ICLR 2026 论文列表

An important aspect subtending language understanding and production is the ability to independently encode positional and symbolic information of the words within a sentence. In Transformers, positional information is typically encoded using Positional Encodings (PEs). One such popular PE, namely Rotary PE (RoPE), has been widely used due to its empirical success. Recently, it has been argued that part of RoPE's success emerges from its ability to encode robust positional and semantic information using large and small frequencies, respectively. In this work, we perform a deeper dive into the positional versus symbolic dichotomy of attention heads behavior, both at the theoretical and empirical level. We provide general definitions of what it means for a head to behave positionally or symbolically, prove that these are two mutually exclusive behaviors and develop a metric to quantify them. We apply our framework to analyze Transformer-based LLMs using RoPE and find that all heads exhibit a strong correspondence between behavior and frequency use. Finally, we introduce canonical tasks designed to be either purely positional or symbolic, and demonstrate that the Transformer performance causally relates to the ability of attention heads to leverage the appropriate frequencies. In particular, we show that we can control the Transformer performance by controlling which frequencies the attention heads can access. Altogether, our work provides a detailed understanding of RoPE, and how its properties relate to model behavior.

The ability to combine linguistic guidance from others with direct experience is central to human development, enabling safe and rapid learning in new environments. How do people integrate these two sources of knowledge, and how might AI systems? We present a computational framework that models human social learning as joint probabilistic inference over structured, executable world models given sensorimotor and linguistic data. We make this possible by turning a pretrained language model into a probabilistic model of how humans share advice conditioned on their beliefs, allowing our agents both to generate advice for others and to interpret linguistic input as evidence during Bayesian inference. Using behavioral experiments and simulations across 10 video games, we show how linguistic guidance can shape exploration and accelerate learning by reducing risky interactions and speeding up key discoveries in both humans and models. We further explore how knowledge can accumulate across generations through iterated learning experiments and demonstrate successful knowledge transfer between humans and models—revealing how structured, language-compatible representations might facilitate human-machine collaborative learning.

Nearest Neighbor Search (NNS) is a fundamental problem in data structures with wide-ranging applications, such as web search, recommendation systems, and, more recently, retrieval-augmented generations (RAG). In such recent applications, in addition to the relevance (similarity) of the returned neighbors, diversity among the neighbors is a central requirement. In this paper, we develop principled welfare-based formulations in NNS for realizing diversity across attributes. Our formulations are based on welfare functions---from mathematical economics---that satisfy central diversity (fairness) and relevance (economic efficiency) axioms. With a particular focus on Nash social welfare, we note that our welfare-based formulations provide objective functions that adaptively balance relevance and diversity in a query-dependent manner. Notably, such a balance was not present in the prior constraint-based approach, which forced a fixed level of diversity and optimized for relevance. In addition, our formulation provides a parametric way to control the trade-off between relevance and diversity, providing practitioners with flexibility to tailor search results to task-specific requirements. We develop efficient nearest neighbor algorithms with provable guarantees for the welfare-based objectives. Notably, our algorithm can be applied on top of any standard ANN method (i.e., use standard ANN method as a subroutine) to efficiently find neighbors that approximately maximize our welfare-based objectives. Experimental results demonstrate that our approach is practical and substantially improves diversity while maintaining high relevance of the retrieved neighbors.

Sparse autoencoders (SAEs) are widely used to extract sparse, interpretable latents from transformer activations. We test whether commonly used SAE quality metrics and automatic explanation pipelines can distinguish trained transformers from randomly initialized ones (e.g., where parameters are sampled i.i.d. from a Gaussian). Over a wide range of Pythia model sizes and multiple randomization schemes, we find that, in many settings, SAEs trained on randomly initialized transformers produce auto-interpretability scores and reconstruction metrics that are similar to those from trained models. These results show that high aggregate auto-interpretability scores do not, by themselves, guarantee that learned, computationally relevant features have been recovered. We therefore recommend treating common SAE metrics as useful but insufficient proxies for mechanistic interpretability and argue for routine randomized baselines and targeted measures of feature 'abstractness'.

Fourier neural operator (FNO) has demonstrated substantial potential in learning mappings between function spaces, such as numerical partial differential equations (PDEs). However, FNO may suffer from inefficiencies when applied to large-scale, high-dimensional function spaces due to the computational overhead associated with high-dimensional Fourier and convolution operators. In this work, we introduce the Tucker-FNO, an efficient neural operator that decomposes the high-dimensional FNO into a series of 1-dimensional FNOs through Tucker decomposition, thereby significantly reducing computational complexity while maintaining expressiveness. Especially, by using the theoretical tools of functional decomposition in Sobolev space, we rigorously establish the universal approximation theorem of Tucker-FNO. Experiments on high-dimensional numerical PDEs such as Navier-Stokes, Plasticity, and Burger's equations show that Tucker-FNO achieves substantial improvement in execution time and performance over FNO. Moreover, by virtue of the compact Tucker decomposition, Tucker-FNO generalizes seamlessly to high-dimensional visual signals by learning mappings from the positional encoding space to the signal's implicit neural representations (INRs). Under this operator INR framework, Tucker-FNO gains consistent improvements on continuous signal restoration over traditional INR methods in terms of efficiency and accuracy.

A key component of in-context reasoning is the ability of language models (LMs) to bind entities for later retrieval. For example, an LM might represent *Ann loves pie* by binding *Ann* to *pie*, allowing it to later retrieve *Ann* when asked *Who loves pie?* Prior research on short lists of bound entities found strong evidence that LMs implement such retrieval via a **positional mechanism**, where *Ann* is retrieved based on its position in context. In this work, we find that this mechanism generalizes poorly to more complex settings; as the number of bound entities in context increases, the positional mechanism becomes noisy and unreliable in middle positions. To compensate for this, we find that LMs supplement the positional mechanism with a **lexical mechanism** (retrieving *Ann* using its bound counterpart *pie*) and a **reflexive mechanism** (retrieving *Ann* through a direct pointer). Through extensive experiments on nine models and ten binding tasks, we uncover a consistent pattern in how LMs mix these mechanisms to drive model behavior. We leverage these insights to develop a causal model combining all three mechanisms that estimates next token distributions with 95\% agreement. Finally, we show that our model generalizes to substantially longer inputs of open-ended text interleaved with entity groups, further demonstrating the robustness of our findings in more natural settings. Overall, our study establishes a more complete picture of how LMs bind and retrieve entities in-context.

The Convolutional Neural Tangent Kernel (CNTK) offers a principled framework for understanding convolutional architectures in the infinite-width regime. However, a comprehensive spectral comparison between CNTK and the classical Neural Tangent Kernel (NTK) remains underexplored. In this work, we present a detailed analysis of the spectral properties of CNTK and NTK, revealing that point cloud data exhibits a stronger alignment with the spectral bias of CNTK than images. This finding suggests that convolutional structures are inherently more suited to such geometric and irregular data formats. Based on this insight, we implement CNTK-based kernel regression for point cloud recognition tasks and demonstrate that it significantly outperforms NTK and other kernel baselines, especially in low-data settings. Furthermore, we derive a closed-form expression that connects CNTK with NTK in hybrid architectures. In addition, we introduce a closed-form of CNTK followed by NTK, while not the main focus, achieves strong empirical performance when applied to point-cloud tasks. Our study not only provides new theoretical understanding of spectral behaviors in neural tangent kernels but also shows that these insights can guide the practical design of CNTK-based regression for structured data such as point clouds.

Influence estimation methods promise to explain and debug machine learning by estimating the impact of individual samples on the final model. Yet, existing methods collapse under training randomness: the same example may appear critical in one run and irrelevant in the next. Such instability undermines their use in data curation or cleanup since it is unclear if we indeed deleted/kept the correct datapoints. To overcome this, we introduce *f-influence* -- a new influence estimation framework grounded in hypothesis testing that explicitly accounts for training randomness, and establish desirable properties that make it suitable for reliable influence estimation. We also design a highly efficient algorithm *f*-*IN*fluence *E*stimation (**f-INE**) that computes f-influence in a **in a single training run**. Finally, we scale up f-INE to estimate influence of instruction tuning data on Llama 3.1 8B and show it can reliably detect poisoned samples that steer model opinions, demonstrating its utility for data cleanup and attributing model behavior.

State-of-the-art machine learning models often incorporate prior knowledge or structural information about the task or data distribution. In some tasks, such knowledge may arise from first principles or emerge as simplified, learned functions that distill essential aspects of the data distribution. Model augmentation has emerged as a strategy to leverage this structured knowledge by coupling it with an auxiliary model to improve predictive performance, while preserving the interpretability offered by the simpler component. In this work, we present a new augmentation framework called the Tutor-Pupil scheme, which is designed to enhance both performance and interpretability. The Pupil is a fixed model, structurally designed for the core task, while the Tutor is a more flexible model trained to apply minimal input-level corrections to improve the Pupil’s performance on the modified input. This strict separation of roles enables the Tutor not only to compensate for the Pupil’s limitations but also to act as a diagnostic instrument. By examining the Tutor’s targeted interventions, we can identify failure modes, detect regions where the Pupil struggles to generalize, and uncover residual patterns or higher-order structures in the data not captured by the original model.

We show that language models’ activations linearly encode when information was learned during training. Our setup involves creating a model with a known training order by sequentially fine-tuning Llama-3.2-1B on six disjoint but otherwise similar datasets about named entities. We find that the average activations of test samples corresponding to the six training datasets encode the training order: when projected into a 2D subspace, these centroids are arranged exactly in the order of training and lie on a straight line. Further, we show that linear probes can accurately (∼90%) distinguish “early” vs. “late” entities, generalizing to entities unseen during the probes’ own training. The model can also be fine-tuned to explicitly report an unseen entity’s training stage (∼80% accuracy). Notably, the training-order encoding does not seem attributable to simple differences in activation magnitudes, losses, or model confidence. Our paper demonstrates that models are capable of differentiating information by its acquisition time, and carries significant implications for how they might manage conflicting data and respond to knowledge modifications.

*Automated circuit discovery* is a central tool in mechanistic interpretability for identifying the internal components of neural networks responsible for specific behaviors. While prior methods have made significant progress, they typically depend on heuristics or approximations and do not offer provable guarantees over continuous input domains for the resulting circuits. In this work, we leverage recent advances in neural network verification to propose a suite of automated algorithms that yield circuits with *provable guarantees*. We focus on three types of guarantees: (1) *input domain robustness*, ensuring the circuit agrees with the model across a continuous input region; (2) *robust patching*, certifying circuit alignment under continuous patching perturbations; and (3) *minimality*, formalizing and capturing a wide array of various notions of succinctness. Interestingly, we uncover a diverse set of novel theoretical connections among these three families of guarantees, with critical implications for the convergence of our algorithms. Finally, we conduct experiments with state-of-the-art verifiers on various vision models, showing that our algorithms yield circuits with substantially stronger robustness guarantees than standard circuit discovery methods, establishing a principled foundation for provable circuit discovery.

Large language models demonstrate the intriguing ability to perform unseen tasks via in-context learning. However, it remains unclear what mechanisms inside the model drive such task-level generalization. In this work, we approach this question through the lens of off-by-one addition (i.e., 1+1=3, 2+2=5, 3+3=?), a two-step, counterfactual task with an unexpected +1 function as a second step. Leveraging circuit-style interpretability techniques such as path patching, we analyze the models' internal computations behind their performance and present three key findings. First, we uncover a function induction mechanism that explains the model's generalization from standard addition to off-by-one addition. This mechanism resembles the structure of the induction head mechanism found in prior work and elevates it to a higher level of abstraction. Second, we show that the induction of the +1 function is governed by multiple attention heads in parallel, each of which emits a distinct piece of the +1 function. Finally, we find that this function induction mechanism is reused in a broader range of tasks, including synthetic tasks such as shifted multiple-choice QA and algorithmic tasks such as base-8 addition. Overall, our findings offer deeper insights into how reusable and composable structures within language models enable task-level generalization.

Generative model ecosystems increasingly operate as competitive multi-platform markets, where platforms strategically select models from a shared pool and users with heterogeneous preferences choose among them. Understanding how platforms interact, when market equilibria exist, how outcomes are shaped by model-provider, platforms, and user behavior, and how social welfare is affected is critical for fostering beneficial market environment. In this paper, we formalize a three-layer *model-platfrom-user* market game and identify conditions for the existence of pure Nash equilibrium. Our analysis shows that market structure, whether platforms converge on similar models or differentiate by selecting distinct ones, depends not only on models’ global average performance but also on their localized attraction to user groups. We further examine welfare outcomes and show that expanding the model pool does not necessarily increase user welfare or market diversity. Finally, we design and evaluate best-response training schemes that allow model-provider to strategically introduce new models into competitive markets.

The high dimensional parameter space of modern deep neural networks — the neuromanifold — is endowed with a unique metric tensor defined by the Fisher information, estimating which is crucial for both theory and practical methods in deep learning. To analyze this tensor for classification networks, we return to a low dimensional space of probability distributions — the core space — and carefully analyze the spectrum of its Riemannian metric. We extend our discoveries there into deterministic bounds of the metric tensor on the neuromanifold. We introduce an unbiased random estimate of the metric tensor and its bounds based on Hutchinson’s trace estimator. It can be evaluated efficiently through a single backward pass, with a standard deviation bounded by the true value up to scaling.

Even when instructed to adhere to source material, language models often generate unsubstantiated content – a phenomenon known as “closed-domain hallucination.” This risk is amplified in processes with multiple generative steps (MGS), compared to processes with a single generative step (SGS). However, due to the greater complexity of MGS processes, we argue that detecting hallucinations in their final outputs is necessary but not sufficient: it is equally important to trace where hallucinated content was likely introduced and how faithful content may have been derived from the source through intermediate outputs. To address this need, we present VeriTrail, the first closed-domain hallucination detection method designed to provide traceability for both MGS and SGS processes. We also introduce the first datasets to include all intermediate outputs as well as human annotations of final outputs’ faithfulness for their respective MGS processes. We demonstrate that VeriTrail outperforms baseline methods on both datasets.

Recent advances in reasoning capabilities of large language models (LLMs) are largely driven by reinforcement learning (RL), yet the underlying parameter dynamics during RL training remain poorly understood. This work identifies two fundamental properties of RL-induced parameter updates in LLMs: (1) Rank-1 Dominance, where the top singular subspace of the parameter update matrix nearly fully determines reasoning improvements, recovering over 99\% of performance gains; and (2) Rank-1 Linear Dynamics, where this dominant subspace evolves linearly throughout training, enabling accurate prediction from early checkpoints. Extensive experiments across 13 LLMs and 10 algorithms validate the generalizability of these properties. More importantly, based on these findings, we propose AlphaRL, a plug-in acceleration framework that extrapolates the final parameter update using a short early training window, achieving up to 2.5× speedup while retaining > 96\% of reasoning performance without extra modules or hyperparameter tuning. This positions our finding as a versatile and practical tool for large-scale RL, opening a path toward principled, interpretable, and efficient training paradigm for LLMs.

Understanding internal representations of neural models is a core interest of mechanistic interpretability. Due to its large dimensionality, the representation space can encode various aspects about inputs. To what extent are different aspects organized and encoded in separate subspaces? Is it possible to find these "natural" subspaces in a purely unsupervised way? Somewhat surprisingly, we can indeed achieve this and find interpretable subspaces by a seemingly unrelated training objective. Our method, neighbor distance minimization (NDM), learns non-basis-aligned subspaces in an unsupervised manner. Qualitative analysis shows subspaces are interpretable in many cases, and encoded information in obtained subspaces tends to share the same abstract concept across different inputs, making such subspaces similar to "variables" used by the model. We also conduct quantitative experiments using known circuits in GPT-2; results show a strong connection between subspaces and circuit variables. We also provide evidence showing scalability to 2B models by finding separate subspaces mediating context and parametric knowledge routing. Viewed more broadly, our findings offer a new perspective on understanding model internals and building circuits.

Modern large language models increasingly use smooth activation functions such as GELU or SiLU, allowing negative pre-activations to carry both signal and gradient. Nevertheless, many neuron-level interpretability analyses have historically focused on large positive activations, often implicitly treating the negative region as less informative, a carryover from the ReLU-era. We challenge this assumption and ask whether and how negative pre-activations are functionally utilized. We address this question by studying a sparse subpopulation of Wasserstein neurons whose output distributions deviate strongly from a Gaussian baseline and that functionally differentiate similar inputs. We show that this negative region plays a functional role rather than reflecting a mere gradient optimization side effect. A minimal, sign-specific intervention that zeroes only the negative pre-activations of a small set of Wasserstein neurons substantially increases perplexity and sharply degrades grammatical performance on BLiMP and TSE, whereas both random and perplexity-matched ablations of many more non-Wasserstein neurons in their negative pre-activations leave grammatical performance largely intact. Conversely, on a suite of non-grammatical benchmarks, the perplexity-matched control ablation is more damaging than the Wasserstein neuron ablation, yielding a double dissociation between syntax and other capabilities. Part-of-speech analysis localizes the excess surprisal to syntactic scaffolding tokens, layer-specific interventions show that small local degradations accumulate across depth, and training-dynamics analysis reveals that the same sign-specific ablation becomes more harmful as Wasserstein neurons emerge and stabilize. Together, these results identify negative pre-activations in a sparse subpopulation of Wasserstein neurons as an actively used substrate for syntax in smooth-activation language models.

Algorithms increasingly serve as information mediators -- from social media feeds and targeted advertising to the increasing ubiquity of LLMs. This engenders a joint process where agents combine private, algorithmically-mediated signals with observational learning from peers to arrive at decisions. To study such settings, we introduce a model of controlled sequential social learning in which an information-mediating planner (e.g., an LLM) controls the information structure of agents while they also learn from the decisions of earlier agents. The planner may seek to improve social welfare (an altruistic planner) or to induce a specific action the planner prefers (a biased planner). Our framework presents a new optimization problem for social learning that combines dynamic programming with decentralized action choices and Bayesian belief updates. In this setting, we prove the convexity of the value function and characterize the optimal policies of altruistic and biased planners, which attain desired tradeoffs between the costs they incur and the payoffs they earn from induced agent choices. The characterization reveals that the optimal planner operates in different modes depending on the range of belief values. The modes include investing the maximum allowed resource, not investing any resource, or the investment increasing or decreasing with increase in the belief. Notably, for some ranges of belief the biased planner even intentionally obfuscates the agents' signals. Even under stringent transparency constraints—information parity with individuals, no lying or cherry‑picking, and full observability—we show that information mediation can substantially shift social welfare in either direction. We complement our theory with simulations in which LLMs act as both planner and agents. Notably, the LLM-based planner in our simulations exhibits emergent strategic behavior in steering public opinion that broadly mirrors the trends predicted, though key deviations suggest the influence of non-Bayesian reasoning—consistent with the cognitive patterns of both human users and LLMs trained on human-like data. Together, we establish our framework as a tractable basis for studying the impact and regulation of LLM information mediators that corresponds to real behavior.