General ML · generalization

A crucial factor for successful human and AI interaction is the ability of language models or chatbots to follow human instructions precisely. A common feature of instructions are output constraints like ``only answer with yes or no" or ``mention the word `abracadabra' at least 3 times" that the user adds to craft a more useful answer. Even today's strongest models struggle with fulfilling such constraints. We find that most models strongly overfit on a small set of verifiable constraints from the benchmarks that test these abilities, a skill called precise instruction following, and are not able to generalize well to unseen output constraints. We introduce a new benchmark, IFBench, to evaluate precise instruction following generalization on 58 new, diverse, and challenging verifiable out-of-domain constraints. In addition, we perform an extensive analysis of how and on what data models can be trained to improve precise instruction following generalization. Specifically, we carefully design constraint verification modules and show that reinforcement learning with verifiable rewards (RLVR) significantly improves instruction following. In addition to IFBench, we release 29 additional new hand-annotated training constraints and verification functions, RLVR training prompts, and code.

The role of hidden units in recurrent neural networks is typically seen as modeling memory, with research focusing on enhancing information retention through gating mechanisms. A less explored perspective views hidden units as active participants in the computation performed by the network, rather than passive memory stores. In this work, we revisit bilinear operations, which involve multiplicative interactions between hidden units and input embeddings. We demonstrate theoretically and empirically that they constitute a natural inductive bias for representing the evolution of hidden states in state tracking tasks. These are the simplest type of tasks that require hidden units to actively contribute to the behavior of the network. We also show that bilinear state updates form a natural hierarchy corresponding to state tracking tasks of increasing complexity, with popular linear recurrent networks such as Mamba residing at the lowest-complexity center of that hierarchy.

Weak-to-strong generalization refers to the phenomenon where a stronger model trained under supervision from a weaker one can outperform its teacher. While prior studies aim to explain this effect, most theoretical insights are limited to abstract frameworks or linear/random feature models. In this paper, we provide a formal analysis of weak-to-strong generalization from a linear CNN (weak) to a two-layer ReLU CNN (strong). We consider structured data composed of label-dependent signals of varying difficulty and label-independent noise, and analyze gradient descent dynamics when the strong model is trained on data labeled by the pretrained weak model. Our analysis identifies two regimes—data-scarce and data-abundant—based on the signal-to-noise characteristics of the dataset, and reveals distinct mechanisms of weak-to-strong generalization. In the data-scarce regime, generalization occurs via benign overfitting or fails via harmful overfitting, depending on the amount of data, and we characterize the transition boundary. In the data-abundant regime, generalization emerges in the early phase through label correction, but we observe that overtraining can subsequently degrade performance.

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 (MS-UDG), 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.

We prove the first generalization bound for large-margin halfspaces that is asymptotically tight in the tradeoff between the margin, the fraction of training points with the given margin, the failure probability and the number of training points.

Benchmarks for out-of-distribution (OOD) generalization often reveal a strong positive correlation between in-distribution (ID) and OOD accuracy across models, a phenomenon known as “accuracy-on-the-line.” This pattern is commonly interpreted as evidence that spurious correlations—relationships that improve ID but harm OOD performance—are rare in practice. We show that this positive correlation can be an artifact of aggregating heterogeneous OOD examples. Using a simple gradient-based method, OODSelect, we identify semantically coherent OOD subsets where accuracy-on-the-line breaks down. Across widely used distribution-shift benchmarks, OODSelect uncovers subsets—sometimes comprising more than half of the standard OOD set—where higher ID accuracy predicts lower OOD accuracy. These results suggest that aggregate metrics can mask critical failure modes in OOD robustness. We release code and the identified subsets to support further research.

In this paper, we propose a new method called \textit{Self-Training with Dynamic Weighting} (STDW), which aims to enhance robustness in Gradual Domain Adaptation (GDA) by addressing the challenge of smooth knowledge migration from the source to the target domain. Traditional GDA methods mitigate domain shift through intermediate domains and self-training but often suffer from inefficient knowledge migration or incomplete intermediate data. Our approach introduces a dynamic weighting mechanism that adaptively balances the loss contributions of the source and target domains during training. Specifically, we design an optimization framework governed by a time-varying hyperparameter $\varrho$ (progressing from 0 to 1), which controls the strength of domain-specific learning and ensures stable adaptation. The method leverages self-training to generate pseudo-labels and optimizes a weighted objective function for iterative model updates, maintaining robustness across intermediate domains. Experiments on rotated MNIST, color-shifted MNIST, portrait datasets, and the Cover Type dataset demonstrate that STDW outperforms existing baselines. Ablation studies further validate the critical role of $\varrho$'s dynamic scheduling in achieving progressive adaptation, confirming its effectiveness in reducing domain bias and improving generalization. This work provides both theoretical insights and a practical framework for robust gradual domain adaptation, with potential applications in dynamic real-world scenarios.

Mixed-integer programming (MIP) provides a powerful framework for optimization problems, with Branch-and-Cut (B&C) being the predominant algorithm in state-of-the-art solvers. The efficiency of B&C critically depends on heuristic policies for making sequential decisions, including node selection, cut selection, and branching variable selection. While traditional solvers often employ heuristics with manually tuned parameters, recent approaches increasingly leverage machine learning, especially neural networks, to learn these policies directly from data. A key challenge is to understand the theoretical underpinnings of these learned policies, particularly their generalization performance from finite data. This paper establishes rigorous sample complexity bounds for learning B&C policies where the scoring functions guiding each decision step (node, cut, branch) have a certain piecewise polynomial structure. This structure generalizes the linear models that form the most commonly deployed policies in practice and investigated recently in a foundational series of theoretical works by Balcan et al. Such piecewise polynomial policies also cover the neural network architectures (e.g., using ReLU activations) that have been the focal point of contemporary practical studies. Consequently, our theoretical framework closely reflects the models utilized by practitioners investigating machine learning within B&C, offering a unifying perspective relevant to both established theory and modern empirical research in this area. Furthermore, our theory applies to quite general sequential decision making problems beyond B&C.

Understanding how the test risk scales with model complexity is a central question in machine learning. Classical theory is challenged by the learning curves observed for large over-parametrized deep networks. Capacity measures based on parameter count typically fail to account for these empirical observations. To tackle this challenge, we consider norm-based capacity measures and develop our study for random features based estimators, widely used as simplified theoretical models for more complex networks. In this context, we provide a precise characterization of how the estimator’s norm concentrates and how it governs the associated test error. Our results show that the predicted learning curve admits a phase transition from under- to over-parameterization, but no double descent behavior. This confirms that more classical U-shaped behavior is recovered considering appropriate capacity measures based on models norms rather than size. From a technical point of view, we leverage deterministic equivalence as the key tool and further develop new deterministic quantities which are of independent interest.

Semantic correspondence (SC) aims to establish semantically meaningful matches across different instances of an object category. We illustrate how recent supervised SC methods remain limited in their ability to generalize beyond sparsely annotated training keypoints, effectively acting as keypoint detectors. To address this, we propose a novel approach for learning dense correspondences by lifting 2D keypoints into a canonical 3D space using monocular depth estimation. Our method constructs a continuous canonical manifold that captures object geometry without requiring explicit 3D supervision or camera annotations. Additionally, we introduce SPair-U, an extension of SPair-71k with novel keypoint annotations, to better assess generalization. Experiments not only demonstrate that our model significantly outperforms supervised baselines on unseen keypoints, highlighting its effectiveness in learning robust correspondences, but that unsupervised baselines outperform supervised counterparts when generalized across different datasets.

Large Vision-Language Action (VLA) models have shown significant potential for embodied AI. However, their predominant training via supervised fine-tuning (SFT) limits generalization due to susceptibility to compounding errors under distribution shifts. Reinforcement learning (RL) offers a path to overcome these limitations by optimizing for task objectives via trial-and-error, yet a systematic understanding of its specific generalization benefits for VLAs compared to SFT is lacking. To address this, our study introduces a comprehensive benchmark for evaluating VLA generalization and systematically investigates the impact of RL fine-tuning across diverse visual, semantic, and execution dimensions. Our extensive experiments reveal that RL fine-tuning, particularly with PPO, significantly enhances generalization in semantic understanding and execution robustness over SFT, while maintaining comparable visual robustness. We identify PPO as a more effective RL algorithm for VLAs than LLM-derived methods like DPO and GRPO. We also develop a simple recipe for efficient PPO training on VLAs, and demonstrate its practical utility for improving VLA generalization. The project page is at https://rlvla.github.io

We study which machine learning algorithms have tight generalization bounds with respect to a given collection of population distributions. Our results build on and extend the recent work of Gastpar et al. (2023). First, we present conditions that preclude the existence of tight generalization bounds. Specifically, we show that algorithms that have certain inductive biases that cause them to be unstable do not admit tight generalization bounds. Next, we show that algorithms that are sufficiently loss-stable do have tight generalization bounds. We conclude with a simple characterization that relates the existence of tight generalization bounds to the conditional variance of the algorithm's loss.

Weak-to-strong generalization—where a student model trained on imperfect labels generated by a weaker teacher nonetheless surpasses that teacher—has been widely observed, but the mechanisms that enable it have remained poorly understood. In this paper, through a theoretical analysis of simple models, we uncover three core mechanisms that can drive this phenomenon. First, by analyzing ridge linear regression, we study the interplay between the teacher and student regularization parameters and prove that a student can compensate for a teacher’s under-regularization and achieve lower test error. We also analyze the role of the parameterization regime of the models and show that qualitatively different phenomena can happen in different regimes. Second, by analyzing weighted ridge linear regression, we show that a student model with a regularization structure better aligned to the target function, can outperform its teacher. Third, in a nonlinear multi‐index learning setting, we demonstrate that a student can learn easy, task-specific features from the teacher while leveraging its own broader pre-training to learn hard‐to‐learn features that the teacher cannot capture.

Advancements in direct-integration fine-tuning frameworks have underscored their potential to enhance the performance of labeled scenarios and tasks. To enhance the generalization of different categories in the same dataset, some methods have added visual loss to these frameworks for unlabeled scenarios. However, the performance of these methods through visual loss does not improve significantly in domain generalization and cross-dataset generalization tasks. This may be attributed to the uncoordinated learning of the two-modalities alignment and visual loss. To mitigate this issue of uncoordinated learning, we propose a novel method called Levelling Paradigm (LePa) to improve performance for unlabeled tasks or scenarios. The proposed LePa, designed as a plug-in module, dynamically constrains and coordinates multiple objective functions, thereby improving the generalization of these baseline methods. Comprehensive experiments have shown that our design can effectively address generalized scenarios and tasks.

It has been recently observed in much of the literature that neural networks exhibit a bottleneck rank property: for larger depths, the activation and weights of neural networks trained with gradient-based methods tend to be of approximately low rank. In fact, the rank of the activations of each layer converges to a fixed value referred to as the ``bottleneck rank", which is the minimum rank required to represent the training data. This perspective is in line with the observation that regularizing linear networks (without activations) with weight decay is equivalent to minimizing the Schatten $p$ quasi norm of the neural network. In this paper we investigate the implications of this phenomenon for generalization. More specifically, we prove generalization bounds for neural networks which exploit the approximate low rank structure of the weight matrices if present. The final results rely on the Schatten $p$ quasi norms of the weight matrices: for small p, the bounds exhibit a sample complexity $ \widetilde{O}(WrL^2)$ where $W$ and $L$ are the width and depth of the neural network respectively and where $r$ is the rank of the weight matrices. As $p$ increases, the bound behaves more like a norm-based bound instead. The proof techniques involve a careful interpolation between the parametric and norm based regimes. We also demonstrate in experiments that this bound outperforms both classic parameter counting and norm based bounds in the typical overparametrized regime.

The emergence of large language models (LLMs) has opened new opportunities for AI-driven chemical problem solving. However, existing chemical LLMs are typically tailored to specific task formats or narrow domains, limiting their capacity to integrate knowledge and generalize across tasks. Model merging offers a promising route for efficiently combining specialized LLMs into a unified model without access to original training data, which is urgently needed in the chemical domain where in-house data and privacy preservation are critical. However, effective model merging in the chemical domain poses unique challenges: (1) significant disparities among chemical LLMs due to task-specific specialization, and (2) a highly imbalanced distribution of chemical LLMs in targeted downstream tasks, where some are over-benchmarked while others remain underexplored. These challenges intensify model inconsistencies such as parameter interference and accumulated fine-tuning noise, which collectively hinder effective model merging. To this end, we propose Curriculum Model Merging (CMM), a curriculum-based framework that progressively merges expert chemical LLMs in a moderate and continual manner. CMM aims to harmonize their inconsistencies while meantime preserve their domain-specific expertise. Comprehensive experiments on two benchmark datasets show that CMM effectively consolidates task-specific expertise and outperforms the state-of-the-art methods by 29.03\% in terms of overall average performance. Moreover, CMM facilitates chemical knowledge generalization across prediction and generative tasks without sacrificing robustness, exhibiting promising merging performance under both expert-abundant and expert-sparse scenarios.

Graph Neural Networks (GNNs) often encounter significant performance degradation under distribution shifts between training and test data, hindering their applicability in real-world scenarios. Recent studies have proposed various methods to address the out-of-distribution (OOD) generalization challenge, with many methods in the graph domain focusing on directly identifying an invariant subgraph that is predictive of the target label. However, we argue that identifying the edges from the invariant subgraph directly is challenging and error-prone, especially when some spurious edges exhibit strong correlations with the targets. In this paper, we propose $\texttt{PrunE}$, the first pruning-based graph OOD method that eliminates spurious edges to improve OOD generalizability. By pruning spurious edges, $\texttt{PrunE}$ retains the invariant subgraph more comprehensively, which is critical for OOD generalization. Specifically, $\texttt{PrunE}$ employs two regularization terms to prune spurious edges: 1) _graph size constraint_ to exclude uninformative spurious edges, and 2) _$\epsilon$-probability alignment_ to further suppress the occurrence of spurious edges. Through theoretical analysis and extensive experiments, we show that $\texttt{PrunE}$ achieves superior OOD performance and outperforms previous state-of-the-art methods significantly.

Neural collapse, i.e., the emergence of highly symmetric, class-wise clustered representations, is frequently observed in deep networks and is often assumed to reflect or enable generalization. In parallel, flatness of the loss landscape has been theoretically and empirically linked to generalization. Yet, the causal role of either phenomenon remains unclear: Are they prerequisites for generalization, or merely by-products of training dynamics? We disentangle these questions using grokking, a training regime in which memorization precedes generalization, allowing us to temporally separate generalization from training dynamics and we find that while both neural collapse and relative flatness emerge near the onset of generalization, only flatness consistently predicts it. Models encouraged to collapse or prevented from collapsing generalize equally well, whereas models regularized away from flat solutions exhibit delayed generalization, resembling grokking, even in architectures and datasets where it does not typically occur. Furthermore, we show theoretically that neural collapse leads to relative flatness under classical assumptions, explaining their empirical co-occurrence. Our results support the view that relative flatness is a potentially necessary and more fundamental property for generalization, and demonstrate how grokking can serve as a powerful probe for isolating its geometric underpinnings.

Recent work studying the generalization of diffusion models with locally linear UNet-based denoisers reveals inductive biases that can be expressed via geometry-adaptive harmonic bases. For such locally linear UNets, these geometry-adaptive harmonic bases can be conveniently visualized through the eigen-decomposition of a UNet’s Jacobian matrix. In practice, however, more recent denoising networks are often transformer-based, e.g., the diffusion transformer (DiT). Due to the presence of nonlinear operations, similar eigen-decomposition analyses cannot be used to reveal the inductive biases of transformer-based denoisers. This motivates our search for alternative ways to explain the strong generalization ability observed in DiT models. Investigating a DiT’s pivotal attention modules, we find that locality of attention maps in a DiT’s early layers are closely associated with generalization. To verify this finding, we modify the generalization of a DiT by restricting its attention windows and observe an improvement in generalization. Furthermore, we empirically find that both the placement and the effective attention size of these local attention windows are crucial factors. Experimental results on the CelebA, ImageNet, MSCOCO, and LSUN data show that strengthening the inductive bias of a DiT can improve both generalization and generation quality when less training data is available. Source code is available at https://github.com/DiT-Generalization/DiT-Generalization.

Modern deep generative models can now produce high-quality synthetic samples that are often indistinguishable from real training data. A growing body of research aims to understand why recent methods, such as diffusion and flow matching techniques, generalize so effectively. Among the proposed explanations are the inductive biases of deep learning architectures and the stochastic nature of the conditional flow matching loss. In this work, we rule out the noisy nature of the loss as a key factor driving generalization in flow matching. First, we empirically show that in high-dimensional settings, the stochastic and closed-form versions of the flow matching loss yield nearly equivalent losses. Then, using state-of-the-art flow matching models on standard image datasets, we demonstrate that both variants achieve comparable statistical performance, with the surprising observation that using the closed-form can even improve performance.