ICML 2025 论文列表

In this paper, we derive refined generalization bounds for the Deep Ritz Method (DRM) and Physics-Informed Neural Networks (PINNs). For the DRM, we focus on two prototype elliptic partial differential equations (PDEs): Poisson equation and static Schrödinger equation on the $d$-dimensional unit hypercube with the Neumann boundary condition. Furthermore, sharper generalization bounds are derived based on the localization techniques under the assumptions that the exact solutions of the PDEs lie in the Barron spaces or the general Sobolev spaces. For the PINNs, we investigate the general linear second order elliptic PDEs with Dirichlet boundary condition using the local Rademacher complexity in the multi-task learning setting. Finally, we discuss the generalization error in the setting of over-parameterization when solutions of PDEs belong to Barron space.

Recent studies have revealed various manifestations of position bias in transformer architectures, from the "lost-in-the-middle" phenomenon to attention sinks, yet a comprehensive theoretical understanding of how attention masks and positional encodings shape these biases remains elusive. This paper presents a graph-theoretic framework for analyzing position bias in multi-layer attention. Modeling attention masks as directed graphs, we quantify how tokens interact with contextual information based on their sequential positions. We uncover two key insights: First, causal masking inherently biases attention toward earlier positions, as tokens in deeper layers attend to increasingly more contextualized representations of earlier tokens. Second, we characterize the competing effects of the causal mask and relative positional encodings, such as the decay mask and rotary positional encoding (RoPE): while both mechanisms introduce distance-based decay within individual attention maps, their aggregate effect across multiple attention layers—coupled with the causal mask—leads to a trade-off between the long-term decay effects and the cumulative importance of early sequence positions. Through controlled numerical experiments, we not only validate our theoretical findings but also reproduce position biases observed in real-world LLMs. Our framework offers a principled foundation for understanding positional biases in transformers, shedding light on the complex interplay of attention mechanism components and guiding more informed architectural design.

Kolmogorov-Arnold Networks (KANs) have seen great success in scientific domains thanks to spline activation functions, becoming an alternative to Multi-Layer Perceptrons (MLPs). However, spline functions may not respect symmetry in tasks, which is crucial prior knowledge in machine learning. In this paper, we propose Equivariant Kolmogorov-Arnold Networks (EKAN), a method for incorporating arbitrary matrix group equivariance into KANs, aiming to broaden their applicability to more fields. We first construct gated spline basis functions, which form the EKAN layer together with equivariant linear weights, and then define a lift layer to align the input space of EKAN with the feature space of the dataset, thereby building the entire EKAN architecture. Compared with baseline models, EKAN achieves higher accuracy with smaller datasets or fewer parameters on symmetry-related tasks, such as particle scattering and the three-body problem, often reducing test MSE by several orders of magnitude. Even in non-symbolic formula scenarios, such as top quark tagging with three jet constituents, EKAN achieves comparable results with state-of-the-art equivariant architectures using fewer than $40\\%$ of the parameters, while KANs do not outperform MLPs as expected. Code and data are available at [https://github.com/hulx2002/EKAN](https://github.com/hulx2002/EKAN).

A key component of transformers is the attention mechanism orchestrating how each token influences the propagation of every other token along the layers of a transformer. In this paper we provide a rigorous, mathematical analysis of the asymptotic properties of attention in transformers. Although we present several results based on different assumptions, all of them point to the same conclusion, all tokens asymptotically converge to each other, a phenomenon that has been empirically reported in the literature. Our findings are carefully compared with existing theoretical results and illustrated by simulations and experimental studies using the GPT-2 model.

This paper studies a family of estimators based on noise-contrastive estimation (NCE) for learning unnormalized distributions. The main contribution of this work is to provide a unified perspective on various methods for learning unnormalized distributions, which have been independently proposed and studied in separate research communities, through the lens of NCE. This unified view offers new insights into existing estimators. Specifically, for exponential families, we establish the finite-sample convergence rates of the proposed estimators under a set of regularity assumptions, most of which are new.

The generalization error (risk) of a supervised statistical learning algorithm quantifies its prediction ability on previously unseen data. Inspired by exponential tilting, Li et al. (2021) proposed the {\it tilted empirical risk} (TER) as a non-linear risk metric for machine learning applications such as classification and regression problems. In this work, we examine the generalization error of the tilted empirical risk in the robustness regime under \textit{negative tilt}. Our first contribution is to provide uniform and information-theoretic bounds on the {\it tilted generalization error}, defined as the difference between the population risk and the tilted empirical risk, under negative tilt for unbounded loss function under bounded $(1+\epsilon)$-th moment of loss function for some $\epsilon\in(0,1]$ with a convergence rate of $O(n^{-\epsilon/(1+\epsilon)})$ where $n$ is the number of training samples, revealing a novel application for TER under no distribution shift. Secondly, we study the robustness of the tilted empirical risk with respect to noisy outliers at training time and provide theoretical guarantees under distribution shift for the tilted empirical risk. We empirically corroborate our findings in simple experimental setups where we evaluate our bounds to select the value of tilt in a data-driven manner.

Continual learning (CL) focuses on the ability of models to learn sequentially from a stream of tasks. A major challenge in CL is catastrophic forgetting (CF). CF is a phenomenon where the model experiences significant performance degradation on previously learned tasks after training on new tasks. Although CF is commonly observed in convolutional neural networks (CNNs), the theoretical understanding about CF within CNNs remains limited. To fill the gap, we present a theoretical analysis of CF in a two-layer CNN. By employing a multi-view data model, we analyze the learning dynamics of different features throughout CL and derive theoretical insights. The findings are supported by empirical results from both simulated and real-world datasets.

Length generalization is the ability of a learning algorithm to learn a hypothesis which generalizes to longer inputs than the inputs in the training set. In this paper, we provide provable guarantees of length generalization for various classes of functions in an idealized setting. First, we formalize the framework of non-asymptotic length generalization, which requires a computable upper bound for the minimum input length that guarantees length generalization, as a function of the complexity of ground-truth function under some given complexity measure. We refer to this minimum input length to length generalize as length complexity. We show the Minimum-Complexity Interpolator learning algorithm achieves optimal length complexity. We further show that whether a function class admits non-asymptotic length generalization is equivalent to the decidability of its language equivalence problem, which implies that there is no computable upper bound for the length complexity of Context-Free Grammars. On the positive side, we show that the length complexity of Deterministic Finite Automata is $2n - 2$ where $n$ is the number of states of the ground-truth automaton. Our main results are upper bounds of length complexity for a subset of a transformer-related function class called C-RASP (Yang & Chiang, 2024). We show that the length complexity of 1-layer C-RASP functions is $O(T^2)$ when the ground-truth function has precision $T$, and that the length complexity of 2-layer C-RASP functions is $O(T^{O(K)})$ when the ground-truth function has precision $T$ and $K$ heads.

While diffusion models generate high-quality images via probability flow, the theoretical understanding of this process remains incomplete. A key question is when probability flow converges to training samples or more general points on the data manifold. We analyze this by studying the probability flow of shallow ReLU neural network denoisers trained with minimal $\ell^2$ norm. For intuition, we introduce a simpler score flow and show that for orthogonal datasets, both flows follow similar trajectories, converging to a training point or a sum of training points. However, early stopping by the diffusion time scheduler allows probability flow to reach more general manifold points. This reflects the tendency of diffusion models to both memorize training samples and generate novel points that combine aspects of multiple samples, motivating our study of such behavior in simplified settings. We extend these results to obtuse simplex data and, through simulations in the orthogonal case, confirm that probability flow converges to a training point, a sum of training points, or a manifold point. Moreover, memorization decreases when the number of training samples grows, as fewer samples accumulate near training points.

Stability is a central property in learning and statistics promising the output of an algorithm $\mathcal{A}$ does not change substantially when applied to similar datasets $S$ and $S'$. It is an elementary fact that any sufficiently stable algorithm (e.g.\ one returning the same result with high probability, satisfying privacy guarantees, etc.) must be randomized. This raises a natural question: can we quantify \textit{how much} randomness is needed for algorithmic stability? We study the randomness complexity of two influential notions of stability in learning: \textit{replicability} (which promises $\mathcal{A}$ usually outputs the same result when run over samples from the same distribution), and \textit{differential privacy} (which promises the output distribution of $\mathcal{A}$ remains similar under neighboring datasets). In particular, building on the ideas of (Dixon, Pavan, Vander Woude, and Vinodchandran ICML 2024) and (Cannone, Su, and Vadhan ITCS 2024), we prove a "weak-to-strong" boosting theorem for stability in these settings: the randomness complexity of a task $\mathcal{M}$ is tightly controlled by the best replication probability of any \textit{deterministic} algorithm solving $\mathcal{M}$, a parameter known as $\mathcal{M}$'s "global stability" (Chase, Moran, Yehudayoff FOCS 2023). Finally, we use this connection to characterize the randomness complexity of PAC Learning: a class has bounded randomness complexity iff it has finite Littlestone dimension, and moreover scales at worst logarithmically in the excess error of the learner. As a corollary, we resolve a question of (Chase, Chornomaz, Moran, and Yehudayoff STOC 2024) about the error-dependent list-replicability of agnostic learning.

We study the task of list-decodable linear regression using batches, recently introduced by Das et al. 2023.. In this setting, we are given $m$ batches with each batch containing $n$ points in $\mathbb R^d$. A batch is called clean if the points it contains are i.i.d. samples from an unknown linear regression distribution. For a parameter $\alpha \in (0, 1/2)$, an unknown $\alpha$-fraction of the batches are clean and no assumptions are made on the remaining batches. The goal is to output a small list of vectors at least one of which is close to the true regressor vector in $\ell_2$-norm. Das et al. 2023 gave an efficient algorithm for this task, under natural distributional assumptions, with the following guarantee. Under the assumption that the batch size satisfies $n \geq \tilde{\Omega}(\alpha^{-1})$ and the total number of batches is $m = \text{poly}(d, n, 1/\alpha)$, their algorithm runs in polynomial time and outputs a list of $O(1/\alpha^2)$ vectors at least one of which is $\tilde{O}(\alpha^{-1/2}/\sqrt{n})$ close to the target regressor. Here we design a new polynomial-time algorithm for this task with significantly stronger guarantees under the assumption that the low-degree moments of the covariates distribution are Sum-of-Squares (SoS) certifiably bounded. Specifically, for any constant $\delta>0$, as long as the batch size is $n \geq \Omega_{\delta}(\alpha^{-\delta})$ and the degree-$\Theta(1/\delta)$ moments of the covariates are SoS certifiably bounded, our algorithm uses $m = \text{poly}((dn)^{1/\delta}, 1/\alpha)$ batches, runs in polynomial-time, and outputs an $O(1/\alpha)$-sized list of vectors one of which is $O(\alpha^{-\delta/2}/\sqrt{n})$ close to the target. That is, our algorithm substantially improves both the minimum batch size and the final error guarantee, while achieving the optimal list size. Our approach leverages higher-order moment information by carefully combining the SoS paradigm interleaved with an iterative method and a novel list pruning procedure for this setting. In the process, we give an SoS proof of the Marcinkiewicz-Zygmund inequality that may be of broader applicability.

This paper introduces $k$-Dynamic Time Warping ($k$-DTW), a novel dissimilarity measure for polygonal curves. $k$-DTW has stronger metric properties than Dynamic Time Warping (DTW) and is more robust to outliers than the Fréchet distance, which are the two gold standards of dissimilarity measures for polygonal curves. We show interesting properties of $k$-DTW and give an exact algorithm as well as a $(1+\varepsilon)$-approximation algorithm for $k$-DTW by a parametric search for the $k$-th largest matched distance. We prove the first dimension-free learning bounds for curves and further learning theoretic results. $k$-DTW not only admits smaller sample size than DTW for the problem of learning the median of curves, where some factors depending on the curves' complexity $m$ are replaced by $k$, but we also show a surprising separation on the associated Rademacher and Gaussian complexities: $k$-DTW admits strictly smaller bounds than DTW, by a factor $\tilde\Omega(\sqrt{m})$ when $k\ll m$. We complement our theoretical findings with an experimental illustration of the benefits of using $k$-DTW for clustering and nearest neighbor classification.

Recent theoretical studies (Kou et al., 2023; Cao et al., 2022) revealed a sharp phase transition from benign to harmful overfitting when the noise-to-feature ratio exceeds a threshold—a situation common in long-tailed data distributions where atypical data is prevalent. However, such harmful overfitting rarely happens in overparameterized neural networks. Further experimental results suggested that memorization is necessary for achieving near-optimal generalization error in long-tailed data distributions (Feldman & Zhang, 2020). We argue that this discrepancy between theoretical predictions and empirical observations arises because previous feature-noise data models overlook the heterogeneous nature of noise across different data classes. In this paper, we refine the feature-noise data model by incorporating class-dependent heterogeneous noise and re-examine the overfitting phenomenon in neural networks. Through a comprehensive analysis of the training dynamics, we establish test loss bounds for the refined model. Our findings reveal that neural networks can leverage "data noise" to learn implicit features that improve the classification accuracy for long-tailed data. Our analysis also provides a training-free metric for evaluating data influence on test performance. Experimental validation on both synthetic and real-world datasets supports our theoretical results.

We propose and investigate probabilistic guarantees for the adversarial robustness of classification algorithms. While traditional formal verification approaches for robustness are intractable and sampling-based approaches do not provide formal guarantees, our approach is able to efficiently certify a probabilistic relaxation of robustness. The key idea is to sample an $\epsilon$-net and invoke a local robustness oracle on the sample. Remarkably, the size of the sample needed to achieve probably approximately global robustness guarantees is independent of the input dimensionality, the number of classes, and the learning algorithm itself. Our approach can, therefore, be applied even to large neural networks that are beyond the scope of traditional formal verification. Experiments empirically confirm that it characterizes robustness better than state-of-the-art sampling-based approaches and scales better than formal methods.

We present `GL-LowPopArt`, a novel Catoni-style estimator for generalized low-rank trace regression. Building on `LowPopArt` (Jang et al., 2024), it employs a two-stage approach: nuclear norm regularization followed by matrix Catoni estimation. We establish state-of-the-art estimation error bounds, surpassing existing guarantees (Fan et al., 2019; Kang et al., 2022), and reveal a novel experimental design objective, $\mathrm{GL}(\pi)$. The key technical challenge is controlling bias from the nonlinear inverse link function, which we address by our two-stage approach. We prove a *local* minimax lower bound, showing that our `GL-LowPopArt` enjoys instance-wise optimality up to the condition number of the ground-truth Hessian. Applications include generalized linear matrix completion, where `GL-LowPopArt` achieves a state-of-the-art Frobenius error guarantee, and **bilinear dueling bandits**, a novel setting inspired by general preference learning (Zhang et al., 2024). Our analysis of a `GL-LowPopArt`-based explore-then-commit algorithm reveals a new, potentially interesting problem-dependent quantity, along with improved Borda regret bound than vectorization (Wu et al., 2024).

Many variants of Optimal Transport (OT) have been developed to address its heavy computation. Among them, notably, Sliced Wasserstein (SW) is widely used for application domains by projecting the OT problem onto one-dimensional lines, and leveraging the closed-form expression of the univariate OT to reduce the computational burden. However, projecting measures onto low-dimensional spaces can lead to a loss of topological information. To mitigate this issue, in this work, we propose to replace one-dimensional lines with a more intricate structure, called \emph{tree systems}. This structure is metrizable by a tree metric, which yields a closed-form expression for OT problems on tree systems. We provide an extensive theoretical analysis to formally define tree systems with their topological properties, introduce the concept of splitting maps, which operate as the projection mechanism onto these structures, then finally propose a novel variant of Radon transform for tree systems and verify its injectivity. This framework leads to an efficient metric between measures, termed Tree-Sliced Wasserstein distance on Systems of Lines (TSW-SL). By conducting a variety of experiments on gradient flows, image style transfer, and generative models, we illustrate that our proposed approach performs favorably compared to SW and its variants.

Diffusion models have emerged as a powerful framework for generative modeling, with guidance techniques playing a crucial role in enhancing sample quality. Despite their empirical success, a comprehensive theoretical understanding of the guidance effect remains limited. Existing studies only focus on case studies, where the distribution conditioned on each class is either isotropic Gaussian or supported on a one-dimensional interval with some extra conditions. How to analyze the guidance effect beyond these case studies remains an open question. Towards closing this gap, we make an attempt to analyze diffusion guidance under general data distributions. Rather than demonstrating uniform sample quality improvement, which does not hold in some distributions, we prove that guidance can improve the whole sample quality, in the sense that the ratio of bad samples (measured by the classifier probability) decreases in the presence of guidance. This aligns with the motivation of introducing guidance.

Length generalization refers to the ability to extrapolate from short training sequences to long test sequences and is a challenge for current large language models. While prior work has proposed some architecture or data format changes to achieve length generalization, these proposals typically apply to a limited set of tasks. Building on prior scratchpad and Chain-of-Thought (CoT) techniques, we propose *Turing Programs*, a novel CoT strategy that decomposes an algorithmic task into steps mimicking the computation of a Turing Machine. This framework is both universal, as it can accommodate any algorithmic task, and simple, requiring only copying text from the context with small modifications. We show that by using Turing Programs, we obtain robust length generalization on a range of algorithmic tasks: addition, multiplication and in-context SGD. We then demonstrate that transformers achieve length generalization on random Turing Programs, suggesting that length generalization is possible for any algorithmic task. Finally, we theoretically prove that transformers can implement Turing Programs, constructing a simple RASP (Weiss et al.) program that simulates an arbitrary Turing machine.

Given independent and identically distributed data from a compactly supported, $\alpha$-Hölder density $f$, we study estimation of the Fisher information of the Gaussian-smoothed density $f*\varphi_t$, where $\varphi_t$ is the density of $N(0, t)$. We derive the minimax rate including the sharp dependence on $t$ and show some simple, plug-in type estimators are optimal for $t > 0$, even though extra debiasing steps are widely employed in the literature to achieve the sharp rate in the unsmoothed ($t = 0$) case. Due to our result's sharp characterization of the scaling in $t$, plug-in estimators of the mutual information and entropy are shown to achieve the parametric rate by way of the I-MMSE and de Bruijn's identities.