ICML 2026 Papers

One of the most pressing challenges in artificial intelligence is to make models more transparent to their users. Recently, explainable artificial intelligence has come up with numerous methods to tackle this challenge. A promising avenue is to use concept-based explanations, that is, high-level concepts instead of plain feature importance scores. Among this class of methods, Concept Activation Vectors (CAVs, Kim et al., 2018) stand out as one of the main protagonists. One interesting aspect of CAVs is that their computation requires sampling random examples from the train set. Therefore, the actual vectors obtained may vary depending on the randomness of this sampling. In this paper, we propose a fine-grained theoretical analysis of CAV construction in order to quantify their variability. Our results, confirmed by experiments on several real-life datasets of four different modalities, point to an universal result: the variance of CAVs declines roughly as $1/N$, where $N$ is the number of random examples. Based on this, we give practical recommendations for a resource-efficient application of the method.

Removing noise is difficult, but adding noise is easy. In this work, we show how to eliminate mean-shift noisy components from PCA by deliberately introducing knockoff mean-shift perturbation. Standard PCA is highly sensitive to shifts in the sample mean: a small fraction of samples from a shifted distribution can cause large deviations in the leading principal components. In high-dimensional regimes, existing Robust PCA approaches cannot handle the mean-shift contamination structure inherent in the mixture model. Using tools from Random Matrix Theory, we prove that the mean-shift spikes are spectrally separable from the stable eigenvalues of the original covariance. Furthermore, the original eigenspace remains asymptotically invariant to the contamination, independent of the mixture weight. Exploiting this spectral stability, we propose a simple, two-stage PCA algorithm by adding knockoff mean that identifies and removes the mean-shift component using only standard PCA operations.

Differentially private Stochastic Gradient Descent (DP-SGD) has become integral to privacy-preserving machine learning, ensuring robust privacy guarantees in sensitive domains. Despite notable empirical advances leveraging features from non-private, pre-trained models to enhance DP-SGD training, a theoretical understanding of feature dynamics in private learning remains underexplored. This paper presents the first theoretical framework to analyze private training through a feature learning perspective. Building on the multi-patch data structure from prior work, our analysis distinguishes between label-dependent feature signals and label-independent noise—a critical aspect overlooked by existing analyses in the DP community. Employing a two-layer CNN with polynomial ReLU activation, we theoretically characterize both feature signal learning and data noise memorization in private training via noisy gradient descent. Our findings reveal that (1) Effective private signal learning requires a higher signal-to-noise ratio (SNR) compared to non-private training, and (2) When data noise memorization occurs in non-private learning, it will also occur in private learning, leading to poor generalization despite small training loss. Our findings highlight the challenges of private learning and prove the benefit of feature enhancement to improve SNR. Experiments on synthetic and real-world datasets also validate our theoretical findings.

Contrastive Representation Learning (CRL) has achieved strong empirical success in multiple machine learning disciplines, yet its theoretical sample complexity remains poorly understood. Existing analyses usually assume that input tuples are identically and independently distributed, an assumption violated in most practical settings where contrastive tuples are constructed from a finite pool of labeled data, inducing dependencies among tuples. While one recent work analyzed this learning setting using U-Statistics to estimate the population risk, the techniques used therein require the risk of each class to concentrate uniformly, making excess risk bounds scale in the order of $\rho_{\min}^{-{1}/{2}}$ where $\rho_{\min}$ denotes the probability of the rarest class. Such a dependency can be overly pessimistic in the extreme multiclass settings where there are many tail classes which contribute minimally to the overall population risk. Our contributions are two-fold. Firstly, we improve upon the previous work and prove a bound with a sample complexity of the same order as the number of classes $R$, regardless of the distribution over classes. Furthermore, we formulate a different estimator that captures the concentration of the risk \textit{across classes}, enabling sharper bounds in extreme multi-class learning scenarios, especially where class distributions are long-tailed. Under mild assumptions on the class distributions, the resulting sample complexity is $\mathcal{{O}}(k)$ where $k$ is the number of samples per tuple.

Scaling test-time computation during language model inference, such as generating intermediate thoughts or sampling multiple candidate answers, has proven effective in improving model performance. While these techniques inherently rely on the stochastic nature of inference to explore diverse reasoning paths, prior theoretical works typically build on a *deterministic decoding* framework, overlooking the stochastic nature of practical language model inference. This work takes an initial step to bridge this gap by establishing a new theoretical framework, incorporating *randomness and sampling* directly into the decoding analysis. To demonstrate the framework's effectiveness, we apply it to the canonical in-context linear regression task with continuous and binary coefficients, simulating decoding via noise injection and sampling to analyze widely adopted inference techniques. We validate our theoretical findings through numerical simulations, with additional experiments on real-world tasks substantiating the framework's potential for practical applications.

Strategic classification examines how decision rules interact with agents who strategically adapt their features. Most existing models focus on maximizing predictive performance, assuming agents best respond to the learned classifier. However, real decision-making systems are rarely optimized solely for accuracy: ethical, economic, and institutional considerations often make some feature changes more desirable than others. At the same time, principals may wish to incentivize these changes fairly across heterogeneous agents. While prior work has studied causal structure between features, notions of desirability, and information disparities in isolation, this work initiates a unified treatment of these components within a single framework. We frame the problem as a constrained optimization problem that captures the trade-offs between optimality, desirability, and fairness. We provide theoretical guarantees on the principal's optimality loss constrained to a particular desirability fairness tolerance for multiple broad classes of fairness measures. Finally, through experiments on real datasets, we show the explicit tradeoff between maximizing accuracy and fairness in desirability effort.

Supervised convolutional neural networks (CNNs) are widely used to solve imaging inverse problems, achieving state-of-the-art performance in numerous applications. However, despite their empirical success, these methods are poorly understood from a theoretical perspective and often treated as black boxes. To bridge this gap, we analyze trained neural networks through the lens of the Minimum Mean Square Error (MMSE) estimator, incorporating functional constraints that capture two fundamental inductive biases of CNNs: translation equivariance and locality via finite receptive fields. Under the empirical training distribution, we derive an analytic, interpretable, and tractable formula for this constrained variant, termed Local-Equivariant MMSE (LE-MMSE). Through extensive numerical experiments across various inverse problems (denoising, inpainting, deconvolution), datasets (FFHQ, CIFAR-10, FashionMNIST), and architectures (U-Net, ResNet, PatchMLP), we demonstrate that our theory matches the neural networks outputs (PSNR $\gtrsim25$ dB). Furthermore, we provide insights into the differences between *physics-aware* and *physics-agnostic* estimators, the impact of high-density regions in the training (patch) distribution, and the influence of other factors (dataset size, patch size, *etc*).

We present algorithms for diffusion model sampling which obtain $\delta$-error in $\mathrm{polylog}(1/\delta)$ steps, given access to $\widetilde O(\delta)$-accurate score estimates in $L^2$. This is an exponential improvement over all previous results. Specifically, under minimal data assumptions, the complexity is $\widetilde O(d\mathrm{polylog}(1/\delta))$ where $d$ is the dimension of the data; under a non-uniform $L$-Lipschitz condition, the complexity is $\widetilde O(\sqrt{dL}\mathrm{polylog}(1/\delta))$; and if the data distribution has intrinsic dimension $d_\star$, then the complexity reduces to $\widetilde O(d_\star\mathrm{polylog}(1/\delta))$. Our approach also yields the first $\mathrm{polylog}(1/\delta)$ complexity sampler for general log-concave distributions using only gradient evaluations.

Motivated by problems in online advertising, we address the task of Learning from Label Proportions (LLP). We introduce a novel and versatile low-variance debiasing methodology to learn from aggregate label information, significantly advancing the state of the art in LLP. Our debiasing approach exhibits remarkable flexibility, seamlessly accommodating a broad spectrum of practically relevant loss functions across both binary and multi-class classification settings. By carefully combining our estimators with standard techniques, we improve sample complexity guarantees for a large class of losses of practical relevance. We also empirically validate the efficacy of our proposed approach across a diverse array of benchmark datasets, demonstrating compelling empirical advantages over standard baselines.

Diffusion models are increasingly used as powerful conditional generators, yet real deployments often involve multiple target distributions arising from different tasks, e.g., diverse prompt domains in text-to-image generation, or multiple environments in robotics with diffusion policies. This naturally leads to a multi-objective learning (MOL) problem. A key challenge is that achieving good Pareto trade-offs can require a generalist model class with substantially larger capacity than what suffices for solving any individual task, thereby increasing statistical cost since sample complexity typically scales with the model complexity. To reconcile this, we develop a principled MOL framework for diffusion models with limited data: a semi-supervised regime where paired (labeled) samples are scarce, but (unlabeled) condition data are abundant. We propose a two-stage training procedure that first fits lightweight specialist models from limited paired data, and then distills them into a generalist model by generating pseudo-samples. We establish generalization bounds showing that the required number of paired samples only depends on the complexity of the specialist model classes. We further extend the theory to diffusion policies for sequential decision making to account for distribution shift in on-policy rollouts. Extensive experiments on robotic control and image restoration tasks are conducted to verify our theoretical results.

Consider the problem of finding the best matching in a weighted graph where we only have access to predictions of the actual stochastic weights, based on an underlying context. If the predictor is the Bayes optimal one, then computing the best matching based on the predicted weights is optimal. However, in practice, this perfect information scenario is not realistic. Given an imperfect predictor, a suboptimal decision rule may compensate for the induced error and thus outperform the standard optimal rule. In this paper, we propose multicalibration as a way to address this problem. This fairness notion requires a predictor to be unbiased on each element of a family of protected sets of contexts. Given a class of matching algorithms $\mathcal C$ and any predictor $\gamma$ of the edge-weights, we show how to construct a specific multicalibrated predictor $\hat \gamma$, with the following property. Picking the best matching based on the output of $\hat \gamma$ is competitive with the best decision rule in $\mathcal C$ applied onto the original predictor $\gamma$. We complement this result by providing sample complexity bounds, and by performing numerical experiments.

We address the problem of causal effect estimation in the presence of hidden confounders using nonparametric instrumental variable (IV) regression. An established approach is to use estimators based on learned \emph{spectral features}, that is, features spanning the top singular subspaces of the operator linking treatments to instruments. While powerful, such features are agnostic to the outcome variable. Consequently, the method can fail when the true causal function is poorly represented by these dominant singular functions. To mitigate, we introduce Augmented Spectral Feature Learning, a framework that makes the feature learning process outcome-aware. Our method learns features by minimizing a novel contrastive loss derived from an augmented operator that incorporates information from the outcome. By learning these task-specific features, our approach remains effective even under spectral misalignment. We provide a theoretical analysis of this framework and validate our approach on challenging benchmarks.

We study the fundamental problem of one-step prediction of a marginally stable unknown nonlinear dynamical system. We describe an algorithm for this problem, based on the technique of spectral filtering, which learns a mapping from past observations to the next based on a spectral representation of the system. Using techniques from online convex optimization, we prove vanishing prediction error for any nonexpansive nonlinear dynamical system with finitely many marginally stable modes, with rates governed by a novel quantitative control-theoretic notion of learnability. The main technical component of our method is a new spectral filtering algorithm for linear dynamical systems, which incorporates past observations and applies to general noisy and marginally stable systems. This generalizes the original spectral filtering algorithm to both asymmetric dynamics as well as incorporating noise correction, and is of independent interest.

This paper investigates theoretical and methodological foundations for stochastic optimal control (SOC) in discrete time. We start formulating the control problem in a general dynamic programming framework, introducing the mathematical structure needed for a detailed convergence analysis. The associate value function is estimated through a sequence of approximations combining nonparametric regression methods and Monte Carlo subsampling. The regression step is performed within reproducing kernel Hilbert spaces (RKHSs), exploiting the classical KRR algorithm, while Monte Carlo sampling methods are introduced to estimate the continuation value. To assess the accuracy of our value function estimator, we propose a natural error decomposition and rigorously control the resulting error terms at each time step. We then analyze how this error propagates backward in time-from maturity to the initial stage-a relatively underexplored aspect of the SOC literature. Finally, we illustrate how our analysis naturally applies to a key financial application: the pricing of American options.

We study strong universal Bayes-consistency in the realizable setting for learning with general metric losses, extending classical characterizations beyond $0$-$1$ classification (Bousquet et al., 2021; Hanneke et al., 2021) and real-valued regression (Attias et al., 2024). Given an instance space $(X,\rho)$, a label space $(Y,\ell)$ with possibly unbounded loss, and a hypothesis class $H \subseteq Y^{X}$, we resolve the realizable case of an open problem presented in Tsir Cohen and Kontorovich (2022). Specifically, we find the necessary and sufficient conditions on the hypothesis class $H$ under which there exists a distribution-free learning rule whose risk converges almost surely to the best-in-class risk (which is zero) for every realizable data-generating distribution. Our main contribution is this sharp characterization in terms of a combinatorial obstruction: Similarly to Attias et al. (2023), we introduce the notion of an infinite non-decreasing $(\gamma_k)$-Littlestone tree, where $\gamma_k \to \infty$. This extends the Littlestone tree structure used in Bousquet et al. (2021) to the metric loss setting.

We work in the adaptive query model, where one is given a point set $P \subset \mathbb{R}^d$ and seeks to construct a data structure that can answer correctly and efficiently a sequence of adaptive queries. In this model, an adversary observes the answers returned by the data structure to previous queries $q_1, \ldots, q_{i-1}$ and, based on this information, chooses the next query point $q_i$. This setting captures strong forms of adaptivity that naturally arise in modern machine learning pipelines, and rules out many classical randomized techniques that assume oblivious queries. Our focus is the problem of furthest neighbor search in this adaptive setting, a fundamental problem in several learning tasks, including diversity maximization, outlier and anomaly detection, adversarial example generation, and more. We present the first adversarially robust data structure for $c$-approximate furthest neighbor queries that achieves query time $\tilde{O}( \min( d n^{1/c^2}, n^{2/c^2} + d))$. This matches the $n$ dependency in the query time of the seminal result by Indyk [SODA'03] for $c$-approximate furthest neighbor in the oblivious setting, and improves upon the $\tilde{O}(n + d)$ query time achieved via the adaptive distance estimation framework of Cherapanamjeri and Nelson [NeurIPS'20] for a wide range of natural parameters. To complement this result, we present an adversarial attack against oblivious approximate furthest neighbor algorithms. Specifically, we show that the data structure from the algorithm by Indyk fails to maintain its guarantees against adaptive queries.

In truncated linear regression, samples $(x,y)$ are shown only when the outcome $y$ falls inside a certain survival set $S^\star$ and the goal is to estimate the unknown $d$-dimensional regressor $w^\star$. This problem has a long history of study in Statistics and Machine Learning going back to the works of (Galton, 1897; Tobin, 1958) and more recently in, e.g., (Daskalakis et al., 2019; 2021; Lee et al., 2023; 2024). Despite this long history, however, most prior works are limited to the special case where $S^\star$ is precisely known. The more practically relevant case, where $S^\star$ is unknown and must be learned from data, remains open: indeed, here the only available algorithms require strong assumptions on the distribution of the feature vectors (e.g., Gaussianity) and, even then, have a $d^{\mathrm{poly} (1/\varepsilon)}$ run time for achieving $\varepsilon$ accuracy. In this work, we give the first algorithm for truncated linear regression with unknown survival set that runs in $\mathrm{poly} (d/\varepsilon)$ time, by only requiring that the feature vectors are sub-Gaussian. Our algorithm relies on a novel subroutine for efficiently learning unions of a bounded number of intervals using access to positive examples (without any negative examples) under a certain smoothness condition. This learning guarantee adds to the line of works on positive-only PAC learning and may be of independent interest.

Equivariance is a powerful inductive bias in neural networks, improving generalisation and physical consistency. Recently, however, non-equivariant models have regained attention, due to their better runtime performance and imperfect symmetries that might arise in real-world applications. This has motivated the development of approximately equivariant models that strike a middle ground between respecting symmetries and fitting the data distribution. Existing approaches in this field either rely on sampling from a group, incurring a high sample complexity, or explicitly parameterise a model as a sum of an equivariant and non-equivariant network. This work instead approaches approximate equivariance via a projection-based regulariser which leverages a layer-wise orthogonal decomposition of a network's layers into equivariant and non-equivariant components. In contrast to existing methods, this penalises non-equivariance at an operator level across the full group orbit, rather than point-wise as in sample-based approaches. We present a mathematical framework for computing the non-equivariance penalty exactly and efficiently in both the spatial and spectral domains. In our experiments, our method consistently outperforms prior approximate equivariance approaches in both model performance and efficiency, achieving substantial runtime gains over sample-based regularisers.

We study a setting where the goal is to learn a target function f(x) with respect to a target distribution D(x), but training is done on i.i.d. samples from a different training distribution D’(x), labeled by the true target f(x). Such a distribution shift (here in the form of covariate shift) is usually viewed negatively, as hurting or making learning harder, and the traditional distribution shift literature is mostly concerned with limiting or avoiding this negative effect. In contrast, we argue that with a well-chosen D'(x), the shift can be positive and make learning easier -- a perspective called Positive Distribution Shift (PDS). Such a perspective is central to contemporary machine learning, where much of the innovation is in finding good training distributions D’(x), rather than changing the training algorithm. We further argue that the benefit is often computational rather than statistical, and that PDS allows computationally hard problems to become tractable even using standard gradient-based training. We formalize different variants of PDS, show how certain hard classes are easily learnable under PDS, and make connections with membership query learning.