Large language models usually face continuous domain shifts during testing, which degrade performance on unseen shifting domains. So, researchers propose continual test-time adaptation (CTTA) to adapt to evolving testing domains while preserving knowledge of previous domains, making adaptability-stability (A-S) balance. Existing CTTA methods are constrained by dense base models that encode knowledge from all domains into a global model, hardly achieving the A-S balance. We observe that the model sparsity of mixture-of-experts (MoE) models is better for achieving A–S balance than dense models. In CTTA, however, MoE faces difficulty in (1) correctly routing samples from unseen shifting domains and (2) capturing domain-level shifts. In this paper, we propose test-time mixture-of-experts routing (TiME) via asymmetric co-optimal transport (As-COOT): we model MoE routing in CTTA as a test-time allocation problem via COOT. To ensure reliable routing, we propose a semantic space alignment to align sample-expert distributions via bidirectional contrastive learning. To address COOT’s limitations in CTTA, we propose As-COOT, relaxing sample-side constraints while enforcing expert-side constraints to ensure noise robustness and balance expert load. Experiments show TiME outperforms baselines. Code is: anonymous.4open.science/r/As-COOT-78FF
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Social Aspects · Alignment
Reward Models (RMs) are crucial for online alignment of language models (LMs) with human preferences. However, RM-based preference-tuning is vulnerable to reward hacking, whereby LM policies learn undesirable behaviors from flawed RMs. By systematically measuring biases in five high-quality RMs, including the state-of-the-art, we find that issues persist despite prior work with respect to length, sycophancy, and overconfidence. We also discover new issues related to bias toward model-specific “styles” and answer-order. We categorize RM failures by complexity and propose a simple post-hoc intervention to mitigate low-complexity biases that arise from spurious correlations. Our proposed $\textbf{mechanistic reward shaping}$ reduces targeted biases without degrading reward quality and while using minimal labeled data. The method is extensible to new biases, model-internal, and generalizes out-of-distribution.
Deep Learning · Large Language Models
Large language models (LLMs) are increasingly applied to symbolic mathematics, yet existing evaluations often conflate pattern memorization with genuine reasoning. To address this gap, we present **ASyMOB**, a high-resolution dataset of **35,368** validated symbolic math problems spanning integration, limits, differential equations, series, and hypergeometrics. Unlike prior benchmarks, **ASyMOB** systematically perturbs each seed problem using symbolic, numeric, and equivalence-preserving transformations, enabling a fine-grained assessment of generalization and robustness. Our evaluation reveals three key findings: (1) most models’ performance collapses under minor perturbations, while top systems exhibit an apparent \textit{regime shift} in robustness; (2) integrated code tools stabilize performance, particularly for weaker models; and (3) we identify examples where Computer Algebra Systems (CAS) fail while LLMs succeed, as well as problems solved only via a hybrid LLM-CAS approach, highlighting a promising integration frontier. **ASyMOB** serves as a principled diagnostic tool for measuring and accelerating progress toward building verifiable, trustworthy AI for scientific discovery.
Deep Learning · Large Language Models
Step-by-step verifiers—also known as process reward models (PRMs)—are a key ingredient for test-time scaling, but training them requires expensive step-level supervision. This work aims to build data-efficient PRMs as verbalized step-wise reward models that verify every step in the solution by generating a verification chain-of-thought (CoT). We propose ThinkPRM, a long CoT verifier fine-tuned on orders of magnitude fewer process labels than those required by discriminative PRMs. Our approach capitalizes on the inherent reasoning abilities of long CoT models, and outperforms LLM-as-a-Judge and discriminative verifiers—using only 1% of the process labels in PRM800K—across several challenging benchmarks. Specifically, ThinkPRM beats the baselines on ProcessBench, MATH-500, and AIME ’24 under best-of-N selection and reward-guided search. In an out-of-domain evaluation over subsets of GPQA-Diamond and LiveCodeBench, our PRM surpasses discriminative verifiers trained with the full PRM800K by 8% and 4.5%, respectively. Lastly, under the same token budget, ThinkPRM scales up verification compute more effectively compared to LLM-as-a-Judge, outperforming it by 7.2% on a subset of ProcessBench. This work highlights the value of generative, long CoT PRMs that can scale test-time compute for verification while requiring minimal supervision for training.
Learning mesh-based operators from data can match training objectives while implicitly violating algebraic consistency constraints that classical discretizations satisfy by construction. Such violations can introduce near-kernel directions, degrade conditioning as resolution increases, and distort the low-frequency spectral structure on which downstream solvers and diagnostics rely. This work introduces a lightweight compatibility certificate for learned operator pairs, together with a closed-form projection that maps a learned pair to its Frobenius-nearest chain-compatible operator. The certificate provides an explicit distance-to-compatibility and yields perturbation bounds for the discrete operator. These bounds imply stability guarantees for elliptic solves and for low-frequency spectral counts, provided a spectral gap separates the kernel from the rest of the spectrum and boundary treatments are well posed. Experiments on standard elliptic problems show that defect-aware training prevents condition-number blow-up at higher resolutions, improves robustness under mesh and topological distribution shifts, and maintains predictive accuracy relative to unconstrained learning. Overall, these results support the use of non-invasive, computable algebraic consistency checks to detect and control failure modes that are not revealed by loss values alone.
General Machine Learning · Online Learning, Active Learning and Bandits
We address the problem of multi-group mean estimation, which seeks to allocate a finite sampling budget across multiple groups to obtain uniformly accurate estimates of their means. Unlike classical multi-armed bandits, whose objective is to minimize regret by identifying and exploiting the best arm, the optimal allocation in this setting requires sampling every group on the order of $\Theta(T)$ times. This fundamental distinction makes exploration-free algorithms both natural and effective. Our work makes three contributions. First, we strengthen the existing results on subgaussian variance concentration using the Hanson-Wright inequality and identify a class of strictly subgaussian distributions that yield sharper guarantees. Second, we design exploration-free non-adaptive and adaptive algorithms, and we establish tighter regret bounds than the existing results. Third, we extend the framework to contextual bandit settings, an underexplored direction, and propose algorithms that leverage side information with provable guarantees. Overall, these results position exploration-free allocation as a principled and efficient approach to multi-group mean estimation, with potential applications in experimental design, personalization, and other domains requiring accurate multi-group inference.
Theory · Deep Learning
This paper investigates the grokking phenomenon, which refers to the sudden transition from a long memorization to generalization observed during neural networks training, in the context of learning multiplication in finite-dimensional algebras (FDA). While prior work on grokking has focused mainly on group operations, we extend the analysis to more general algebraic structures, including non-associative, non-commutative, and non-unital algebras. We show that learning group operations is a special case of learning FDA, and that learning multiplication in FDA amounts to learning a bilinear product specified by the algebra’s structure tensor. For algebras over the reals, we connect the learning problem to matrix factorization with an implicit low-rank bias, and for algebras over finite fields, we show that grokking emerges naturally as models must learn discrete representations of algebraic elements. This leads us to experimentally investigate the following core questions: (i) how do algebraic properties such as commutativity, associativity, and unitality influence both the emergence and timing of grokking, (ii) how structural properties of the structure tensor of the FDA, such as sparsity and rank, influence generalization, and (iii) to what extent generalization correlates with the model learning latent embeddings aligned with the algebra’s representation. Our work provides a unified framework for grokking across algebraic structures and new insights into how mathematical structure governs neural network generalization dynamics.
Applications · Computer Vision
Open-Vocabulary Segmentation(OVS) aims to achieve pixel-level semantic recognition from arbitrary text queries. Existing large-scale visual-linguistic models, such as CLIP, perform well in zero-shot generalization, but their image-level training objectives and real-valued cross-modal alignment mix amplitude and phase information, limiting fine-grained segmentation and often causing blurred boundaries and fragmented structures. Inspired by the ability of electromagnetic wave phase to control interference independently of amplitude, we propose PhaseAlign, an OVS framework based on Complex Phase Alignment (CPA). CPA explicitly decouples the magnitude and phase of visual and textual embeddings in the complex domain, refining effective features for stable cross-modal alignment. To further enhance structural awareness, we introduce spatial-aware cross-modal projection, which models local neighborhood relations via multi-scale spatial contrast normalization, and attention-guided affinity modeling, which leverages pre-trained ViT self-attention to propagate category activations, improving boundary clarity and region integrity. Experiments show that PhaseAlign achieves state-of-the-art performance on multiple OVS benchmarks.
General Machine Learning · Representation Learning
This paper studies the minimal dimension required to embed subset memberships ($m$ elements and ${m\choose k}$ subsets of at most $k$ elements) into vector spaces, denoted as Minimal Embeddable Dimension (MED). The tight bounds of MED are derived theoretically and supported empirically for various notions of "distances" or "similarities", including $\ell_2$ metric, inner product, and cosine similarity. In addition, we conduct numerical simulation in a more achievable setting, where the ${m\choose k}$ subset embeddings are chosen as the centroid of the embeddings of the contained elements. Our simulation easily realizes a logarithmic dependency between the MED and the number of elements to embed. These findings imply that embedding-based retrieval limitations stem primarily from learnability challenges, not geometric constraints, guiding future algorithm design.
Mode collapse is a persistent challenge in generative modeling and appears in autoregressive text generation as behaviors ranging from explicit looping to gradual loss of diversity and premature trajectory convergence. We take a dynamical-systems view and reinterpret mode collapse as reduced state-space accessibility caused by *geometric collapse*: during generation, the model's internal trajectory becomes confined to a low-dimensional region of its representation space. This implies mode collapse is not purely a token-level phenomenon and cannot be reliably solved by symbolic constraints or probability-only decoding heuristics. Guided by this perspective, we propose *Reinforced Mode Regulation* (RMR), a lightweight, online state-space intervention that regulates dominant self-reinforcing directions in the Transformer value cache (implemented as low-rank damping). Across multiple large language models, RMR substantially reduces mode collapse and enables stable generation at extremely low entropy rates (down to 0.8 nats/step), whereas standard decoding typically collapses near 2.0 nats/step.
Applications · Chemistry, Physics, and Earth Sciences
Data-driven dynamical system models often fail to recover the long-term structure of the underlying system, as their behavior is weakly constrained off the data manifold. Conjugacy-based approaches address this limitation by learning a diffeomorphism that pushes forward a source vector field to match observed dynamics, inheriting qualitative topology from the source. However, such methods typically presuppose that the chosen source system is topologically compatible with the target data. When this assumption is violated, the conjugacy problem becomes ill-posed, and arbitrary corrections can be traded off against diffeomorphic variation, leading to non-identifiability. We propose a framework that relaxes this assumed prior by jointly learning the diffeomorphic conjugacy together with controlled adjustments to the source dynamics via low-dimensional context modulation. Inspired by versal unfolding theory, we enforce the modulation space to be orthogonal to the worst-case orbit-tangent directions, obtained by adversarially searching over a class of parameterized diffeomorphisms. This promotes an identifiable decomposition of dynamical variation into diffeomorphic and intrinsic, topology-changing components, enabling interpretable corrections that recover the canonical structure such as normal forms and symmetries.
Deep Learning · Graph Neural Networks
We show that contrary to conventional wisdom in the community, graph neural networks (GNNs) are not continuous with respect to all natural modes of graph convergence. As a result, GNNs may generate substantially different latent representations for graphs that are very similar. In particular they assign vastly different latent embeddings to graphs that represent the same underlying object at different resolution scales. We trace this failure of continuity back to a structural obstruction arising from commonly used information-propagation schemes. Building on this insight we then derive a principled modification to standard GNN architectures which equips models with continuity across scales. The proposed modification enables consistent integration of distinct resolutions and reliable generalization between them. We systematically validate our theoretical findings in a wide range of numerical experiments.
Deep Learning · Algorithms
We introduce Conditional Clifford-Steerable CNNs (C-CSCNNs), a unified framework that incorporates equivariance to arbitrary pseudo-Euclidean groups and significantly improves the expressivity of standard CSCNNs. We show that the kernel basis of the standard formulation is incomplete, limiting model capacity. To address this, we augment the kernels with equivariant representations of the input feature field. We derive the equivariance constraint for these input-dependent kernels and show how it can be solved efficiently via implicit parameterization. We empirically validate on multiple PDE forecasting tasks, including fluid dynamics and relativistic electrodynamics, where our method consistently outperforms standard CSCNNs and performs on par with state-of-the-art baselines.
Deep Learning · Foundation Models
Vision-Language Models (VLMs) perform well on commonsense reasoning tasks but struggle with visual spatial reasoning. Most existing solutions introduce extra 3D priors or external spatial encoders, which increase complexity and degrade the underlying VLMs' general-purpose capabilities after spatial fine-tuning. To this end, we propose a parameter-efficient \textit{\textbf{Spatio}-vision \textbf{L}anguage \textbf{M}odels (SpatioLM)}, that enhances spatial intelligence without extra 3D priors or third-party spatial encoders. Concretely, we design a plug-and-play and non-invasive spatio-vision module that elicits the spatial knowledge inherent in VLMs. Furthermore, we innovatively leverage pseudo depth and camera information as supervision to guide the model in learning physically coherent representations. Extensive experiments show that SpatioLM achieves significant improvements in diverse tasks, including spatial perception and understanding while maintains the general-purpose capabilities. Notably, the model achieves an impressive score of 71.6 on the VSI-Bench (the first model to surpass 70). In addition, it attains competitive performance when transferred to embodied manipulation tasks.
Deep Learning · Large Language Models
Diffusion-based large language models (dLLMs) have emerged as a promising alternative to autoregressive models, leveraging simultaneous denoising to enable global planning and iterative refinement. These properties make dLLMs particularly attractive for long-context generation. However, deploying dLLMs faces a prohibitive memory capacity barrier, as existing inference systems are inefficient for the diffusion paradigm. We observe that current inference systems are misaligned with dLLMs. Unlike autoregressive models, whose memory footprint is dominated by the cumulative KV-Cache, dLLMs are bottlenecked by transient activations rematerialized per step. Moreover, generic memory reuse mechanisms lack the global visibility to handle dynamic memory peaks of dLLMs, which alternate between logits and feed-forward networks. To address these challenges, we present Mosaic, a memory-efficient inference system that shifts dLLM execution from local, static memory management to a global, dynamic paradigm. Mosaic integrates (i) a mask-only logits kernel eliminating redundant activation materialization, (ii) a lazy chunking optimizer using online heuristics to adaptively tame dynamic memory peaks, and (iii) a global memory manager leveraging virtual addressing to mitigate memory fragmentation. Extensive evaluations show that Mosaic reduces the memory peak-to-average ratio by 2.71$\times$ on average and increases the maximum supportable inference sequence length on identical hardware by 15.30--32.34$\times$. Crucially, Mosaic is training-free and preserves exact model outputs, while simultaneously reducing end-to-end latency by 2.5\%--55.4\%.
Online advertising platforms rely on machine learning models to predict click-through rates (pCTR) and conversion rates (pCVR) for auction mechanisms. We introduce a novel framework to study the interaction between recommender system model quality, auction format, and au- tobidder behavior. We formalize when model improvements—defined via a refinement relation inspired by filtrations in probability theory—lead to improvements in platform-level Evaluation Cri- teria Metrics (ECM) such as revenue, welfare, or liquid welfare. Our main contributions are: (1) a formal definition of model improvement based on cluster refinement, and (2) a complete charac- terization of ECM monotonicity across different combinations of bidder types (tCPA, max-CPA), auction formats (first-price, second-price, VCG), and budget constraints. We show that first-price auctions with uniform bidding guarantee revenue monotonicity for tCPA bidders without budgets (via Jensen’s inequality), while second-price auc- tions and budget constraints can break this prop- erty. We provide full numerical counterexamples for all negative results. Our findings have practi- cal implications for advertising platforms seeking to align model improvements with business out- comes.
Deep Learning · Large Language Models
Visual agents employ tools such as zoom-in cropping within visual chains of thought to access fine-grained details. Prior work has primarily demonstrated the effectiveness of these tools on visual search tasks, leaving their applicability to more diverse and complex visual problems underexplored. In this paper, we move beyond visual search and study challenging visual tasks that require advanced spatial understanding and reasoning, such as 3D spatial reasoning, where agents must not only crop or zoom in on relevant regions but also understand how these local details relate to the global context. We identify a tool-use collapse phenomenon: models progressively stop using tools while still achieving higher task accuracy. Moreover, we observe a clear asymmetry: (i) completely eliminating tool use degrades performance, whereas (ii) incentivizing tool use yields only marginal gains despite substantially increasing usage. We find that vanilla training and rewarding tool-use encouragement reduce rollout diversity during training, explaining why higher tool-use does not yield stronger reasoning performance. Motivated by these findings, we encourage diverse rollout exploration by adding an entropy-regularization term to the reinforcement learning objective, which results in the best performance despite tool usage gradually declining during training. Overall, our findings suggest a training-time view of tools as scaffolding, where broader exploration in both text and vision shapes representations that improve despite tool-use collapse.
Deep Learning · Graph Neural Networks
Semidefinite programs (SDPs) are a powerful framework for convex optimization and for constructing strong relaxations of hard combinatorial problems. However, solving large SDPs can be computationally expensive, motivating the use of machine learning models as fast computational surrogates. Graph neural networks (GNNs) are a natural candidate in this setting due to their sparsity-awareness and ability to model variable-constraint interactions. In this work, we study what expressive power is sufficient to recover optimal SDP solutions. We first prove negative results showing that standard GNN architectures fail on recovering linear SDP solutions. We then identify a more expressive architecture that captures the key structure of SDPs and can, in particular, emulate the updates of a standard first-order solver. Empirically, on both synthetic and SDPLIB benchmarks of various classes of SDPs, this more expressive architecture achieves consistently lower prediction error and objective gap than theoretically weaker baselines. Finally, using the learned high-quality predictions to warm-start the first-order solver yields practical speedups of up to 80%.
Deep Learning · Large Language Models
Large Language Models (LLMs) have shown significant potential in scientific discovery but struggle to bridge the gap between theoretical reasoning and verifiable physical simulation. Existing solutions operate in a passive "execute-then-response" loop and thus lack runtime perception, obscuring agents to transient anomalies (e.g., numerical instability or diverging oscillations). To address this limitation, we propose EmbodiedAct, a framework that transforms established scientific software into active embodied agents by grounding LLMs in embodied actions with a tight perception-execution loop. We instantiate EmbodiedAct within MATLAB and evaluate it on complex engineering design and scientific modeling tasks. Extensive experiments show that EmbodiedAct significantly outperforms existing baselines, achieving SOTA performance by ensuring satisfactory reliability and stability in long-horizon simulations and enhanced accuracy in scientific modeling.
Deep Learning · Graph Neural Networks
Graph neural networks (GNNs) have been demonstrated to be powerful tools for analyzing structural graph data. However, most existing methods usually rely on fixed adjacency structures for information propagation, lacking strong adaptability to the latent semantic relationships that exist but are not explicitly connected in graph, especially in complementary high-pass and low-pass filtering views. To this end, this paper proposes a novel Dual-channel Dynamic Graph Neural Network (DCD-GNN), mainly consisting of parallel representation learning channels: a static structure-preserving channel and a dynamic adjacency-enhancing channel. The dynamic channel exploits both low-pass structural filtering and high-pass personalized detail via self-attention adjacency learning and then integrates them for comprehensive semantic modeling, while the static channel maintains structural stability. Both channels employ a multi-scale representation fusion mechanism and are finally fused into a unified and discriminative node embedding representation. Extensive experiments on various graph benchmark datasets verify the superiority of DCD-GNN in discriminative graph representation learning.