ICML 2025 Papers

Little is known about the training dynamics of equivariant neural networks, in particular how it compares to data augmented training of their non-equivariant counterparts. Recently, neural tangent kernels (NTKs) have emerged as a powerful tool to analytically study the training dynamics of wide neural networks. In this work, we take an important step towards a theoretical understanding of training dynamics of equivariant models by deriving neural tangent kernels for a broad class of equivariant architectures based on group convolutions. As a demonstration of the capabilities of our framework, we show an interesting relationship between data augmentation and group convolutional networks. Specifically, we prove that they share the same expected prediction over initializations at all training times and even off the data manifold. In this sense, they have the same training dynamics. We demonstrate in numerical experiments that this still holds approximately for finite-width ensembles. By implementing equivariant NTKs for roto-translations in the plane ($G=C_{n}\ltimes\mathbb{R}^{2}$) and 3d rotations ($G=\mathrm{SO}(3)$), we show that equivariant NTKs outperform their non-equivariant counterparts as kernel predictors for histological image classification and quantum mechanical property prediction.

Humans develop _world models_ that capture the underlying generation process of data. Whether neural networks can learn similar world models remains an open problem. In this work, we present the first theoretical results for this problem, showing that in a _multi-task_ setting, models with a _low-degree bias_ provably recover latent data-generating variables under mild assumptions--even if proxy tasks involve complex, non-linear functions of the latents. However, such recovery is sensitive to model architecture. Our analysis leverages Boolean models of task solutions via the Fourier-Walsh transform and introduces new techniques for analyzing invertible Boolean transforms, which may be of independent interest. We illustrate the algorithmic implications of our results and connect them to related research areas, including self-supervised learning, out-of-distribution generalization, and the linear representation hypothesis in large language models.

We show that deep neural networks can achieve dimension-independent rates of convergence for learning structured densities typical of image, audio, video, and text data. For example, in images, where each pixel becomes independent of the rest of the image when conditioned on pixels at most $t$ steps away, a simple $L^2$-minimizing neural network can attain a rate of $n^{-1/((t+1)^2+4)}$, where $t$ is independent of the ambient dimension $d$, i.e. the total number of pixels. We further provide empirical evidence that, in real-world applications, $t$ is often a small constant, thus effectively circumventing the curse of dimensionality. Moreover, for sequential data (e.g., audio or text) exhibiting a similar local dependence structure, our analysis shows a rate of $n^{-1/(t+5)}$, offering further evidence of dimension independence in practical scenarios.

Data heterogeneity across multiple sources is common in real-world machine learning (ML) settings. Although many methods focus on enabling a single model to handle diverse data, real-world markets often comprise multiple competing ML providers. In this paper, we propose a game-theoretic framework—the Heterogeneous Data Game—to analyze how such providers compete across heterogeneous data sources. We investigate the resulting pure Nash equilibria (PNE), showing that they can be non-existent, homogeneous (all providers converge on the same model), or heterogeneous (providers specialize in distinct data sources). Our analysis spans monopolistic, duopolistic, and more general markets, illustrating how factors such as the ``temperature'' of data-source choice models and the dominance of certain data sources shape equilibrium outcomes. We offer theoretical insights into both homogeneous and heterogeneous PNEs, guiding regulatory policies and practical strategies for competitive ML marketplaces.

We study the classical optimization problem $\min_{x \in \mathbb{R}^d} f(x)$ and analyze the gradient descent (GD) method in both nonconvex and convex settings. It is well-known that, under the $L$–smoothness assumption ($\|\| \nabla^2 f(x) \|\| \leq L$), the optimal point minimizing the quadratic upper bound $f(x_k) + \langle \nabla f(x_k), x_{k+1} - x_k \rangle + \frac{L}{2} \|\| x_{k+1} - x_k \|\|^2$ is $x_{k+1} = x_k - \gamma_k \nabla f(x_k)$ with step size $\gamma_k = \frac{1}{L}$. Surprisingly, a similar result can be derived under the $\ell$-generalized smoothness assumption ($\|\| \nabla^2 f(x) \|\| \leq \ell( \|\| \nabla f(x) \|\| )$). In this case, we derive the step size $$\gamma_k = \int_{0}^{1} \frac{d v}{\ell( \|\| \nabla f(x_k) \|\| + \|\| \nabla f(x_k) \|\| v)}.$$ Using this step size rule, we improve upon existing theoretical convergence rates and obtain new results in several previously unexplored setups.

In this position paper, we promote the study of function spaces parameterized by machine learning models through the lens of algebraic geometry. To this end, we focus on algebraic models, such as neural networks with polynomial activations, whose associated function spaces are semi-algebraic varieties. We outline a dictionary between algebro-geometric invariants of these varieties, such as dimension, degree, and singularities, and fundamental aspects of machine learning, such as sample complexity, expressivity, training dynamics, and implicit bias. Along the way, we review the literature and discuss ideas beyond the algebraic domain. This work lays the foundations of a research direction bridging algebraic geometry and deep learning, that we refer to as neuroalgebraic geometry.

In this paper, we investigate the existence of online learning algorithms with bandit feedback that simultaneously guarantee $O(1)$ regret compared to a given comparator strategy, and $\tilde{O}(\sqrt{T})$ regret compared to any fixed strategy, where $T$ is the number of rounds. We provide the first affirmative answer to this question whenever the comparator strategy supports every action. In the context of zero-sum games with min-max value zero, both in normal- and extensive form, we show that our results allow us to guarantee to risk at most $O(1)$ loss while being able to gain $\Omega(T)$ from exploitable opponents, thereby combining the benefits of both no-regret algorithms and minimax play.

In recent times, there have been major developments in two distinct yet connected domains of quantum information. On the one hand, substantial progress has been made in so-called randomized measurement protocols. Here, a number of properties of unknown quantum states can be deduced from surprisingly few measurement outcomes, using schemes such as classical shadows. On the other hand, significant progress has been made in quantum machine learning. For example, exponential advantages have been proven when the data consists of quantum states and quantum algorithms can coherently measure multiple copies of input states. In this work, we aim to understand the implications and limitations of combining randomized measurement protocols with quantum machine learning, although the implications are broader. Specifically, we investigate quantum machine learning algorithms that, when dealing with quantum data, can either process it entirely using quantum methods or measure the input data through a fixed measurement scheme and utilize the resulting classical information. We prove limitations for the general class of quantum machine learning algorithms that use fixed measurement schemes on the input quantum states. Our results have several implications. From the perspective of randomized measurement procedures, we show limitations of measure-first protocols in the average case, improving on the state-of-the-art which only focuses on worst-case scenarios. Additionally, previous lower bounds were only known for physically unrealizable states. We improve upon this by employing quantum pseudorandom functions to prove that a learning separation also exists when dealing with physically realizable states, which may be encountered in experiments. From a machine learning perspective, our results are crucial for defining a physically meaningful task that shows fully quantum machine learning processing is not only more efficient but also necessary for solving certain problems. The tasks at hand are also realistic, as the algorithms and proven separations hold when working with efficiently preparable states and remain robust in the presence of measurement and preparation errors.

Deep learning with noisy labels presents significant challenges. In this work, we theoretically characterize the role of label noise from a feature learning perspective. Specifically, we consider a signal-noise data distribution, where each sample comprises a label-dependent signal and label-independent noise, and rigorously analyze the training dynamics of a two-layer convolutional neural network under this data setup, along with the presence of label noise. Our analysis identifies two key stages. In Stage I, the model perfectly fits all the clean samples (i.e., samples without label noise) while ignoring the noisy ones (i.e., samples with noisy labels). During this stage, the model learns the signal from the clean samples, which generalizes well on unseen data. In Stage II, as the training loss converges, the gradient in the direction of noise surpasses that of the signal, leading to overfitting on noisy samples. Eventually, the model memorizes the noise present in the noisy samples and degrades its generalization ability. Furthermore, our analysis provides a theoretical basis for two widely used techniques for tackling label noise: early stopping and sample selection. Experiments on both synthetic and real-world setups validate our theory.

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.

Contrastive Representation Learning (CRL) has achieved impressive success in various domains in recent years. Nevertheless, the theoretical understanding of the generalization behavior of CRL has remained limited. Moreover, to the best of our knowledge, the current literature only analyzes generalization bounds under the assumption that the data tuples used for contrastive learning are independently and identically distributed. However, in practice, we are often limited to a fixed pool of reusable labeled data points, making it inevitable to recycle data across tuples to create sufficiently large datasets. Therefore, the tuple-wise independence condition imposed by previous works is invalidated. In this paper, we provide a generalization analysis for the CRL framework under non-$i.i.d.$ settings that adheres to practice more realistically. Drawing inspiration from the literature on U-statistics, we derive generalization bounds which indicate that the required number of samples in each class scales as the logarithm of the covering number of the class of learnable feature representations associated to that class. Next, we apply our main results to derive excess risk bounds for common function classes such as linear maps and neural networks.

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.

A modern paradigm for generalization in machine learning and AI consists of pre-training a task-agnostic foundation model, generally obtained using self-supervised and multimodal contrastive learning. The resulting representations can be used for prediction on a downstream task for which no labeled data is available. We present a theoretical framework to better understand this approach, called zero-shot prediction. We identify the target quantities that zero-shot prediction aims to learn, or learns in passing, and the key conditional independence relationships that enable its generalization ability.

We study the complexity of learning $k$-mixtures of Gaussians ($k$-GMMs) on $\mathbb R^d$. This task is known to have complexity $d^{\Omega(k)}$ in full generality. To circumvent this exponential lower bound on the number of components, research has focused on learning families of GMMs satisfying additional structural properties. A natural assumption posits that the component weights are not exponentially small and that the components have the same unknown covariance. Recent work gave a $d^{O(\log(1/w_{\min}))}$-time algorithm for this class of GMMs, where $w_{\min}$ is the minimum weight. Our first main result is a Statistical Query (SQ) lower bound showing that this quasi-polynomial upper bound is essentially best possible, even for the special case of uniform weights. Specifically, we show that it is SQ-hard to distinguish between such a mixture and the standard Gaussian. We further explore how the distribution of weights affects the complexity of this task. Our second main result is a quasi-polynomial upper bound for the aforementioned testing task when most of the weights are uniform while a small fraction of the weights are potentially arbitrary.

We address the regression problem under the constraint of demographic parity, a commonly used fairness definition. Recent studies have revealed fair minimax optimal regression algorithms, the most accurate algorithms that adhere to the fairness constraint. However, these analyses are tightly coupled with specific data generation models. In this paper, we provide meta-theorems that can be applied to various situations to validate the fair minimax optimality of the corresponding regression algorithms. Furthermore, we demonstrate that fair minimax optimal regression can be achieved through post-processing methods, allowing researchers and practitioners to focus on improving conventional regression techniques, which can then be efficiently adapted for fair regression.

We obtain an analytic, interpretable and predictive theory of creativity in convolutional diffusion models. Indeed, score-matching diffusion models can generate highly original images that lie far from their training data. However, optimal score-matching theory suggests that these models should only be able to produce memorized training examples. To reconcile this theory-experiment gap, we identify two simple inductive biases, locality and equivariance, that: (1) induce a form of combinatorial creativity by preventing optimal score-matching; (2) result in fully analytic, completely mechanistically interpretable, local score (LS) and equivariant local score (ELS) machines that, (3) after calibrating a single time-dependent hyperparameter can quantitatively predict the outputs of trained convolution only diffusion models (like ResNets and UNets) with high accuracy (median $r^2$ of $0.95, 0.94, 0.94, 0.96$ for our top model on CIFAR10, FashionMNIST, MNIST, and CelebA). Our model reveals a {\it locally consistent patch mosaic} mechanism of creativity, in which diffusion models create exponentially many novel images by mixing and matching different local training set patches at different scales and image locations. Our theory also partially predicts the outputs of pre-trained self-attention enabled UNets (median $r^2 \sim 0.77$ on CIFAR10), revealing an intriguing role for attention in carving out semantic coherence from local patch mosaics.

Although transformers have demonstrated impressive capabilities for in-context learning (ICL) in practice, theoretical understanding of the underlying mechanism that allows transformers to perform ICL is still in its infancy. This work aims to theoretically study the training dynamics of transformers for in-context classification tasks. We demonstrate that, for in-context classification of Gaussian mixtures under certain assumptions, a single-layer transformer trained via gradient descent converges to a globally optimal model at a linear rate. We further quantify the impact of the training and testing prompt lengths on the ICL inference error of the trained transformer. We show that when the lengths of training and testing prompts are sufficiently large, the prediction of the trained transformer approaches the ground truth distribution of the labels. Experimental results corroborate the theoretical findings.

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.

In this paper, we investigate how randomness and uncertainty influence learning in games. Specifically, we examine a perturbed variant of the dynamics of “follow-the-regularized-leader” (FTRL), where the players’ payoff observations and strategy updates are continually impacted by random shocks. Our findings reveal that, in a fairly precise sense, “uncertainty favors extremes”: in any game, regardless of the noise level, every player’s trajectory of play reaches an arbitrarily small neighborhood of a pure strategy in finite time (which we estimate). Moreover, even if the player does not ultimately settle at this strategy, they return arbitrarily close to some (possibly different) pure strategy infinitely often. This prompts the question of which sets of pure strategies emerge as robust predictions of learning under uncertainty. We show that (a) the only possible limits of the FTRL dynamics under uncertainty are pure Nash equilibria; and (b) a span of pure strategies is stable and attracting if and only if it is closed under better replies. Finally, we turn to games where the deterministic dynamics are recurrent—such as zero-sum games with interior equilibria—and show that randomness disrupts this behavior, causing the stochastic dynamics to drift toward the boundary on average.