bounds via $f$-information in the supersample setting, applicable to both bounded and unbounded loss functions. Unlike previous MI-based bounds, our proof strategy does not rely on upper bounding the cumulant-generating function (CGF) in the variational formula of MI. Instead, we set the CGF or its upper bound to zero by carefully selecting the measurable function invoked in the variational formula. Although some of our techniques are partially inspired by recent advances in the coin-betting framework (e.g., Jang et al. (2023)), our results are independent of any previous findings from regret guarantees of online gambling algorithms. Additionally, our newly derived MI-based bound recovers many previous results and improves our understanding of their potential limitations. Finally, we empirically compare various $f$-information measures for generalization, demonstrating the improvement of our new bounds over the previous bounds.
Bayesian inference provides a principled probabilistic framework for quantifying uncertainty by updating beliefs based on prior knowledge and observed data through Bayes' theorem. In Bayesian deep learning, neural network weights are treated as random variables with prior distributions, allowing for a probabilistic interpretation and quantification of predictive uncertainty. However, Bayesian methods lack theoretical generalization guarantees for unseen data. PAC-Bayesian analysis addresses this limitation by offering a frequentist framework to derive generalization bounds for randomized predictors, thereby certifying the reliability of Bayesian methods in machine learning. Temperature $T$, or inverse-temperature $\lambda = \frac{1}{T}$, originally from statistical mechanics in physics, naturally arises in various areas of statistical inference, including Bayesian inference and PAC-Bayesian analysis. In Bayesian inference, when $T < 1$ (``cold'' posteriors), the likelihood is up-weighted, resulting in a sharper posterior distribution. Conversely, when $T > 1$ (``warm'' posteriors), the likelihood is down-weighted, leading to a more diffuse posterior distribution. By balancing the influence of observed data and prior regularization, temperature adjustments can address issues of underfitting or overfitting in Bayesian models, bringing improved predictive performance.
Large Language Models (LLMs), built on Transformer architectures, exhibit remarkable generalization across a wide range of tasks. However, fine-tuning these models for specific tasks remains resource-intensive due to their extensive parameterization. In this paper, we explore two remarkable phenomena related to the attention mechanism during the fine-tuning of LLMs (where $\mathbf{W}_q$, $\mathbf{W}_k$, and $\mathbf{W}_v$ denote the weights of the query, key, and value layers, respectively). The first phenomenon, termed "Unequal Importance of Attention Matrices", highlights the impact of fine-tuning different weight matrices. It shows that optimizing the $\mathbf{W}_v$ matrix yields significantly better performance than optimizing the $\mathbf{W}_k$ matrix. Fine-tuning only the $\mathbf{W}_q$ and $\mathbf{W}_v$ matrices is computationally efficient while delivering results comparable to, or even better than fine-tuning all three matrices ($\mathbf{W}_q$, $\mathbf{W}_k$, and $\mathbf{W}_v$). The second phenomenon,"Attention Matrices with Customized Learning Rate Lead to Better Convergence", emphasizes the importance of assigning distinct learning rates to these matrices. Specifically, a higher learning rate for the $\mathbf{W}_v$ matrix compared to $\mathbf{W}_q$ and $\mathbf{W}_k$ accelerates convergence and improves performance. Building on these insights, we propose a new strategy that improves fine-tuning efficiency in terms of both storage and time. Experimental results on benchmark datasets validate the effectiveness of this approach, supporting our theoretical findings. Our analysis lays the theoretical groundwork for configuring and improving algorithms in LLMs fine-tuning.
The sample compression theory provides generalization guarantees for predictors that can be fully defined using a subset of the training dataset and a (short) message string, generally defined as a binary sequence. Previous works provided generalization bounds for the zero-one loss, which is restrictive notably when applied to deep learning approaches. In this paper, we present a general framework for deriving new sample compression bounds that hold for real-valued unbounded losses. Using the Pick-To-Learn (P2L) meta-algorithm, which transforms the training method of any machine-learning predictor to yield sample-compressed predictors, we empirically demonstrate the tightness of the bounds and their versatility by evaluating them on random forests and multiple types of neural networks.
The ability of machine learning (ML) algorithms to generalize well to unseen data has been studied through the lens of information theory, by bounding the generalization error with the input-output mutual information (MI), i.e., the MI between the training data and the learned hypothesis. Yet, these bounds have limited practicality for modern ML applications (e.g., deep learning), due to the difficulty of evaluating MI in high dimensions. Motivated by recent findings on the compressibility of neural networks, we consider algorithms that operate by slicing the parameter space, i.e., trained on random lower-dimensional subspaces. We introduce new, tighter information-theoretic generalization bounds tailored for such algorithms, demonstrating that slicing improves generalization. Our bounds offer significant computational and statistical advantages over standard MI bounds, as they rely on scalable alternative measures of dependence, i.e., disintegrated mutual information and $k$-sliced mutual information. Then, we extend our analysis to algorithms whose parameters do not need to exactly lie on random subspaces, by leveraging rate-distortion theory. This strategy yields generalization bounds that incorporate a distortion term measuring model compressibility under slicing, thereby tightening existing bounds without compromising performance or requiring model compression. Building on this, we propose a regularization scheme enabling practitioners to control generalization through compressibility. Finally, we empirically validate our results and achieve the computation of non-vacuous information-theoretic generalization bounds for neural networks, a task that was previously out of reach.
Pre-trained large language models (LLMs) based on Transformer have demonstrated striking in-context learning (ICL) abilities. With a few demonstration input-label pairs, they can predict the label for an unseen input without any parameter updates. In this paper, we show an exciting phenomenon that SVD-based weight pruning can enhance ICL performance, and more surprising, pruning weights in deep layers often results in more stable performance improvements than in shallow layers. However, the underlying mechanism of those findings still remains an open question. To reveal those findings, we conduct an in-depth theoretical analysis by presenting the implicit gradient descent (GD) trajectories of ICL and giving the mutual information based generalization bounds of ICL via full implicit GD trajectories. This helps us reasonably explain the surprising experimental findings. Besides, based on all our experimental and theoretical insights, we intuitively propose a simple, model-compression and derivative-free algorithm for downstream tasks in enhancing ICL inference. Experiments on benchmark datasets and open source LLMs display the method effectiveness\footnote{The code is available at \url{https://github.com/chen123CtrlS/EnhancingICL_SVDPruning}.}.
The generalization bound is a crucial theoretical tool for assessing the generalizability of learning methods and there exist vast literatures on generalizability of normal learning, adversarial learning, and data poisoning. Unlike other data poison attacks, the backdoor attack has the special property that the poisoned triggers are contained in both the training set and the test set and the purpose of the attack is two-fold. To our knowledge, the generalization bound for the backdoor attack has not been established. In this paper, we fill this gap by deriving algorithm-independent generalization bounds in the clean-label backdoor attack scenario. Precisely, based on the goals of backdoor attack, we give upper bounds for the clean sample population errors and the poison population errors in terms of the empirical error on the poisoned training dataset. Furthermore, based on the theoretical result, a new clean-label backdoor attack is proposed that computes the poisoning trigger by combining adversarial noise and indiscriminate poison. We show its effectiveness in a variety of settings.
Bounding and predicting the generalization gap of overparameterized neural networks remains a central open problem in theoretical machine learning. There is a recent and growing body of literature that proposes the framework of fractals to model optimization trajectories of neural networks, motivating generalization bounds and measures based on the fractal dimension of the trajectory. Notably, the persistent homology dimension has been proposed to correlate with the generalization gap. This paper performs an empirical evaluation of these persistent homology-based generalization measures, with an in-depth statistical analysis. Our study reveals confounding effects in the observed correlation between generalization and topological measures due to the variation of hyperparameters. We also observe that fractal dimension fails to predict generalization of models trained from poor initializations. We lastly reveal the intriguing manifestation of model-wise double descent in these topological generalization measures. Our work forms a basis for a deeper investigation of the causal relationships between fractal geometry, topological data analysis, and neural network optimization.
Many algorithms have been recently proposed for causal machine learning. Yet, there is little to no theory on their quality, especially considering finite samples. In this work, we propose a theory based on generalization bounds that provides such guarantees. By introducing a novel change-of-measure inequality, we are able to tightly bound the model loss in terms of the deviation of the treatment propensities over the population, which we show can be empirically limited. Our theory is fully rigorous and holds even in the face of hidden confounding and violations of positivity. We demonstrate our bounds on semi-synthetic and real data, showcasing their remarkable tightness and practical utility.
While a number of knowledge graph representation learning (KGRL) methods have been proposed over the past decade, very few theoretical analyses have been conducted on them. In this paper, we present the first PAC-Bayesian generalization bounds for KGRL methods. To analyze a broad class of KGRL models, we propose a generic framework named ReED (Relation-aware Encoder-Decoder), which consists of a relation-aware message passing encoder and a triplet classification decoder. Our ReED framework can express at least 15 different existing KGRL models, including not only graph neural network-based models such as R-GCN and CompGCN but also shallow-architecture models such as RotatE and ANALOGY. Our generalization bounds for the ReED framework provide theoretical grounds for the commonly used tricks in KGRL, e.g., parameter-sharing and weight normalization schemes, and guide desirable design choices for practical KGRL methods. We empirically show that the critical factors in our generalization bounds can explain actual generalization errors on three real-world knowledge graphs.