ICML 2026 Papers

Symbolic regression (SR) traditionally balances accuracy and complexity, implicitly assuming that simpler formulas are structurally more rational. We argue that this assumption is insufficient: existing algorithms often exploit this metric to discover accurate and compact but structurally irrational formulas that are numerically ill-conditioned and physically inexplicable. Inspired by the structural stability of real physical laws, we propose the Effective Information Criterion (EIC) to quantify formula rationality. EIC models formulas as information channels and measures the amplification of inherent rounding noise during recursive calculation, effectively distinguishing physically plausible structures from pathological ones without relying on ground truth. Our analysis reveals a stark structural stability gap between human-derived equations and SR-discovered results. By integrating EIC into SR workflows, we provide explicit structural guidance: for heuristic search, EIC steers algorithms toward stable regions to yield superior Pareto frontiers; for generative models, EIC-based filtering improves pre-training sample efficiency by 2–4 times and boosts generalization $R^2$ by 22.4\%. Finally, an extensive study with 108 human experts shows that EIC aligns with human preferences in 70\% of cases, validating structural stability as a critical prerequisite for human-perceived interpretability. We release our code at https://github.com/tsinghua-fib-lab/EIC

We study flow-matching transformers when data lie on a low-dimensional manifold. Our key insight is a flow decomposition that splits motion along the manifold from motion off the manifold. The scheme works for first and higher-order flow matching and ties model complexity to the intrinsic manifold dimension. Building on these, we establish tighter sample-complexity bounds for velocity approximation, velocity estimation, and distribution estimation. Our results show how flow-matching transformers escape the curse of dimensionality by utilizing intrinsic data structure.

Listwise reranking is a critical yet costly component in vision-centric retrieval and multimodal retrieval-augmented generation (M-RAG) over long documents. Although recent VLM-based rerankers achieve strong accuracy, they are often impractical due to long visual-token inputs and autoregressive decoding, resulting in high latency. We propose ZipRerank, a very efficient listwise multimodal reranker that directly addresses both bottlenecks: it shortens the input via query-image early interaction and eliminates multi-step generation by scoring all candidates in a single forward pass. ZipRerank is trained with a two-stage recipe: listwise pretraining on large-scale text reranking data rendered as images, followed by multimodal finetuning with VLM-teacher supervision and a soft-ranking objective to handle noisy rankings. Extensive experiments on the MMDocIR benchmark demonstrate that ZipRerank matches or surpasses state-of-the-art multimodal rerankers while reducing LLM inference latency by up to an order of magnitude, making it well-suited for latency-sensitive real-world systems. Source code is available at https://anonymous.4open.science/r/ZipRerank.

Generalizing pretrained models to unseen datasets without retraining is a central challenge toward foundation models. Achieving fully inductive inference on numerical data is particularly difficult due to large variations in feature dimensionality and semantics across datasets. We observe that, in the presence of graph structure, numerical data admits a distinct structure-induced representational axis beyond the feature space, which we formalize as the *view space*. This view space enables a unified representation of graphs with heterogeneous features and motivates *Graph View Transformation* (GVT), a class of parametric mappings that can be shared across arbitrary graphs. We instantiate this framework with Recurrent GVT, an architecture for fully inductive node representation learning in node classification. Pretrained on OGBN-Arxiv and evaluated on 27 benchmarks, Recurrent GVT outperforms GraphAny, the prior fully inductive graph model, by +8.93%, and surpasses 12 individually tuned GNNs by at least +3.30%. These results establish the view space as a principled and practical foundation for learning across graphs with heterogeneous feature spaces. Code and checkpoints are available in https://github.com/dooho00/graph-view-space.

A key challenge in normalizing flows is finding expressive invertible scalar bijections. Existing approaches face trade-offs: affine transformations are smooth and analytically invertible but lack expressivity; monotonic splines offer local control but are only piecewise smooth and act on bounded domains; residual flows achieve smoothness but need numerical inversion. We introduce three families of *analytic bijections* that are globally smooth ($C^\infty$), defined on all of $\mathbb{R}$, and analytically invertible in closed form, combining the favorable properties of prior approaches. Beyond serving as drop-in replacements in coupling flows, where they match or exceed spline performance, we develop *radial flows*: a novel architecture using direct parametrization that transforms the radial coordinate while preserving angular direction. Radial flows exhibit exceptional training stability, produce geometrically interpretable transformations, and on targets with radial structure can achieve comparable quality to coupling flows with $1000\times$ fewer parameters. We provide comprehensive evaluation on 1D and 2D benchmarks, and demonstrate applicability to higher-dimensional physics problems through experiments on $\phi^4$ lattice field theory, where our bijections outperform affine baselines and enable problem-specific designs that address mode collapse.

Extending traditional graph anomaly detection (GAD) from one-for-one to one-for-all paradigms, generalist GAD aims to learn a universal detector for identifying anomalies on unseen graphs without graph-specific retraining. Nevertheless, existing approaches primarily focus on aligning heterogeneous features across different data domains via PCA-based projection, which harmonizes feature dimensions but neglects semantic alignment. As a result, GAD models fail to acquire semantically transferable knowledge from source-domain pre-training, and even exhibit negative transfer on unseen graphs. To address this issue, we propose a Relational Fingerprint-based generalist GAD approach (REFI-GAD for short), aligning heterogeneous raw features with a universal and semantics-aware relational fingerprint (REFI) that encodes anomaly-indicative cues from both contextual and structural perspectives. Building on REFI, we design a fingerprint-grounded generalist GAD model, which combines a transformer-based encoder to capture domain-invariant knowledge with an SNR-guided refinement module for domain-specific adaptation. Extensive experiments on 14 datasets demonstrate that REFI-GAD significantly outperforms state-of-the-art methods.

Learning beneficial treatment allocations for a patient population is an important problem in precision medicine. For such allocations, a certain proportion of treated patients may not receive any benefit. This proportion of unnecessary treated represents a `treatment risk' which is a waste of resources and may, in addition, expose patients to unnecessary adverse effects. Therefore, we aim to control the treatment risk when learning beneficial allocations. This learning problem is complicated by the fact that the treatment risk is generally not identifiable from either randomized trial or observational data. We propose a certifiable learning method that controls treatment risk, using finite samples in the partially identified setting. The method is illustrated using both simulated and real data.

As Large Reasoning Models (LRMs) are increasingly deployed, auditing their chain-of-thought (CoT) traces for safety becomes critical. Recent work has reported that monitorability—the degree to which CoT faithfully and informatively reflects internal computation—can appear as a "free gift" during the early stages of Reinforcement Learning with Verifiable Rewards (RLVR). We make this observation concrete through a systematic evaluation across model families and training domains. Our results show that this effect is not universal: monitorability improvements are strongly data-dependent. In particular, we demonstrate the critical role of data diversity and instruction-following data during RLVR training. We further show that monitorability is orthogonal to capability—improvements in reasoning performance do not imply increased transparency. Through mechanistic analysis, we attribute monitorability gains primarily to response distribution sharpening (entropy reduction) and increased attention to the prompt, rather than stronger causal reliance on reasoning traces. We also reveal how monitorability dynamics vary with controlled training and evaluation difficulty. Together, these findings provide a holistic view of how monitorability emerges under RLVR, clarifying when gains are likely to occur and when they are not.

State-space language models such as Mamba and gated linear attention (GLA) offer efficient alternatives to transformers due to their linear complexity and parallel training, but often lack the expressivity and robust state-tracking needed for complex reasoning. We address these limitations by reframing sequence modelling through a probabilistic lens, using Bayesian filters as a core primitive. While classical filters such as Kalman filters provide principled state estimation and uncertainty tracking, they are typically viewed as inherently sequential. We show that reparameterising the Kalman filter in information form enables its updates to be computed via an associative scan, allowing efficient parallel training. Building on this insight, we introduce the Kalman Linear Attention (KLA) layer, a neural sequence-modelling primitive that performs time-parallel probabilistic inference while maintaining explicit belief-state uncertainty. KLA offers strictly more expressive non-linear updates and gating than GLA variants while retaining their computational advantages. On language modelling tasks, KLA matches or outperforms modern SSMs and GLAs across representative benchmarks for discrete token manipulation and state tracking.

Foundation models have become a dominant paradigm in machine learning, achieving remarkable performance across diverse tasks through large-scale pretraining. However, they often yield overconfident, uncalibrated predictions. The standard approach to quantifying epistemic uncertainty are ensembles of multiple independently trained models. But their computational cost scales linearly with ensemble size, making them impractical for large foundation models. We propose Singular Value Ensemble (SVE), a parameter-efficient implicit ensembling method. SVE builds on a simple, but powerful core assumption: namely, that the singular vectors of the weight matrices correspond to meaningful directions in the representation space. If the singular vectors are indeed meaningful (orthogonal) "knowledge directions", then a model ensemble can be obtained by modulating only how strongly each direction contributes to the output. Rather than learning new parameters for each ensemble member, we freeze the singular vectors and only train per-member singular values that rescale the contribution of each direction in that shared knowledge basis. Ensemble diversity emerges naturally during joint training as stochastic initialization and random batch sampling cause different members to converge to different combinations of the same underlying knowledge. SVE performs comparable to an explicit ensemble, while increasing the parameter count of the base model by <1%, making principled uncertainty estimation accessible in resource-constrained settings. We validate SVE on NLP and vision tasks with various different backbones and show that it improves calibration while maintaining predictive accuracy.

Spatial understanding remains a key challenge in vision-language models. Yet it is still unclear whether such understanding is truly acquired, and if so, through what mechanisms. We present a controllable 1D image–text testbed to probe how left–right relational understanding emerges in Transformer-based vision and text encoders trained with a CLIP-style contrastive objective. We train lightweight Transformer-based vision and text encoders end-to-end on paired descriptions of one- and two-object scenes and evaluate generalization to unseen object pairs while systematically varying label and layout diversity. We find that contrastive training learns left–right relations and that label diversity, more than layout diversity, is the primary driver of generalization in this setting. To gain the mechanistic understanding, we perform an attention decomposition and show that interactions between positional and token embeddings induce a horizontal attention gradient that breaks left–right symmetry in the encoders; ablating this contribution substantially reduces left–right discrimination. Our results provide a mechanistic insight of when and how CLIP-style models acquire relational competence.

Fair graph clustering has emerged as a critical research area for addressing algorithmic bias in machine learning. The objective is to ensure that the proportion of each protected group within a cluster is consistent with its representation in the entire dataset. However, most existing spectral solutions rely on computationally expensive eigendecompositions of the graph Laplacian, limiting their scalability. In this paper, we propose Riemannian Fair Spectral Clustering (R-FairSC), a novel method that formulates fair spectral clustering as a constrained optimization problem on a Riemannian manifold. We develop a Riemannian alternating direction method of multipliers employing a variable-splitting strategy to efficiently solve the associated subproblems. Numerical experiments on large synthetic and real-world graphs demonstrate that R-FairSC significantly improves computational efficiency over state-of-the-art methods while maintaining high clustering quality and fairness.

Spectral clustering is hindered by its $O(n^{3})$ time complexity, and although anchor-based approximations reduce this to $O(nm^{2})$ ($m \ll n$), they often yield degenerate or unbalanced partitions in the absence of explicit cluster-size control. To address this limitation, we propose \textbf{SC-FAGC (Size-Constrained Fast Anchor Graph Clustering)}, a unified formulation that integrates entropy regularization and bilateral cardinality constraints within an anchor-based spectral clustering framework, enforcing user-specified lower and upper bounds on anchor-level cluster sizes to avoid trivial or highly unbalanced solutions. To solve the resulting non-convex problem efficiently, we develop an \textbf{Iteratively Re-weighted (IRW)} optimization scheme that linearizes the objective and solves each subproblem via a \textbf{Double-Bounded Optimal Transport (DB-OT)} solver based on the \textbf{Sinkhorn--Knopp} algorithm, guaranteeing convergence while maintaining scalability. Extensive experiments on benchmark datasets demonstrate that SC-FAGC consistently achieves state-of-the-art performance across six standard clustering metrics while strictly enforcing the prescribed anchor-level constraints. Moreover, it induces effective data-level size control after label transmission, offering a principled and scalable solution for large-scale graph clustering with controllable partition structure.

The success of Large Language Models (LLMs) hinges on the stable training of deep Transformer architectures. A critical design choice is the placement of normalization layers, leading to a fundamental trade-off: the ''PreNorm'' architecture ensures training stability at the cost of potential performance degradation in deep models, while the ''PostNorm'' architecture offers strong performance but suffers from severe training instability. In this work, we propose SpanNorm, a novel technique designed to resolve this dilemma by integrating the strengths of both paradigms. SpanNorm adopts the clean residual path of PreNorm to stabilize signal propagation while employing a PostNorm-style computation that normalizes the output of the residual connection, thereby enhancing model performance. We provide a theoretical analysis demonstrating that SpanNorm, combined with a principled scaling strategy, maintains bounded signal variance throughout the network, preventing the gradient issues that plague PostNorm models, and alleviating the representation collapse of PreNorm. Empirically, SpanNorm consistently outperforms standard normalization schemes in both dense and Mixture-of-Experts (MoE) scenarios, paving the way for more powerful and stable Transformer architectures.

In recent years, decentralized optimization has gained significant attention for solving machine learning problems where data are distributed across multiple devices. However, existing decentralized optimization algorithms are primarily designed for single-level and two-level optimization tasks, limiting their application to more complex problems such as decentralized stochastic $K$-level optimization, where $K>2$. In this work, we propose a novel decentralized stochastic $K$-level variance-reduced gradient descent algorithm to address the significant computation and communication overhead caused by the multi-level structure in decentralized stochastic $K$-level optimization problems. Moreover, we propose a novel theoretical analysis to tackle the recursive dependence issue caused by the multi-level structure when establishing the convergence rate of our algorithm. Finally, the experimental results confirm the effectiveness of our proposed algorithm.

Fine-tuning Large Language Models (LLMs) as autonomous agents on domain-specific data has emerged as a promising paradigm for tackling interactive, real-world tasks. However, existing studies have overlooked the critical coordination between long-term planning and multi-step execution in optimizing agent capabilities. This oversight leads to the propagation of impractical plans and plan-deviated trajectories within the optimization process, resulting in suboptimal task performance and hindering the further development of LLM-based agents in long-horizon tasks. To bridge this gap, we propose $\textbf{CoPE}$, a novel framework that explicitly integrates planning–execution coordination into LLM-based agent optimization. CoPE employs Self-Refining MCTS to generate task plans and multiple execution trajectories through environment interactions. By quantifying the coordination between planning and execution, CoPE assigns higher optimization weights to well-coordinated samples, enabling LLM-based agents to learn better planning and execution policies. Extensive experiments demonstrate that CoPE substantially improves agent coordination, outperforming state-of-the-art baselines on benchmarks comprising two long-horizon multi-step tasks. Codes and data are available at https://github.com/Octobrist/CoPE.

Machine learning interatomic potentials (MLIPs) have brought substantial gains in the extrapolation capability in computational chemistry. However, most equivariant models are typically built with spherical tensors (STs), while Cartesian tensor formulations remain less developed despite their natural alignment with atomic coordinates and tensorial targets. In this work, we develop a Cartesian framework for irreducible Cartesian tensors (ICTs) by introduce the Cartesian-3j symbol and Cartesian Generalized Clebsch-Gordan Coefficients, which serve as direct analogues of the Wigner-3j symbol and Generalized Clebsch-Gordan coefficients defined for ST coupling. We extend the e3nn library to support ICT product, and use this framework to build Cartesian counterparts of MACE, NequIP, and Allegro, allowing the first controlled comparison where architectures are held fixed and only the tensor basis is changed. Our experiments show that irreducible Cartesian models can achieve accuracy comparable to spherical counterparts, but direct Cartesianization incurs unfavorable compute and memory scaling, motivating dedicated Cartesian architectural choices. Leveraging ICTs and our framework, we introduce TACE-v1-OAM-M and demonstrate that it achieves competitive performance on Matbench Discovery compared to state-of-the-art ST models.

A central insight in lossless data compression is the close connection between probabilistic next- symbol prediction and efficient sequence compression, whereby predictive models can be combined with classical coding techniques to achieve strong compression performance. Applying this approach with powerful modern learned models, such as LLMs, has been shown to achieve markedly better compression than traditional techniques across a wide range of domains. However, significant practical challenges remain, including model non-determinism, in which a model produces different predictions on different machines despite identical parameters and inputs; such mismatches between the encoder and decoder can lead to complete decoding failure. Probability Matching Interval Coding (PMATIC) was recently introduced as a drop-in framework for mismatch-robust coding and shown to enable reliable compression and decompression in the presence of bounded prediction mismatch (Adler & Tang, 2026). In this work, we present a generalization of PMATIC that allows the incorpo- ration of tight theoretical results into the design and more flexible parameter optimization, resulting in substantial improvements in compression efficiency and robustness.

Clustering in a big data setting is an intensively studied problem, with coresets emerging as one of the important paradigms in this line of work. Given a cost function $\text{cost}(P,S)$ mapping input points $P$ and a solution $S$ to an objective value, a coreset is a typically weighted sketch $\Omega\subseteq P$ such that $\text{cost}(\Omega,S)\approx \text{cost}(P,S)$. In practice, coreset sizes much smaller than those suggested by theoretical guarantees are often found to be sufficient. In this paper, we offer an explanation for this phenomenon. Smaller coreset sizes suffice if we only wish to preserve the costs of \emph{good} solutions, i.e., solutions with low cost. We define and devise \emph{approximation-preserving coresets}, which provide a weaker guarantee than strong coresets, which apply to all solutions, while providing stronger guarantees than weak coresets, which apply only to the optimum solution. We complement this result by showing that even a very small distortion in the approximation factor cannot admit coresets of this size.