We study a fundamental question of domain generalization: given a family of domains (i.e., data distributions), how many randomly sampled domains do we need to collect data from in order to learn a model that performs reasonably well on every seen and unseen domain in the family? We model this problem in the PAC framework and introduce a new combinatorial measure, which we call the domain shattering dimension. We show that this dimension characterizes the domain sample complexity. Furthermore, we establish a tight quantitative relationship between the domain shattering dimension and the classic VC dimension, demonstrating that every hypothesis class that is learnable in the standard PAC setting is also learnable in our setting.

Distribution shifts between training and test data undermine the reliability of deep neural networks, challenging real-world applications across domains and subpopulations. While distributionally robust optimization (DRO) methods like GroupDRO aim to improve robustness by optimizing worst-case performance over predefined groups, their use of a single global classifier can be restrictive when facing substantial inter-environment variability. We propose Multi-Expert Distributionally Robust Optimization (MEDRO), a novel extension of GroupDRO designed to address such complex shifts. MEDRO employs a shared feature extractor with m environment-specific expert classifier heads, and introduces a min-max objective over all m2 expert-environment pairings, explicitly modeling cross-environment risks. This expanded uncertainty set captures fine-grained distributional variations that a single classifier might overlook. Empirical evaluations on a range of standard distribution shift benchmarks demonstrate that MEDRO often achieves robust predictive performance compared to existing methods. Furthermore, MEDRO offers practical inference strategies, such as ensembling or gating mechanisms, for typical scenarios where environment labels are unavailable at test time. Our findings suggest MEDRO as a promising step toward resilient and generalizable machine learning under real-world distribution shifts.

Many confounding-resistant domain generalization methods for image classification have been developed based on causal interventions. However, their reliance on strong assumptions limits their effectiveness in handling unobserved confounders. Although recent work introduces instrumental variables (IVs) to overcome this limitation, the reliance on manually predefined instruments, particularly in the context of visual data, may result in severe bias or invalidity when IV conditions are violated. To address these issues, we propose a novel approach to automatically learning Visual Instrumental Variables for confounding-resistant Domain Generalization (VIV-DG). We observe that certain non-causal visual attributes in image data naturally satisfy the basic conditions required for valid IVs. Motivated by this insight, we propose the visual instrumental variable, a novel concept that extends classical IV theory to the visual domain. Furthermore, we develop an automatic visual instrumental variable learner that enforces IV conditions on learned representations, enabling the automatic learning of valid visual instrumental variables from image data. Ultimately, VIV-DG inherits the strengths of classical IVs to mitigate unobserved confounding and avoids the significant bias caused by violations of IV conditions in predefined IVs. Extensive experiments on multiple benchmarks verify that VIV-DG achieves superior generalization ability.

Real-world data distributions often shift continuously across multiple latent factors such as time, geography, and socioeconomic contexts. However, existing domain generalization approaches typically treat domains as discrete or as evolving along a single axis (e.g., time). This oversimplification fails to capture the complex, multidimensional nature of real-world variation. This paper introduces the task of Continuous Domain Generalization (CDG), which aims to generalize predictive models to unseen domains defined by arbitrary combinations of continuous variations. We present a principled framework grounded in geometric and algebraic theories, showing that optimal model parameters across domains lie on a low-dimensional manifold. To model this structure, we propose a Neural Lie Transport Operator (NeuralLio), which enables structure-preserving parameter transitions by enforcing geometric continuity and algebraic consistency. To handle noisy or incomplete domain variation descriptors, we introduce a gating mechanism to suppress irrelevant dimensions and a local chart-based strategy for robust generalization. Extensive experiments on synthetic and real-world datasets, including remote sensing, scientific documents, and traffic forecasting, demonstrate that our method significantly outperforms existing baselines in both generalization accuracy and robustness.

The generalization ability of deep learning has been extensively studied in supervised settings, yet it remains less explored in unsupervised scenarios. Recently, the Unsupervised Domain Generalization (UDG) task has been proposed to enhance the generalization of models trained with prevalent unsupervised learning techniques, such as Self-Supervised Learning (SSL). UDG confronts the challenge of distinguishing semantics from variations without category labels. Although some recent methods have employed domain labels to tackle this issue, such domain labels are often unavailable in real-world contexts. In this paper, we address these limitations by formalizing UDG as the task of learning a Minimal Sufficient Semantic Representation: a representation that (i) preserves all semantic information shared across augmented views (sufficiency), and (ii) maximally removes information irrelevant to semantics (minimality). We theoretically ground these objectives from the perspective of information theory, demonstrating that optimizing representations to achieve sufficiency and minimality directly reduces out-of-distribution risk. Practically, we implement this optimization through Minimal-Sufficient UDG (MSUDG), a learnable model by integrating (a) an InfoNCE-based objective to achieve sufficiency; (b) two complementary components to promote minimality: a novel semantic-variation disentanglement loss and a reconstruction-based mechanism for capturing adequate variation. Empirically, MS-UDG sets a new state-of-the-art on popular unsupervised domain-generalization benchmarks, consistently outperforming existing SSL and UDG methods, without category or domain labels during representation learning.

Traditional federated learning (FL) algorithms operate under the assumption that the data distributions at training (source domains) and testing (target domain) are the same. The fact that domain shifts often occur in practice necessitates equipping FL methods with a domain generalization (DG) capability. However, existing DG algorithms face fundamental challenges in FL setups due to the lack of samples/domains in each client's local dataset. In this paper, we propose StableFDG, a style and attention based learning strategy for accomplishing federated domain generalization, introducing two key contributions. The first is style-based learning, which enables each client to explore novel styles beyond the original source domains in its local dataset, improving domain diversity based on the proposed style sharing, shifting, and exploration strategies. Our second contribution is an attention-based feature highlighter, which captures the similarities between the features of data samples in the same class, and emphasizes the important/common characteristics to better learn the domain-invariant characteristics of each class in data-poor FL scenarios. Experimental results show that StableFDG outperforms existing baselines on various DG benchmark datasets, demonstrating its efficacy.

Magnification shift is a major obstacle to robust histopathology classification, because models trained on one imaging scale often generalize poorly to another. Here, we evaluated this problem on the BreaKHis dataset using a strict patient-disjoint leave-one-magnification-out protocol, comparing supervised baseline, baseline augmented with DCGAN-generated patches, and a gradient-reversal domain-general model designed to preserve discriminative information while suppressing magnification-specific variation. Across held-out magnifications, the domain-general model achieved the strongest overall discrimination and its clearest gain was observed when 200X was held out. By contrast, GAN augmentation produced inconsistent effects, improving some folds but degrading others, particularly at 400X. The domain-general model also yielded the lowest Brier score at 0.063 vs 0.089 at baseline. Sparse embedding analysis further revealed that domain-general training reduced average signature size more than three-fold (306 versus 1,074 dimensions) while preserving equivalent predictive performance (AUC: 0.967 vs 0.965; F1: 0.930 vs 0.931). It also increased cross-fold signature reproducibility from near-zero Jaccard overlap in the baseline to 0.99 between the 100X and 200X folds. These findings show that calibrated, compact, and transferable representations can be learned without added architectural complexity, with clear implications for the reliable deployment of computational pathology models across heterogeneous acquisition settings.