ICLR 2025 Papers

The infinite width limit of random neural networks is known to result in Neural Networks as Gaussian Process (NNGP) (Lee et al. (2018)), characterized by task-independent kernels. It is widely accepted that larger network widths contribute to improved generalization (Park et al. (2019)). However, this work challenges this notion by investigating the narrow width limit of the Bayesian Parallel Branching Neural Network (BPB-NN), an architecture that resembles neural networks with residual blocks. We demonstrate that when the width of a BPB-NN is significantly smaller compared to the number of training examples, each branch exhibits more robust learning due to a symmetry breaking of branches in kernel renormalization. Surprisingly, the performance of a BPB-NN in the narrow width limit is generally superior to or comparable to that achieved in the wide width limit in bias-limited scenarios. Furthermore, the readout norms of each branch in the narrow width limit are mostly independent of the architectural hyperparameters but generally reflective of the nature of the data. We demonstrate such phenomenon primarily in the branching graph neural networks, where each branch represents a different order of convolutions of the graph; we also extend the results to other more general architectures such as the residual-MLP and demonstrate that the narrow width effect is a general feature of the branching networks. Our results characterize a newly defined narrow-width regime for parallel branching networks in general.

Deep classifiers are known to be sensitive to data distribution shifts, primarily due to their reliance on spurious correlations in training data. It has been suggested that these classifiers can still find useful features in the network's last layer that hold up under such shifts. In this work, we question the use of last-layer representations for out-of-distribution (OOD) generalisation and explore the utility of intermediate layers. To this end, we introduce \textit{Intermediate Layer Classifiers} (ILCs). We discover that intermediate layer representations frequently offer substantially better generalisation than those from the penultimate layer. In many cases, zero-shot OOD generalisation using earlier-layer representations approaches the few-shot performance of retraining on penultimate layer representations. This is confirmed across multiple datasets, architectures, and types of distribution shifts. Our analysis suggests that intermediate layers are less sensitive to distribution shifts compared to the penultimate layer. These findings highlight the importance of understanding how information is distributed across network layers and its role in OOD generalisation, while also pointing to the limits of penultimate layer representation utility. Code is available at https://github.com/oshapio/intermediate-layer-generalization.

We study goal-conditioned RL through the lens of generalization, but not in the traditional sense of random augmentations and domain randomization. Rather, we aim to learn goal-directed policies that generalize with respect to the horizon: after training to reach nearby goals (which are easy to learn), these policies should succeed in reaching distant goals (which are quite challenging to learn). In the same way that invariance is closely linked with generalization is other areas of machine learning (e.g., normalization layers make a network invariant to scale, and therefore generalize to inputs of varying scales), we show that this notion of horizon generalization is closely linked with invariance to planning: a policy navigating towards a goal will select the same actions as if it were navigating to a waypoint en route to that goal. Horizon generalization and invariance to planning are appealing because of their potential reach: they imply that a policy trained to reach nearby goals would succeed at reaching goals that are arbitrarily more distant.Our theoretical analysis proves that both horizon generalization and planning invariance are possible, under some assumptions. We present new experimental results, as well as recalling results from prior work, in support of our theoretical results. Taken together, our results open the door to studying how techniques for invariance and generalization developed in other areas of machine learning might be adapted to achieve this alluring property.

The primary objective of learning methods is generalization. Classic generalization bounds, based on VC-dimension or Rademacher complexity, are uniformly applicable to all networks in the hypothesis space. On the other hand, algorithm-dependent generalization bounds, like stability bounds, address more practical scenarios and provide generalization conditions for neural networks trained using SGD. However, these bounds often rely on strict assumptions, such as the NTK hypothesis or convexity of the empirical loss, which are typically not met by neural networks. In order to establish generalizability under less stringent assumptions, this paper investigates generalizability of neural networks that minimize the empirical risk. A lower bound for population accuracy is established based on the expressiveness of these networks, which indicates that with adequately large training sample and network sizes, these networks can generalize effectively. Additionally, we provide a lower bound necessary for generalization, demonstrating that, for certain data distributions, the quantity of data required to ensure generalization exceeds the network size needed to represent that distribution. Finally, we provide theoretical insights into several phenomena in deep learning, including robust overfitting, importance of over-parameterization networks, and effects of loss functions.

Sharpness-Aware Minimization (SAM) has emerged as a powerful method for improving generalization in machine learning models by minimizing the sharpness of the loss landscape. However, despite its success, several important questions regarding the convergence properties of SAM in non-convex settings are still open, including the benefits of using normalization in the update rule, the dependence of the analysis on the restrictive bounded variance assumption, and the convergence guarantees under different sampling strategies. To address these questions, in this paper, we provide a unified analysis of SAM and its unnormalized variant (USAM) under one single flexible update rule (Unified SAM), and we present convergence results of the new algorithm under a relaxed and more natural assumption on the stochastic noise. Our analysis provides convergence guarantees for SAM under different step size selections for non-convex problems and functions that satisfy the Polyak-Lojasiewicz (PL) condition (a non-convex generalization of strongly convex functions). The proposed theory holds under the arbitrary sampling paradigm, which includes importance sampling as special case, allowing us to analyze variants of SAM that were never explicitly considered in the literature. Experiments validate the theoretical findings and further demonstrate the practical effectiveness of Unified SAM in training deep neural networks for image classification tasks.

Domain Generalization (DG) aims to learn from multiple known source domains a model that can generalize well to unknown target domains. One of the key approaches in DG is training an encoder which generates domain-invariant representations. However, this approach is not applicable in Federated Domain Generalization (FDG), where data from various domains are distributed across different clients. In this paper, we introduce a novel approach, dubbed Federated Learning via On-server Matching Gradient (FedOMG), which can efficiently leverage domain information from distributed domains. Specifically, we utilize the local gradients as information about the distributed models to find an invariant gradient direction across all domains through gradient inner product maximization. The advantages are two-fold: 1) FedOMG can aggregate the characteristics of distributed models on the centralized server without incurring any additional communication cost, and 2) FedOMG is orthogonal to many existing FL/FDG methods, allowing for additional performance improvements by being seamlessly integrated with them. Extensive experimental evaluations on various settings demonstrate the robustness of FedOMG compared to other FL/FDG baselines. Our method outperforms recent SOTA baselines on four FL benchmark datasets (MNIST, EMNIST, CIFAR-10, and CIFAR-100), and three FDG benchmark datasets (PACS, VLCS, and OfficeHome). The reproducible code is publicly available~\footnote[1]{\url{https://github.com/skydvn/fedomg}}.

How diffusion models generalize beyond their training set is not known, and is somewhat mysterious given two facts: the optimum of the denoising score matching (DSM) objective usually used to train diffusion models is the score function of the training distribution; and the networks usually used to learn the score function are expressive enough to learn this score to high accuracy. We claim that a certain feature of the DSM objective—the fact that its target is not the training distribution's score, but a noisy quantity only equal to it in expectation—strongly impacts whether and to what extent diffusion models generalize. In this paper, we develop a mathematical theory that partly explains this 'generalization through variance' phenomenon. Our theoretical analysis exploits a physics-inspired path integral approach to compute the distributions typically learned by a few paradigmatic under- and overparameterized diffusion models. We find that the distributions diffusion models effectively learn to sample from resemble their training distributions, but with `gaps' filled in, and that this inductive bias is due to the covariance structure of the noisy target used during training. We also characterize how this inductive bias interacts with feature-related inductive biases.

Choice of training data distribution greatly influences model behavior. Yet, in large-scale settings, precisely characterizing *how* changes in training data affects predictions is often difficult due to model training costs. Current practice is to instead extrapolate from scaled down, inexpensive-to-train proxy models. However, changes in data do not influence smaller and larger models identically. Therefore, understanding how choice of data affects large-scale models raises the question: how does training data distribution influence model behavior across compute scale? We find that small- and large-scale language model predictions (generally) *do* highly correlate across choice of training data. Equipped with these findings, we characterize how proxy scale affects effectiveness in two downstream proxy model applications: data attribution and dataset selection.

Benign overfitting refers to the phenomenon where an over-parameterized model fits the training data perfectly, including noise in the data, but still generalizes well to the unseen test data. While prior work provides some theoretical understanding of this phenomenon under the in-distribution setup, modern machine learning often operates in a more challenging Out-of-Distribution (OOD) regime, where the target (test) distribution can be rather different from the source (training) distribution. In this work, we take an initial step towards understanding benign overfitting in the OOD regime by focusing on the basic setup of over-parameterized linear models under covariate shift. We provide non-asymptotic guarantees proving that benign overfitting occurs in standard ridge regression, even under the OOD regime when the target covariance satisfies certain structural conditions. We identify several vital quantities relating to source and target covariance, which govern the performance of OOD generalization. Our result is sharp, which provably recovers prior in-distribution benign overfitting guarantee (Tsigler & Bartlett, 2023), as well as under-parameterized OOD guarantee (Ge et al., 2024) when specializing to each setup. Moreover, we also present theoretical results for a more general family of target covariance matrix, where standard ridge regression only achieves a slow statistical rate of $\mathcal{O}(1/\sqrt{n})$ for the excess risk, while Principal Component Regression (PCR) is guaranteed to achieve the fast rate $\mathcal{O}(1/n)$, where $n$ is the number of samples.

Modular neural networks outperform nonmodular neural networks on tasks ranging from visual question answering to robotics. These performance improvements are thought to be due to modular networks' superior ability to model the compositional and combinatorial structure of real-world problems. However, a theoretical explanation of how modularity improves generalizability, and how to leverage task modularity while training networks remains elusive. Using recent theoretical progress in explaining neural network generalization, we investigate how the amount of training data required to generalize on a task varies with the intrinsic dimensionality of a task's input. We show theoretically that when applied to modularly structured tasks, while nonmodular networks require an exponential number of samples with task dimensionality, modular networks' sample complexity is independent of task dimensionality: modular networks can generalize in high dimensions. We then develop a novel learning rule for modular networks to exploit this advantage and empirically show the improved generalization of the rule, both in- and out-of-distribution, on high-dimensional, modular tasks.

Recent work has shown that Transformers trained from scratch can successfully solve various arithmetic and algorithmic tasks, such as adding numbers and computing parity. While these Transformers generalize well on unseen inputs of the same length, they struggle with length generalization, i.e., handling inputs of unseen lengths. In this work, we demonstrate that looped Transformers with an adaptive number of steps significantly improve length generalization. We focus on tasks with a known iterative solution, involving multiple iterations of a RASP-L operation—a length-generalizable operation that can be expressed by a finite-sized Transformer. We train looped Transformers using our proposed learning algorithm and observe that they learn highly length-generalizable solutions for various tasks.

In this paper, we present a novel stability analysis of adversarial training and prove generalization upper bounds in terms of an expansiveness property of adversarial perturbations used during training and used for evaluation. These expansiveness parameters appear not only govern the vanishing rate of the generalization error but also govern its scaling constant. Our proof techniques do not rely on artificial assumptions of the adversarial loss, as are typically used in previous works. Our bound attributes the robust overfitting in PGD-based adversarial training to the sign function used in the PGD attack, resulting in a bad expansiveness parameter. The peculiar choice of sign function in the PGD attack appears to impact adversarial training both in terms of (inner) optimization and in terms of generalization, as shown in this work. This aspect has been largely overlooked to date. Going beyond the sign-function based PGD attacks, we further show that poor expansiveness properties exist in a wide family of PGD-like iterative attack algorithms, which may highlight an intrinsic difficulty in adversarial training.