General ML · generalization

The expressive power of message-passing graph neural networks (MPNNs) is reasonably well understood, primarily through combinatorial techniques from graph isomorphism testing. However, MPNNs' generalization abilities---making meaningful predictions beyond the training set---remain less explored. Current generalization analyses often overlook graph structure, limit the focus to specific aggregation functions, and assume the impractical, hard-to-optimize $0$-$1$ loss function. Here, we extend recent advances in graph similarity theory to assess the influence of graph structure, aggregation, and loss functions on MPNNs' generalization abilities. Our empirical study supports our theoretical insights, improving our understanding of MPNNs' generalization properties.

Self-attention has emerged as a core component of modern neural architectures, yet its theoretical underpinnings remain elusive. In this paper, we study self-attention through the lens of *interacting entities*, ranging from agents in multi-agent reinforcement learning to alleles in genetic sequences, and show that a single layer linear self-attention can *efficiently* represent, learn, and generalize functions capturing pairwise interactions, including out-of-distribution scenarios. Our analysis reveals that self-attention acts as a *mutual interaction learner* under minimal assumptions on the diversity of interaction patterns observed during training, thereby encompassing a wide variety of real-world domains. In addition, we validate our theoretical insights through experiments demonstrating that self-attention learns interaction functions and generalizes across both population distributions and out-of-distribution scenarios. Building on our theories, we introduce *HyperFeatureAttention*, a novel neural network module designed to learn couplings of different feature-level interactions between entities. Furthermore, we propose *HyperAttention*, a new module that extends beyond pairwise interactions to capture multi-entity dependencies, such as three-way, four-way, or general $n$-way interactions.

The paradigm of weak-to-strong generalization constitutes the training of a strong AI model on data labeled by a weak AI model, with the goal that the strong model nevertheless outperforms its weak supervisor on the target task of interest. For the setting of real-valued regression with the squared loss, recent work quantitatively characterizes the gain in performance of the strong model over the weak model in terms of the misfit between the strong and weak model. We generalize such a characterization to learning tasks whose loss functions correspond to arbitrary Bregman divergences when the strong class is convex. This extends the misfit-based characterization of performance gain in weak-to-strong generalization to classification tasks, as the cross-entropy loss can be expressed in terms of a Bregman divergence. In most practical scenarios, however, the strong model class may not be convex. We therefore weaken this assumption and study weak-to-strong generalization for convex combinations of $k$ strong models in the strong class, in the concrete setting of classification. This allows us to obtain a similar misfit-based characterization of performance gain, up to an additional error term that vanishes as $k$ gets large. Our theoretical findings are supported by thorough experiments on synthetic as well as real-world datasets.

In the bandits with knapsacks framework (BwK) the learner has $m$ resource-consumption (i.e., packing) constraints. We focus on the generalization of BwK in which the learner has a set of general long-term constraints. The goal of the learner is to maximize their cumulative reward, while at the same time achieving small cumulative constraints violations. In this scenario, there exist simple instances where conventional methods for BwK fail to yield sublinear violations of constraints. We show that it is possible to circumvent this issue by requiring the primal and dual algorithm to be weakly adaptive. Indeed, even without any information on the Slater's parameter $\rho$ characterizing the problem, the interaction between weakly adaptive primal and dual regret minimizers leads to a ``self-bounding'' behavior of dual variables. In particular, their norm remains suitably upper bounded across the entire time horizon even without explicit projection steps. By exploiting this property, we provide best-of-both-worlds guarantees for stochastic and adversarial inputs. In the first case, we show that the algorithm guarantees sublinear regret. In the latter case, we establish a tight competitive ratio of $\rho/(1+\rho)$. In both settings, constraints violations are guaranteed to be sublinear in time. Finally, this results allow us to obtain new result for the problem of contextual bandits with linear constraints, providing the first no-$\alpha$-regret guarantees for adversarial contexts.

Curvature regularization techniques like Sharpness Aware Minimization (SAM) have shown great promise in improving generalization on vision tasks. However, we find that SAM performs poorly in domains like natural language processing (NLP), often degrading performance --- even with twice the compute budget. We investigate the discrepancy across domains and find that in the NLP setting, SAM is dominated by regularization of the logit statistics --- instead of improving the geometry of the function itself. We use this observation to develop an alternative algorithm we call Functional SAM, which regularizes curvature only through modification of the statistics of the overall function implemented by the neural network, and avoids spurious minimization through logit manipulation. Furthermore, we argue that preconditioning the SAM perturbation also prevents spurious minimization, and when combined with Functional SAM, it gives further improvements. Our proposed algorithms show improved performance over AdamW and SAM baselines when trained for an equal number of steps, in both fixed-length and Chinchilla-style training settings, at various model scales (including billion-parameter scale). On the whole, our work highlights the importance of more precise characterizations of sharpness in broadening the applicability of curvature regularization to large language models (LLMs)

A central goal of machine learning is generalization. While the No Free Lunch Theorem states that we cannot obtain theoretical guarantees for generalization without further assumptions, in practice we observe that simple models which explain the training data generalize best—a principle called Occam's razor. Despite the need for simple models, most current approaches in machine learning only minimize the training error, and at best indirectly promote simplicity through regularization or architecture design. Here, we draw a connection between Occam's razor and in-context learning—an emergent ability of certain sequence models like Transformers to learn at inference time from past observations in a sequence. In particular, we show that the next-token prediction loss used to train in-context learners is directly equivalent to a data compression technique called prequential coding, and that minimizing this loss amounts to jointly minimizing both the training error and the complexity of the model that was implicitly learned from context. Our theory and the empirical experiments we use to support it not only provide a normative account of in-context learning, but also elucidate the shortcomings of current in-context learning methods, suggesting ways in which they can be improved. We make our code available at https://github.com/3rdCore/PrequentialCode.

Composition—the ability to generate myriad variations from finite means—is believed to underlie powerful generalization. However, compositional generalization remains a key challenge for deep learning. A widely held assumption is that learning disentangled (factorized) representations naturally supports this kind of extrapolation. Yet, empirical results are mixed, with many generative models failing to recognize and compose factors to generate out-of-distribution (OOD) samples. In this work, we investigate a controlled 2D Gaussian "bump" generation task with fully disentangled $(x,y)$ inputs, demonstrating that standard generative architectures still fail in OOD regions when training with partial data, by re-entangling latent representations in subsequent layers. By examining the model's learned kernels and manifold geometry, we show that this failure reflects a "memorization" strategy for generation via data superposition rather than via composition of the true factorized features. We show that when models are forced—through architectural modifications with regularization or curated training data—to render the disentangled latents into the full-dimensional representational (pixel) space, they can be highly data-efficient and effective at composing in OOD regions. These findings underscore that disentangled latents in an abstract representation are insufficient and show that if models can represent disentangled factors directly in the output representational space, it can achieve robust compositional generalization.

Information-theoretic bounds, while achieving significant success in analyzing the generalization of randomized learning algorithms, have been criticized for their slow convergence rates and overestimation. This paper presents novel bounds that bridge the expected empirical and population risks through a binarized variant of the Jensen-Shannon divergence. Leveraging our foundational lemma that characterizes the interaction between an arbitrary and a binary variable, we derive hypothesis-based bounds that enhance existing conditional mutual information bounds by reducing the number of conditioned samples from $2$ to $1$. We additionally establish prediction-based bounds that surpass prior bounds based on evaluated loss mutual information measures. Thereafter, through a new binarization technique for the evaluated loss variables, we obtain exactly tight generalization bounds broadly applicable to general randomized learning algorithms for any bounded loss functions. Our results effectively address key limitations of previous results in analyzing certain stochastic convex optimization problems, without requiring additional stability or compressibility assumptions about the learning algorithm.

Weak-to-strong (W2S) generalization is a type of finetuning (FT) where a strong (large) student model is trained on pseudo-labels generated by a weak teacher. Surprisingly, W2S FT often outperforms the weak teacher. We seek to understand this phenomenon through the observation that FT often occurs in intrinsically low-dimensional spaces. Leveraging the low intrinsic dimensionality of FT, we analyze W2S in the ridgeless regression setting from a variance reduction perspective. For a strong student-weak teacher pair with sufficiently expressive low-dimensional feature subspaces $\mathcal{V}_s, \mathcal{V}_w$, we provide an exact characterization of the variance that dominates the generalization error of W2S. This unveils a virtue of discrepancy between the strong and weak models in W2S: the variance of the weak teacher is inherited by the strong student in $\mathcal{V}_s \cap \mathcal{V}_w$, while reduced by a factor of $\mathrm{dim}(\mathcal{V}_s)/N$ in the subspace of discrepancy $\mathcal{V}_w \setminus \mathcal{V}_s$ with $N$ pseudo-labels for W2S. Our analysis further casts light on the sample complexities and the scaling of performance gap recovery in W2S. The analysis is supported by experiments on synthetic regression problems, as well as real vision and NLP tasks.

We introduce Flat Hilbert Bayesian Inference (FHBI), an algorithm designed to enhance generalization in Bayesian inference. Our approach involves an iterative two-step procedure with an adversarial functional perturbation step and a functional descent step within the reproducing kernel Hilbert spaces. This methodology is supported by a theoretical analysis that extends previous findings on generalization ability from finite-dimensional Euclidean spaces to infinite-dimensional functional spaces. To evaluate the effectiveness of FHBI, we conduct comprehensive comparisons against nine baseline methods on the VTAB-1K benchmark, which encompasses 19 diverse datasets across various domains with diverse semantics. Empirical results demonstrate that FHBI consistently outperforms the baselines by notable margins, highlighting its practical efficacy.

The out-of-distribution (OOD) generalization problem in federated learning (FL) has recently attracted significant research interest. A common approach, derived from centralized learning, is to extract causal features which exhibit causal relationships with the label. However, in FL, the global feature extractor typically captures only invariant causal features shared across clients and thus discards many other causal features that are potentially useful for OOD generalization. To address this problem, we propose FedUni, a simple yet effective architecture trained to extract all possible causal features from any input. FedUni consists of a comprehensive feature extractor, designed to identify a union of all causal feature types in the input, followed by a feature compressor, which discards potential \textit{inactive} causal features. With this architecture, FedUni can benefit from collaborative training in FL while avoiding the cost of model aggregation (i.e., extracting only invariant features). In addition, to further enhance the feature extractor's ability to capture causal features, FedUni add a causal intervention module on the client side, which employs a counterfactual generator to generate counterfactual examples that simulate distributions shifts. Extensive experiments and theoretical analysis demonstrate that our method significantly improves OOD generalization performance.

Federated learning (FL) often struggles with generalization due to heterogeneous client data. Local models are prone to overfitting their local data distributions, and even transferable features can be distorted during aggregation. To address these challenges, we propose FedCONST, an approach that adaptively modulates update magnitudes based on the global model’s parameter strength. This prevents over-emphasizing well-learned parameters while reinforcing underdeveloped ones. Specifically, FedCONST employs linear convex constraints to ensure training stability and preserve locally learned generalization capabilities during aggregation. A Gradient Signal-to-Noise Ratio (GSNR) analysis further validates FedCONST's effectiveness in enhancing feature transferability and robustness. As a result, FedCONST effectively aligns local and global objectives, mitigating overfitting and promoting stronger generalization across diverse FL environments, achieving state-of-the-art performance.

It is well-known that the Hessian of deep loss landscape matters to optimization and generalization of deep learning. Previous studies reported a rough Hessian structure in deep learning, which consists of two components, a small number of large eigenvalues and a large number of nearly-zero eigenvalues. To the best of our knowledge, we are the first to report that a simple but overlooked power-law Hessian structure exists in well-trained deep neural networks, including Convolutional Neural Networks (CNNs) and Large Language Models (LLMs). Moreover, we provide a maximum-entropy theoretical interpretation for the power-law Hessian structure and theoretically demonstrate the existence of robust and low-dimensional subspace of deep neural networks. Our extensive experiments using the proposed power-law spectral method demonstrate that the power-law Hessian spectra critically relate to multiple important behaviors of deep learning, including optimization, generalization, and overparameterization. Notably, we discover that the power-law Hessian structure of a given LLM can effectively predict generalization during training, while conventional sharpness-based generalization measures that often works well on CNNs become nearly useless for as a generalization predictor of LLMs.

In many real-world applications, predictive tasks inevitably involve low-quality input features (Weak Features; WFs) which arise due to factors such as misobservations, missingness, or partial observations. While several methods have been proposed to estimate the true values of specific types of WFs and to solve a downstream task, a unified theoretical framework that comprehensively addresses these methods remains underdeveloped. In this paper, we propose a unified framework called Weak Features Learning (WFL), which accommodates arbitrary discrete WFs and a broad range of learning algorithms, and we demonstrate its validity. Furthermore, we introduce a class of algorithms that learn both the estimation model for WFs and the predictive model for a downstream task and perform a generalization error analysis under finite-sample conditions. Our results elucidate the interdependencies between the estimation errors of WFs and the prediction error of a downstream task, as well as the theoretical conditions necessary for the learning approach to achieve consistency. This work establishes a unified theoretical foundation, providing generalization error analysis and performance guarantees, even in scenarios where WFs manifest in diverse forms.

Federated learning (FL) is a widely adopted privacy-preserving distributed learning framework, yet its generalization performance remains less explored compared to centralized learning. In FL, the generalization error consists of two components: the out-of-sample gap, which measures the gap between the empirical and true risk for participating clients, and the participation gap, which quantifies the risk difference between participating and non-participating clients. In this work, we apply an information-theoretic analysis via the conditional mutual information (CMI) framework to study FL's two-level generalization. Beyond the traditional supersample-based CMI framework, we introduce a superclient construction to accommodate the two-level generalization setting in FL. We derive multiple CMI-based bounds, including hypothesis-based CMI bounds, illustrating how privacy constraints in FL can imply generalization guarantees. Furthermore, we propose fast-rate evaluated CMI bounds that recover the best-known convergence rate for two-level FL generalization in the small empirical risk regime. For specific FL model aggregation strategies and structured loss functions, we refine our bounds to achieve improved convergence rates with respect to the number of participating clients. Empirical evaluations confirm that our evaluated CMI bounds are non-vacuous and accurately capture the generalization behavior of FL algorithms.

Large language models (LLMs) exhibit remarkable task generalization, solving tasks they were never explicitly trained on with only a few demonstrations. This raises a fundamental question: When can learning from a small set of tasks generalize to a large task family? In this paper, we investigate task generalization through the lens of autoregressive compositional structure, where each task is a composition of T operations, and each operation is among a finite family of D subtasks. This yields a total class of size~D^T. We first show that generalization to all D^T tasks is theoretically achievable by training on only \tilde{O}(D) tasks. Empirically, we demonstrate that Transformers achieve such exponential task generalization on sparse parity functions via In-context Learning (ICL) and chain-of-thought (CoT) reasoning. We further demonstrate this exponential generalization in arithmetic and language translation, extending beyond parity functions.

Understanding and improving generalization capabilities is crucial for both classical and quantum machine learning (QML). Recent studies have revealed shortcomings in current generalization theories, particularly those relying on uniform bounds, across both classical and quantum settings. In this work, we present a margin-based generalization bound for QML models, providing a more reliable framework for evaluating generalization. Our experimental studies on the quantum phase recognition dataset demonstrate that margin-based metrics are strong predictors of generalization performance, outperforming traditional metrics like parameter count. By connecting this margin-based metric to quantum information theory, we demonstrate how to enhance the generalization performance of QML through a classical-quantum hybrid approach when applied to classical data.

Large language models (LLMs) have demonstrated remarkable potential in handling natural language processing (NLP) tasks and beyond. LLMs usually can be categorized as transformer decoder-only models (DOMs), utilizing Next-Token-Prediction (NTP) as their pre-training methodology. Despite their tremendous empirical successes, the theoretical understanding of how NTP pre-training affects the model's generalization behavior is lacking. To fill this gap, we establish the fine-grained generalization analysis for NTP pre-training based on Rademacher complexity, where the dependence between tokens is also addressed. Technically, a novel decomposition of Rademacher complexity is developed to study DOMs from the representation learner and the token predictor, respectively. Furthermore, the upper bounds of covering number are established for multi-layer and multi-head transformer-decoder models under the Frobenius norm, which theoretically pioneers the incorporation of mask matrix within the self-attention mechanism. Our results reveal that the generalization ability of NTP pre-training is affected quantitively by the number of token sequences $N$, the maximum length of sequence $m$, and the count of parameters in the transformer model $\Theta$. Additionally, experiments on public datasets verify our theoretical findings.

Selecting high-quality candidates from large datasets is critical in applications such as drug discovery, precision medicine, and alignment of large language models (LLMs). While Conformal Selection (CS) provides rigorous uncertainty quantification, it is limited to univariate responses and scalar criteria. To address this, we propose Multivariate Conformal Selection (mCS), a generalization of CS designed for multivariate response settings. Our method introduces regional monotonicity and employs multivariate nonconformity scores to construct conformal $p$-values, enabling finite-sample False Discovery Rate (FDR) control. We present two variants: $\texttt{mCS-dist}$, using distance-based scores, and $\texttt{mCS-learn}$, which learns optimal scores via differentiable optimization. Experiments on simulated and real-world datasets demonstrate that mCS significantly improves selection power while maintaining FDR control, establishing it as a robust framework for multivariate selection tasks.

Sharp generalization bound for neural networks trained by gradient descent (GD) is of central interest in statistical learning theory and deep learning. In this paper, we consider nonparametric regression by an over-parameterized two-layer NN trained by GD. We show that, if the neural network is trained by GD with early stopping, then the trained network renders a sharp rate of the nonparametric regression risk of $O(\epsilon_n^2)$, which is the same rate as that for the classical kernel regression trained by GD with early stopping, where $\epsilon_n$ is the critical population rate of the Neural Tangent Kernel (NTK) associated with the network and $n$ is the size of the training data. It is remarked that our result does not require distributional assumptions on the covariate as long as the covariate lies on the unit sphere, in a strong contrast with many existing results which rely on specific distributions such as the spherical uniform data distribution or distributions satisfying certain restrictive conditions. As a special case of our general result, when the eigenvalues of the associated NTK decay at a rate of $\lambda_j \asymp j^{-\frac{d}{d-1}}$ for $j \ge 1$ which happens under certain distributional assumption such as the training features follow the spherical uniform distribution, we immediately obtain the minimax optimal rate of $O(n^{-\frac{d}{2d-1}})$, which is the major results of several existing works in this direction. The neural network width in our general result is lower bounded by a function of only $d$ and $\epsilon_n$, and such width does not depend on the minimum eigenvalue of the empirical NTK matrix whose lower bound usually requires additional assumptions on the training data. Our results are built upon two significant technical results which are of independent interest. First, uniform convergence to the NTK is established during the training process by GD, so that we can have a nice decomposition of the neural network function at any step of the GD into a function in the Reproducing Kernel Hilbert Space associated with the NTK and an error function with a small $L^{\infty}$-norm. Second, local Rademacher complexity is employed to tightly bound the Rademacher complexity of the function class comprising all the possible neural network functions obtained by GD. Our result formally fills the gap between training a classical kernel regression model and training an over-parameterized but finite-width neural network by GD for nonparametric regression without distributional assumptions about the spherical covariate.