ICLR 2026 Papers

We revisit Group Relative Policy Optimization (GRPO) in both on-policy and off-policy optimization regimes. Our motivation comes from recent work on off-policy Proximal Policy Optimization (PPO), which improves training stability, sampling efficiency, and memory usage. In addition, a recent analysis of GRPO suggests that estimating the advantage function with off-policy samples could be beneficial. Building on these observations, we adapt GRPO to the off-policy setting. We show that both on-policy and off-policy GRPO objectives yield an improvement in the reward. This result motivates the use of clipped surrogate objectives in the off-policy version of GRPO. We then compare the empirical performance of reinforcement learning with verifiable rewards in post-training using both GRPO variants. Our results show that off-policy GRPO either significantly outperforms or performs on par with its on-policy counterpart.

Most familiar equilibrium concepts, such as Nash and correlated equilibrium, guarantee only that no single player can improve their utility by deviating unilaterally. They offer no guarantees against profitable coordinated deviations by coalitions. Although the literature proposes notions to address multilateral deviations (\emph{e.g.}, strong Nash and coalition-proof equilibrium), these generally fail to exist. In this paper, we study a solution concept that accommodates multi-player deviations and is guaranteed to exist. We prove a fixed-parameter lower bound on the complexity of computing such an equilibrium and present an algorithm that matches this bound.

We investigate the trade-off between expressive power and privacy guarantees in graph representation learning. Privacy-preserving machine learning faces growing regulatory demands that pose a fundamental challenge: safeguarding sensitive data while maintaining expressive power. To address this challenge, we propose homomorphism density vectors to obtain graph embeddings that are private and expressive. Homomorphism densities are provably highly discriminative and offer a powerful tool for distinguishing non-isomorphic graphs. By adding noise calibrated to each density’s sensitivity, we ensure that the resulting embeddings satisfy formal differential privacy guarantees. Our theoretical construction preserves expressivity in expectation, as each private embedding remains unbiased with respect to the true homomorphism densities. Our embeddings match, in expectation, the expressive power of a broad range of graph neural networks (GNNs), such as message-passing and subgraph GNNs, while providing formal privacy guarantees.

Industrial anomaly detection has been significantly advanced by large multimodal models (LMMs), enabling diverse human instructions beyond detection, particularly through visual-grounded reasoning for better image understanding. However, the lack of domain-specific knowledge of LMMs limits the accurate generation of responses in complex industrial scenarios. In this work, we present JUDO, Juxtaposed Domain-Oriented Multi-modal Reasoner, a framework that efficiently incorporates domain knowledge and context in visual and text reasoning. The visual reasoning provides detailed inspection by segmenting the defect region in the query image by juxtaposing it with the normal image as visual domain context, enabling a fine-grained visual comparative analysis. Furthermore, we inject domain knowledge through supervised fine-tuning (SFT) to enhance context understanding and subsequently guide domain reasoning through reinforcement learning (GRPO) with three tailored rewards. Experimental results demonstrate that JUDO achieves superior performance on the MMAD benchmark, surpassing models such as Qwen2.5-VL-7B and GPT4o. These results highlight the importance of enhancing domain knowledge and context for effective reasoning in anomaly understanding. The implementation of JUDO can be found in https://anonymous.4open.science/r/JUDO-9C8B.

Coarse data arise when learners observe only partial information about samples; namely, a set containing the sample rather than its exact value. This occurs naturally through measurement rounding, sensor limitations, and lag in economic systems. We study Gaussian mean estimation from coarse data, where each true sample $x$ is drawn from a $d$-dimensional Gaussian distribution with identity covariance, but is revealed only through the set of a partition containing $x$. When the coarse samples, roughly speaking, have ``low'' information, the mean cannot be uniquely recovered from observed samples (i.e., the problem is not *identifiable*). Recent work by Fotakis et al. (2021) established that *sample*-efficient mean estimation is possible when the unknown mean is *identifiable* and the partition consists of only *convex* sets. Moreover, they showed that without convexity, mean estimation becomes NP-hard. However, two fundamental questions remained open: 1. When is the mean identifiable under convex partitions? 2. Is *computationally* efficient estimation possible under identifiability and convex partitions? This work resolves both questions. We provide a geometric characterization of when a convex partition is identifiable, showing it depends on whether the convex sets form ``slabs'' in a direction. Second, we give the first polynomial-time algorithm for finding $\varepsilon$-accurate estimates of the Gaussian mean given coarse samples from an unknown convex partition, matching the optimal $\widetilde{O}(d/\varepsilon^2)$ sample complexity. Our results have direct applications to robust machine learning, particularly robustness to observation rounding. As a concrete example, we derive a sample- and computationally- efficient algorithm for linear regression with market friction, a canonical problem in using ML in economics, where exact prices are unobserved and one only sees a range containing the price (Rosett, 1959).

Arena-based evaluation has become a key method for assessing large language models (LLMs) through head-to-head model comparisons, closely reflecting human preferences. However, current arena rating systems (e.g., ELO rating system) often suffer from inefficiencies due to exhaustive or random model pair annotations, leading to redundant evaluations, longer evaluation times, and lower overall efficiency. To address these challenges, we propose a novel adaptive model-pair selection algorithm. By leveraging the asymptotic normality of LLM ability estimation under sparse conditions, our approach strategically selects high-value model pairs, focusing on confrontations with the lowest variance. Specifically, we introduce Fisher information as a metric to guide model pair selection, optimizing the evaluation process through A-optimality and D-optimality. A-optimality minimizes estimation variance, ensuring balanced reliability across models, while D-optimality reduces uncertainty by maximizing the determinant of the Fisher Information Matrix. Extensive experiments on both simulated and real-world datasets demonstrate that our method outperforms existing approaches in terms of information efficiency and result reliability. Notably, our method offers a flexible, general toolkit that can be easily integrated into existing arena-based platforms, greatly improving scalability and efficiency for large-scale LLM evaluations.

Large Reasoning Models (LRMs) have shown impressive capabilities in multi-step reasoning tasks. However, alongside these successes, a more deceptive form of model error has emerged—**Reasoning Hallucination**—where logically coherent but factually incorrect reasoning traces lead to persuasive yet faulty conclusions. Unlike traditional hallucinations, these errors are embedded within structured reasoning, making them more difficult to detect and potentially more harmful. In this work, we investigate reasoning hallucinations from a mechanistic perspective. We propose the **Reasoning Score**, which quantifies the depth of reasoning by measuring the divergence between logits obtained from projecting late layers of LRMs to the vocabulary space, effectively distinguishing shallow pattern-matching from genuine deep reasoning. Using this score, we conduct an in-depth analysis on the ReTruthQA dataset and identify two key reasoning hallucination patterns: early-stage fluctuation in reasoning depth and incorrect backtracking to flawed prior steps. These insights motivate our **R**easoning **H**allucination **D**etection (**RHD**) framework, which achieves state-of-the-art performance across multiple domains. To mitigate reasoning hallucinations, we further introduce **GRPO-R**, an enhanced reinforcement learning algorithm that incorporates step-level deep reasoning rewards via potential-based shaping. Our theoretical analysis establishes stronger generalization guarantees, and experiments demonstrate improved reasoning quality and reduced hallucination rates.

The task of learning a diffusion-based neural sampler for drawing samples from an unnormalized target distribution can be viewed as a stochastic optimal control problem on path measures. However, the training of neural samplers can be challenging when the target distribution is multimodal with significant barriers separating the modes, potentially leading to mode collapse. We propose a framework named **Proximal Diffusion Neural Sampler (PDNS)** that addresses these challenges by tackling the stochastic optimal control problem via proximal point method on the space of path measures. PDNS decomposes the learning process into a series of simpler subproblems that create a path gradually approaching the desired distribution. This staged procedure traces a progressively refined path to the desired distribution and promotes thorough exploration across modes. For a practical and efficient realization, we instantiate each proximal step with a proximal weighted denoising cross-entropy (WDCE) objective. We demonstrate the effectiveness and robustness of PDNS through extensive experiments on both continuous and discrete sampling tasks, including challenging scenarios in molecular dynamics and statistical physics.

High-dimensional linear mappings, or linear layers, dominate both the parameter count and the computational cost of most modern deep-learning models. We introduce lookup multivariate Kolmogorov-Arnold Networks (lmKANs), which deliver a substantially better trade-off between capacity and inference cost. Our construction expresses a general high-dimensional mapping through trainable low-dimensional multivariate functions. These functions can carry dozens or hundreds of trainable parameters each, and yet it takes only a few multiplications to compute them because they are implemented as spline lookup tables. Empirically, lmKANs reduce inference FLOPs by up to 6.0× while matching the flexibility of MLPs in general high-dimensional function approximation. In another feedforward fully connected benchmark, on the tabular-like dataset of randomly displaced methane configurations, lmKANs enable more than 10× higher H100 throughput at equal accuracy. Within the framework of Convolutional Neural Networks, lmKAN-based CNNs cut inference FLOPs at matched accuracy by 1.6–2.1× and by 1.7× on the CIFAR-10 and ImageNet-1k datasets, respectively.

We study instrumental variable regression (IVaR) under differential privacy constraints. Classical IVaR methods (like two-stage least squares regression) rely on solving moment equations that directly use sensitive covariates and instruments, creating significant risks of privacy leakage and posing challenges in designing algorithms that are both statistically efficient and differentially private. We propose a noisy two-stage gradient descent algorithm that ensures $\rho$-zero-concentrated differential privacy by injecting carefully calibrated noise into the gradient updates. Our analysis establishes finite-sample convergence rates for the proposed method, showing that the algorithm achieves consistency while preserving privacy. In particular, we derive precise bounds quantifying the trade-off among optimization, privacy, and sampling error. To the best of our knowledge, this is the first work to provide both privacy guarantees and provable convergence rates for instrumental variable regression in linear models. We further validate our theoretical findings with experiments on both synthetic and real datasets, demonstrating that our method offers practical accuracy-privacy trade-offs.

The deployment of large language models (LLMs) is increasingly constrained by memory and latency bottlenecks, motivating the need for quantization techniques that flexibly balance accuracy and efficiency. Recent work has introduced multi-precision models, which enable inference at multiple precisions within a single model depending on runtime constraints. To support such flexibility, quantized weights are often stored as bit-planes, where hardware efficiency improves when the compute operates directly at the bit-plane level and activates only the precision required by each request. In this work, we present AnyBCQ, a hardware-friendly multi-precision extension of Binary-Coded Quantization (BCQ) that supports direct bit-plane operations. By representing weights as binary bit-planes with corresponding scale factors, AnyBCQ enables bit-plane–level computation and maps naturally to accelerator-friendly, bit-parallel arithmetic. Our progressive precision expansion mechanism incrementally refines scaling factors while reusing previously assigned binary codes, yielding monotonic improvements in accuracy as additional bits are enabled. We further co-design a specialized kernel that exploits the BCQ structure to support dynamic per-request precision selection with negligible overhead. Experiments on recent LLMs demonstrate that AnyBCQ significantly narrows the accuracy drop in the low-bit regime (e.g. 2-bit), remains competitive at higher precision, and achieves throughput gains of up to $3.0\times$ over half precision and $1.2\times$ over state-of-the-art multi-precision methods. By aligning algorithmic flexibility with hardware efficiency, AnyBCQ provides a practical foundation for multi-precision LLM deployment across diverse service-level objectives.

The impressive performance gains of modern language models currently rely on scaling parameters: larger models store more world knowledge and reason better. Yet compressing all world knowledge into parameters is unnecessary, as only a fraction is used per prompt, and impractical for edge devices with limited inference-time memory and compute. We address this shortcoming by a memory-augmented architecture and a pretraining strategy aligned with existing hardware paradigms. We introduce small language models that access large hierarchical parametric memory banks encoding world knowledge. Our pretraining learns to store long-tail world knowledge in the memory parameters, while the small language model acts as an anchor capturing common knowledge and general reasoning abilities. Through trillion-token-scale experiments, we show significant gains: a 160M-parameters model augmented with an 18M-parameters memory fetched from a 4.6B memory bank obtains comparable performance to a regular model with more than 2x the parameters. We study the optimal type and size of parametric memories in transformers, scaling them to over 21B parameters. We find that our proposed hierarchical feed-forward memories work robustly across transformer architectures, whether added during pretraining or post-hoc.

Human-centered assessment tasks, which are essential for systematic decision-making, rely heavily on human judgment and typically lack verifiable ground truth. Existing approaches face a dilemma: methods using only human judgments suffer from heterogeneous expertise and inconsistent rating scales, while methods using only model-generated scores must learn from imperfect proxies or incomplete features. We propose Aggregate-then-Calibrate (AtC), a two-stage framework that combines these complementary sources. Stage-1 aggregates heterogeneous comparative judgments into a consensus ranking $\hat{\pi}$ using a rank-aggregation model that accounts for annotator reliability. Stage-2 calibrates any predictive model’s scores by an isotonic projection onto the order $\hat{\pi}$, enforcing ordinal consistency while preserving as much of the model’s quantitative information as possible. Theoretically, we show: (1) modeling annotator heterogeneity yields strictly more efficient consensus estimation than homogeneity; (2) isotonic calibration enjoys risk bounds even when the consensus ranking is misspecified; and (3) AtC asymptotically outperforms model-only assessment. Across semi-synthetic and real-world datasets, AtC consistently improves accuracy and robustness over human-only or model-only assessments. Our results bridge judgment aggregation with model-free calibration, providing a principled recipe for human-centered assessment when ground truth is costly, scarce, or unverifiable. The data and code are available at \url{https://anonymous.4open.science/r/12500_AtC_supp-4F50}.

Currently the most powerful AI systems are aligned with human values via reinforcement learning from human feedback. Yet, reinforcement learning from human feedback models human preferences as noisy samples from a single linear ordering of shared human values and is unable to incorporate democratic AI alignment. In particular, the standard approach fails to represent and reflect diverse and conflicting perspectives of pluralistic human values. Recent research introduced the theoretically principled notion of quantile fairness for training a reinforcement learning policy in the presence of multiple, competing sets of values from different agents. Quite recent work provided an algorithm for achieving quantile fairness in the tabular setting with explicit access to the full set of states, actions and transition probabilities in the MDP. These current methods require solving linear programs with the size of the constraint set given by the number of states and actions, making it unclear how to translate this into practical training algorithms that can only take actions and observe individual transitions from the current state. In this paper, we design and prove the correctness of a new algorithm for quantile fairness that makes efficient use of standard policy optimization as a black-box without any direct dependence on the number of states or actions. We further empirically validate our theoretical results and demonstrate that our algorithm achieves competitive fairness guarantees to the prior work, while being orders of magnitude more efficient with respect to computation and the required number of samples. Our algorithm opens a new avenue for provable fairness guarantees in any setting where standard policy optimization is possible.

Metanetworks are neural architectures designed to operate directly on pretrained weights to perform downstream tasks. However, the parameter space serves only as a proxy for the underlying function class, and the parameter-function mapping is inherently non-injective: distinct parameter configurations may yield identical input-output behaviors. As a result, metanetworks that rely solely on raw parameters risk overlooking the intrinsic symmetries of the architecture. Reasoning about functional identity is therefore essential for effective metanetwork design, motivating the development of equivariant metanetworks, which incorporate equivariance principles to respect architectural symmetries. Existing approaches, however, typically enforce strict equivariance, which imposes rigid constraints and often leads to sparse and less expressive models. To address this limitation, we introduce the novel concept of quasi-equivariance, which allows metanetworks to move beyond the rigidity of strict equivariance while still preserving functional identity. We lay down a principled basis for this framework and demonstrate its broad applicability across diverse neural architectures, including feedforward, convolutional, and transformer networks. Through empirical evaluation, we show that quasi-equivariant metanetworks achieve good trade-offs between symmetry preservation and representational expressivity. These findings advance the theoretical understanding of weight-space learning and provide a principled foundation for the design of more expressive and functionally robust metanetworks.

Vision-language models (VLMs), such as CLIP and SigLIP, have found remarkable success in classification, retrieval, and generative tasks. For this, VLMs deterministically map images and text descriptions to a joint latent space in which their similarity is assessed using the cosine similarity. However, a deterministic mapping of inputs fails to capture uncertainties over concepts arising from domain shifts when used in downstream tasks. In this work, we propose post-hoc uncertainty estimation in VLMs that does not require additional training. Our method leverages a Bayesian posterior approximation over the last layers in VLMs and analytically quantifies uncertainties over cosine similarities. We demonstrate its effectiveness for uncertainty quantification and support set selection in active learning. Compared to baselines, we obtain improved and well-calibrated predictive uncertainties, interpretable uncertainty estimates, and sample-efficient active learning. Our results show promise for safety-critical applications of large-scale models.

Contrastive learning has achieved remarkable success in self-supervised representation learning, often guided by information-theoretic objectives such as mutual information maximization. Motivated by the limitations of static augmentations and rigid invariance constraints, we propose IE-CL (Incremental-Entropy Contrastive Learning), a framework that explicitly optimizes the entropy gain between augmented views while preserving semantic consistency. Our theoretical framework reframes the challenge by identifying the encoder as an information bottleneck and proposes a joint optimization of two components: a learnable transformation for entropy generation and an encoder regularizer for its preservation. Experiments on CIFAR-10/100, STL-10, and ImageNet demonstrate that IE-CL consistently improves performance under small-batch settings. Moreover, our core modules can be seamlessly integrated into existing frameworks. This work bridges theoretical principles and practice, offering a new perspective in contrastive learning.

Deep generative models have advanced text-to-online handwriting generation (TOHG), which aims to synthesize realistic pen trajectories conditioned on textual input and style references. However, most existing methods still primarily focus on character- or word-level generation, resulting in inefficiency and a lack of holistic structural modeling when applied to full text lines. To address these issues, we propose DiffInk, the first latent diffusion Transformer framework for full-line handwriting generation. We first introduce InkVAE, a novel sequential variational autoencoder enhanced with two complementary latent-space regularization losses: (1) an OCR-based loss enforcing glyph-level accuracy, and (2) a style-classification loss preserving writing style. This dual regularization yields a semantically structured latent space where character content and writer styles are effectively disentangled. We then introduce InkDiT, a novel latent diffusion Transformer that integrates target text and reference styles to generate coherent pen trajectories. Experimental results demonstrate that DiffInk outperforms existing state-of-the-art (SOTA) methods in both glyph accuracy and style fidelity, while significantly improving generation efficiency.

Understanding the implicit bias of optimization algorithms is key to explaining and improving the generalization of deep models. The hyperbolic implicit bias induced by pointwise overparameterization promotes sparsity, but also yields a small inverse Riemannian metric near zero, slowing down parameter movement and impeding meaningful parameter sign flips. To overcome this obstacle, we propose Hyperbolic Aware Minimization (HAM), which alternates a standard optimizer step with a lightweight hyperbolic mirror step. The mirror step incurs less compute and memory than pointwise overparameterization, reproduces its beneficial hyperbolic geometry for feature learning, and mitigates the small–inverse-metric bottleneck. Our characterization of the implicit bias in the context of underdetermined linear regression provides insights into the mechanism how HAM consistently increases performance --even in the case of dense training, as we demonstrate in experiments with standard vision benchmarks. HAM is especially effective in combination with different sparsification methods, advancing the state of the art.