ICLR 2026 Papers

Squared tensor networks (TNs) and their extension as computational graphs---squared circuits---have been used as expressive distribution estimators, yet supporting closed-form marginalization. However, the squaring operation introduces additional complexity when computing the partition function or marginalizing variables, which hinders their applicability in ML. To solve this issue, canonical forms of TNs are parameterized via unitary matrices to simplify the computation of marginals. However, these canonical forms do not apply to circuits, as they can represent factorizations that do not directly map to a known TN. Inspired by the ideas of orthogonality in canonical forms and determinism in circuits enabling tractable maximization, we show how to parameterize squared circuits to overcome their marginalization overhead. Our parameterizations unlock efficient marginalization even in factorizations different from TNs, but encoded as circuits, whose structure would otherwise make marginalization computationally hard. Finally, our experiments on distribution estimation show how our proposed conditions in squared circuits come with no expressiveness loss, while enabling more efficient learning.

As Vision Transformers (ViTs) become standard backbones across vision, a mechanistic account of their computational phenomenology is now essential. Despite architectural cues that hint at dynamical structure, there is no settled framework that interprets Transformer depth as a well-characterized flow. In this work, we introduce the $\textbf{Block-Recurrent Hypothesis (BRH)}$, arguing that trained ViTs admit a block-recurrent depth structure such that the computation of the original $L$ blocks can be accurately rewritten using only $k \ll L$ distinct blocks applied recurrently. Across diverse ViTs, between-layer representational similarity matrices suggest few contiguous phases. To determine whether this reflects reusable computation, we operationalize our hypothesis in the form of block recurrent surrogates of pretrained ViTs, which we call Recurrent Approximations to Phase-structured TransfORmers ($\texttt{Raptor}$). Using small-scale ViTs, we demonstrate that phase-structure metrics correlate with our ability to accurately fit $\texttt{Raptor}$ and identify the role of stochastic depth in promoting the recurrent block structure. We then provide an empirical existence proof for BRH in foundation models by showing that we can train a $\texttt{Raptor}$ model to recover $94$\% of DINOv2 ImageNet-1k linear probe accuracy in only 2 blocks. To provide a mechanistic account of these observations, we leverage our hypothesis to develop a program of $\textbf{Dynamical Interpretability}$. We find $\textit{\textbf{(i)}}$ directional convergence into class-dependent angular basins with self-correcting trajectories under small perturbations $\textit{\textbf{(ii)}}$ token-specific dynamics, where $\texttt{cls}$ executes sharp late reorientations while $\texttt{patch}$ tokens exhibit strong late-stage coherence reminiscent of a mean-field effect and converge rapidly toward their mean direction and $\textit{\textbf{(iii)}}$ a collapse of the update field to low rank in late depth, consistent with convergence to low-dimensional attractors. Altogether, we find that a compact recurrent program emerges along the depth of ViTs, pointing to a low-complexity normative solution that enables these models to be studied through principled dynamical systems analysis.

Long-range dependencies are critical for effective graph representation learning, yet most existing datasets focus on small graphs tailored to inductive tasks, offering limited insight into long-range interactions. Current evaluations primarily compare models employing global attention (e.g., graph transformers) with those using local neighborhood aggregation (e.g., message-passing neural networks) without a direct measurement of long-range dependency. In this work, we introduce $\texttt{City-Networks}$, a novel large-scale transductive learning dataset derived from real-world city road networks. This dataset features graphs with over $10^5$ nodes and significantly larger diameters than those in existing benchmarks, naturally embodying long-range information. We annotate the graphs based on local node eccentricities, ensuring that the classification task inherently requires information from distant nodes. Furthermore, we propose a generic measurement based on the Jacobians of neighbors from distant hops, offering a principled quantification of long-range dependencies. Finally, we provide theoretical justifications for both our dataset design and the proposed measurement—particularly by focusing on over-smoothing and influence score dilution—which establishes a robust foundation for further exploration of long-range interactions in graph neural networks.

Current safety evaluations of language models rely on benchmark-based assessments that may miss targeted vulnerabilities. We present RepIt, a simple and data-efficient framework for isolating concept-specific representations in LM activations. While existing steering methods already achieve high attack success rates through broad interventions, RepIt enables a more concerning capability: selective suppression of refusal on targeted concepts while preserving refusal elsewhere. Across five frontier LMs, RepIt produces evaluation-evading models that answer questions related to weapons of mass destruction while still scoring as safe on standard benchmarks. We find the edit of the steering vector localizes to just 100-200 neurons, and robust concept vectors can be extracted from as few as a dozen examples on a single A6000, highlighting how targeted, hard-to-detect modifications can exploit evaluation blind spots with minimal resources. By demonstrating precise concept disentanglement, this work exposes critical vulnerabilities in current safety evaluation practices and demonstrates an immediate need for more comprehensive, representation-aware assessments.

This paper analyzes the generalization error of minimum-norm interpolating solutions in linear regression using spiked covariance data models. The paper characterizes how varying spike strengths and target-spike alignments can affect risk, especially in overparameterized settings. The study presents an exact expression for the generalization error, leading to a comprehensive classification of benign, tempered, and catastrophic overfitting regimes based on spike strength, the aspect ratio $c=d/n$ (particularly as $c \to \infty$), and target alignment. Notably, in well-specified aligned problems, increasing spike strength can surprisingly induce catastrophic overfitting before achieving benign overfitting. The paper also reveals that target-spike alignment is not always advantageous, identifying specific, sometimes counterintuitive, conditions for its benefit or detriment. Alignment with the spike being detrimental is empirically demonstrated to persist in nonlinear models.

Unlike the foundation model scaling laws seen in natural language processing and computer vision, biological foundation models scale relatively poorly. For example, the ESM-2 family of protein language models plateaus at 650M-3B parameters on ProteinGym benchmarks. We address this limitation by introducing Reverse Distillation, a principled framework that decomposes large protein language model representations into orthogonal subspaces guided by smaller models of the same family. We hypothesize that this decomposition matches the natural hierarchy of protein properties, where broad features like secondary structure are robustly captured by compact, smaller models while the residual capacity of larger models specializes in protein-family specific functions. Our method is theoretically grounded and enables monotonic scaling---larger reverse-distilled models consistently outperform their smaller counterparts, overcoming the scaling plateau. Moreover, on ProteinGym benchmarks, reverse-distilled ESM-2 variants broadly outperform their respective baseline models at the same embedding dimensionality. Our approach offers a generalizable framework for disentangling hierarchical feature spaces in foundation model embeddings, with potential applications across biology and other domains where scaling challenges persist.

Hyperparameter tuning can dramatically impact training stability of large-scale models. Recent works on neural network parameterisations, such as μP, have shown that layer types and sizes should dictate how global hyperparameters should be rescaled in order to achieve efficient transfer across model sizes. On the other hand, the established practice for hyperparameter optimisation search is to look for optimal global base values that apply at some fixed model scale. We transfer hyperparameters across all scaling axes: width and depth, using an extension of CompleteP (Dey et al., 2025), training horizon, and batch size. Our study covers all optimisation hyperparameters of modern models: learning rates, Adam parameters, weight decay, initialisation scales, and residual block multipliers. Lastly, we demonstrate that hyperparameter transfer holds even in the per-layer hyperparameter regime. We characterise the empirical challenges of navigating the high-dimensional hyperparameter landscape, and propose practical guidelines for tackling this optimisation problem. We suggest a simplified parameterisation of the hyperparameter space that reduces the dimensionality of the search-space at no performance cost. Our experiments demonstrate training speed improvements when applying transferred hyperparameters to Large Language Models.

Loss curves are smooth during most of model training, so visible discontinuities stand out as possible conceptual breakthroughs. Studying these breakthroughs enables a deeper understanding of learning dynamics, but only when they are properly identified. This paper argues that similar breakthroughs occur frequently throughout training but they are obscured by a loss metric that collapses all variation into a single scalar. To find these hidden transitions, we introduce POLCA, a method for decomposing changes in loss along arbitrary bases of the low-rank training subspace. We use our method to identify clusters of samples that share similar changes in loss during training, disaggregating the overall loss into that of smaller groups of conceptually similar data. We validate our method on synthetic arithmetic and natural language tasks, showing that POLCA recovers clusters that represent interpretable breakthroughs in the model's capabilities. We demonstrate the promise of these hidden phase transitions as a tool for unsupervised interpretability.

Trained Transformers have been shown to compute abstract features that appear redundant for predicting the immediate next token. We identify which components of the gradient signal from the next-token prediction objective give rise to this phenomenon, and we propose a method to estimate the influence of those components on the emergence of specific features. After validating our approach on toy tasks, we use it to interpret the origins of the world model in OthelloGPT and syntactic features in a small language model. Finally, we apply our framework to a pretrained LLM, showing that features with extremely high or low influence on future tokens tend to be related to formal reasoning domains such as code. Overall, our work takes a step toward understanding hidden features of Transformers through the lens of their development during training.

The discontinuous operations inherent in quantization and sparsification introduce a long-standing obstacle to backpropagation, particularly in ultra-low precision and sparse regimes. While the community has long viewed quantization as unfriendly to gradient descent due to its lack of smoothness, we pinpoint—for the first time—that the key issue is the absence of a proper gradient path that allows training to learn robustness to quantization noise. The standard Straight-Through Estimator (STE) exacerbates this with its well-understood mismatch: a quantization-aware forward pass but oblivious backward pass, leading to unmanaged error and instability. We solve this by explicitly modeling quantization as additive noise, making the full forward-backward path well-defined without heuristic gradient estimation. As one natural solution, we introduce a denoising dequantization transform derived from a principled ridge regression objective, creating an explicit, corrective gradient path that makes learning robust to the noise STE bypasses. We extend this to sparsification by treating it as a special form of quantization that zeros out small values. Our unified framework trains models at arbitrary precisions and sparsity levels with off-the-shelf recipes, enabling stable A1W1 and sub-1-bit networks where others falter. It yields state-of-the-art results, mapping efficiency frontiers for modern LLMs and providing a theoretically grounded path to hyper-efficient neural networks.

Brain-like intelligent systems need brain-like learning methods. Equilibrium Propagation (EP) is a biologically plausible learning framework with strong potential for brain-inspired computing hardware. However, existing implementations of EP suffer from instability and prohibitively high computational costs. Inspired by the structure and dynamics of the brain, we propose a biologically plausible Feedback-regulated REsidual recurrent neural network (FRE-RNN) and study its learning performance in EP framework. Feedback regulation enables rapid convergence by attenuating feedback signals and reducing the disturbance of feedback path to feedforward path. The improvement in convergence property reduces the computational cost and training time of EP by orders of magnitude, delivering performance on par with backpropagation (BP) in benchmark tasks. Meanwhile, residual connections with brain-inspired topologies help alleviate the vanishing gradient problem that arises when feedback pathways are weak in deep RNNs. Our approach substantially enhances the applicability and practicality of EP. The techniques developed here also offer guidance to implementing in-situ learning in physical neural networks.

Large language models (LLMs) are increasingly paired with activation-based monitoring to detect and prevent harmful behaviors that may not be apparent at the surface-text level. However, existing activation safety approaches, trained on broad misuse datasets, struggle with poor precision, limited flexibility, and lack of interpretability. This paper introduces a new paradigm: rule-based activation safety, inspired by rule-sharing practices in cybersecurity. We propose modeling activations as cognitive elements (CEs), fine-grained, interpretable factors such as "making a threat" and "payment processing", that can be composed to capture nuanced, domain-specific behaviors with higher precision. Building on this representation, we present a practical framework that defines predicate rules over CEs and detects violations in real time. This enables practitioners to configure and update safeguards without retraining models or detectors, while supporting transparency and auditability. Our results show that compositional rule-based activation safety improves precision, supports domain customization, and lays the groundwork for scalable, interpretable, and auditable AI governance. We open source GAVEL and provide an automated rule creation tool.

The extent to which decoder-only language models (LMs) engage in planning, that is, organizing intermediate computations to support coherent long-range generation, remains an important question, with implications for interpretability, reliability, and principled model design. Planning involves structuring computations over long horizons, and considering multiple possible continuations, but how far transformer-based LMs exhibit them without external scaffolds, e.g., chain-of-thought prompting, is unclear. We address these questions by analyzing the hidden states at the core of transformer computations, which capture intermediate results and act as carriers of information. Since these hidden representations are redundant and encumbered with fine-grained details, we develop a pipeline based on vector-quantized variational autoencoders that compresses them into compact summary codes. These codes enable measuring mutual information and analyzing the computational structure of the underlying model behavior. Using this framework, we study planning in LMs across synthetic grammar, path-finding tasks, and natural language datasets, focusing on two planning properties: (i) the planning horizon of pre-output computations, and (ii) the extent to which the model considers alternative valid continuations. As a separate downstream use of the same pipeline, we also analyze how decision-relevant information is distributed across layers and earlier prefix blocks when producing next-token predictions. Together, these analyses advance our understanding of planning in LMs and provide a general-purpose pipeline for inspecting internal model dynamics. Our results reveal that the effective planning horizon is task-dependent, that models implicitly preserve information about unused correct continuations, and that predictions draw most on recent computations, though earlier blocks remain informative.

Large Language Models (LLMs) have demonstrated outstanding capabilities in various code-related tasks, including code completion, translation, or summarization. However, these pretrained models are static, posing a challenge to incorporate new knowledge into an LLM to correct erroneous behavior. Approaches such as retraining or fine-tuning demand extensive labeled datasets and might be computationally expensive, while prompt engineering fails to change models permanently. Knowledge Editing (KE) techniques offer a more efficient alternative, enabling model updates with minimal data, even just a single example. Nevertheless, existing KE methods often manipulate parameters within the Transformer's multi-layer perceptrons (MLPs), where neuronal polysemanticity hinders both the precision and interpretability of the edits. To address these limitations, we exploit TransCoder, an MLP-like model component with a wide and sparsely activated hidden feature vector. Specifically, we introduce **TransCoder-based Precise Editing** (**TCPE**), a novel method that leverages the sparsity and monosemanticity of the TransCoder’s neurons for highly localized knowledge editing. TCPE exhibits neuron-level mechanistic interpretability characteristics, revealing the correspondence between the edited neurons and the specific code-related knowledge. Furthermore, we present KECode, a new evaluation benchmark for code-to-code translation based on functional equivalence. Using KECode, we conduct a systematic evaluation of representative KE methods in the context of code-to-code translation. Our experimental results demonstrate that TCPE outperforms existing KE methods, achieving a substantial improvement of translation accuracy of CodeLlama-7b-Instruct from 57.5% to 64.0% in a low-resource scenario of Java-to-D translation.

Attention is foundational to large language models (LLMs), enabling different heads to have diverse focus on relevant input tokens. However, learned behaviors like attention sinks, where the first token receives the most attention despite limited semantic importance, suggest some heads may be inactive, and point to a significant source of computational redundancy. To analyze this phenomenon, we propose a taxonomy of 12 score functions that measure different ways a head can be inactive. Thresholding these scores allows us to analyze different sets of potentially inactive attention heads. We evaluate whether identified heads are inactive through model interventions, finding that more than 12% of attention heads are inactive on average, and can be ablated in specific contexts while maintaining MMLU accuracy to within 1% of the pretrained LLM. Across 3 model families, our score functions that measure the average norm of a head's output consistently identify inactive heads that would not have been found by score functions that rely solely on attention weights. We establish that relying on a score function that measures a first token attention sink would underestimate the prevalence of inactive heads, failing to identify more than 7% of inactive heads on average. We also show how measuring score distributions can provide insights into attention behavior. For instance, we find evidence that finetuning causes little to no change in attention behavior, and that even within the same model family, large model scales present markedly different attention behaviors.

Confidence calibration has been widely studied to improve the trustworthiness of predictions in vision-language models (VLMs). However, we theoretically reveal that standard confidence calibration inherently _impairs_ the ability to distinguish between correct and incorrect predictions (i.e., Misclassification Detection, MisD), which is crucial for reliable deployment of VLMs in high-risk applications. In this paper, we investigate MisD in VLMs and propose confidence recalibration to enhance MisD. Specifically, we design a new confidence calibration objective to replace the standard one. This modification theoretically achieves higher precision in the MisD task and reduces the mixing of correct and incorrect predictions at every confidence level, thereby overcoming the limitations of standard calibration for MisD. As the calibration objective is not differentiable, we introduce a differentiable surrogate loss to enable better optimization. Moreover, to preserve the predictions and zero-shot ability of the original VLM, we develop a post-hoc framework, which employs a lightweight meta network to predict sample-specific temperature factors, trained with the surrogate loss. Extensive experiments across multiple metrics validate the effectiveness of our approach on MisD.

With the increasing demand for interpretability in machine learning, functional ANOVA decomposition has gained renewed attention as a principled tool for breaking down high-dimensional function into low-dimensional components that reveal the contributions of different variable groups. Recently, Tensor Product Neural Network (TPNN) has been developed and applied as basis functions in the functional ANOVA model, referred to as ANOVA-TPNN. A disadvantage of ANOVA-TPNN, however, is that the components to be estimated must be specified in advance, which makes it difficult to incorporate higher-order TPNNs into the functional ANOVA model due to computational and memory constraints. In this work, we propose Bayesian-TPNN, a Bayesian inference procedure for the functional ANOVA model with TPNN basis functions, enabling the detection of higher-order components with reduced computational cost compared to ANOVA-TPNN. We develop an efficient MCMC algorithm and demonstrate that Bayesian-TPNN performs well by analyzing multiple benchmark datasets. Theoretically, we prove that the posterior of Bayesian-TPNN is consistent.

While Retrieval-Augmented Generation (RAG) enables large language models (LLMs) to generate contextually grounded responses, contextual faithfulness remains challenging as LLMs may not consistently trust provided context, leading to hallucinations that undermine reliability. We observe an inverse correlation between response copying degree and context-unfaithful hallucinations on RAGTruth, suggesting higher copying degrees reduce hallucinations by fostering genuine contextual belief. We propose Copy-Paste, a generation paradigm that directly embeds contextual fragments to ensure faithfulness, and instantiate it through CopyPasteLLM via two-stage high-copying preference training. We design three prompting methods to enhance copying degree, demonstrating that high-copying responses achieve superior contextual faithfulness and hallucination control. These approaches enable a fully automated pipeline that transforms generated responses into high-copying preference data for training CopyPasteLLM. On FaithEval, ConFiQA and PubMedQA, CopyPasteLLM achieves best performance in both counterfactual and original contexts, remarkably with 12.2\% to 24.5\% accuracy improvements on FaithEval over the best baseline, while requiring only 365 training samples—1/50th of baseline data. To elucidate CopyPasteLLM's effectiveness, we propose the Context-Parameter Copying Capturing algorithm. Interestingly, this reveals that CopyPasteLLM recalibrates reliance on internal parametric knowledge rather than external knowledge during generation. All codes are available at https://github.com/longyongchao/CopyPasteLLM

Data filtering plays a central role in improving model performance, particularly for vision language models that are pretrained on large, noisy, and redundant image-caption datasets. Existing filtering techniques assess every sample individually and retain those that exceed a certain quality threshold, but such strategies fail to capture higher-order interactions. In this work, we propose a novel submodular framework for data selection that addresses this limitation. Our method, Submodular Distribution Matching (SDM), selects a subset by: (1) training a type of sparse autoencoder to learn disentangled and \emph{monotone} features; (2) estimating a target feature distribution from a target dataset; and (3) selecting a subset of samples whose feature distribution closely matches the target via submodular maximization. Given the DataComp-medium training set and no external models, SDM achieves state-of-the-art accuracy on both ImageNet-1K and average performance across 38 downstream tasks. On the full DataComp-medium benchmark, SDM delivers performance within 1\% of the state-of-the-art results while using over \textbf{\emph{5×}} fewer GPU hours than the leading approach.