Riemannian Lyapunov Optimizer: A Unified Framework for Optimization

We introduce Riemannian Lyapunov Optimizers (RLOs), a family of optimization algorithms that unifies classic optimizers within one geometric framework. Unlike heuristic improvements to existing optimizers, RLOs are systematically derived from a novel control-theoretic framework that reinterprets optimization as an extended state discrete-time controlled dynamical system on a Riemannian parameter manifold. Central to this framework is the identification of a Normally Attracting Invariant Manifold (NAIM), which organizes training dynamics into two distinct stages: rapid alignment of the speed state to a target graph, followed by controlled evolution within it. We formalize this by constructing a strict Lyapunov function that certifies convergence to a target manifold. This perspective yields a constructive ``optimizer generator" that not only recovers classic algorithms but enables the principled design of RLOs. We validate our theory via geometric diagnostics and demonstrate that grounding optimizer design in control theory yields state-of-the-art performance in large-scale benchmarks. Overall, RLOs bridge control theory and modern machine learning optimization, providing a unified language and a systematic toolkit for designing stable, effective optimizers.

Latent-Space Contrastive Reinforcement Learning for Stable and Efficient LLM Reasoning

While Large Language Models (LLMs) demonstrate exceptional performance in surface-level text generation, their nature in handling complex multi-step reasoning tasks often remains one of ``statistical fitting'' rather than systematic logical deduction. Traditional Reinforcement Learning (RL) attempts to mitigate this by introducing a ``think-before-speak'' paradigm. However, applying RL directly in high-dimensional, discrete token spaces faces three inherent challenges: sample-inefficient rollouts, high gradient estimation variance, and the risk of catastrophic forgetting. To fundamentally address these structural bottlenecks, we propose \textbf{DeepLatent Reasoning (DLR)}, a latent-space bidirectional contrastive reinforcement learning framework. This framework shifts the trial-and-error cost from expensive token-level full sequence generation to the continuous latent manifold. Specifically, we introduce a lightweight assistant model to efficiently sample $K$ reasoning chain encodings within the latent space. These encodings are filtered via a dual reward mechanism based on correctness and formatting; only high-value latent trajectories are fed into a \textbf{frozen main model} for single-pass decoding. To maximize reasoning diversity while maintaining coherence, we design a contrastive learning objective to enable directed exploration within the latent space. Since the main model parameters remain frozen during optimization, this method mathematically eliminates catastrophic forgetting. Experiments demonstrate that under comparable GPU computational budgets, DLR achieves more stable training convergence, supports longer-horizon reasoning chains, and facilitates the sustainable accumulation of reasoning capabilities, providing a viable path toward reliable and scalable reinforcement learning for LLMs.

When Helpers Become Hazards: A Benchmark for Analyzing Multimodal LLM-Powered Safety in Daily Life

As Multimodal Large Language Models (MLLMs) become an indispensable assistant in human life, the unsafe content generated by MLLMs poses a danger to human behavior, perpetually overhanging human society like a sword of Damocles. To investigate and evaluate the safety impact of MLLMs responses on human behavior in daily life, we introduce SaLAD, a multimodal safety benchmark which contains 2,013 real-world image-text samples across 10 common categories, with a balanced design covering both unsafe scenarios and cases of oversensitivity. It emphasizes realistic risk exposure, authentic visual inputs, and fine-grained cross-modal reasoning, ensuring that safety risks cannot be inferred from text alone. We further propose a safety-warning-based evaluation framework that encourages models to provide clear and informative safety warnings, rather than generic refusals. Results on 18 MLLMs demonstrate that the top-performing models achieve a safe response rate of only 57.2% on unsafe queries. Moreover, even popular safety alignment methods limit effectiveness of the models in our scenario, revealing the vulnerabilities of current MLLMs in identifying dangerous behaviors in daily life. Our dataset is available at https://github.com/xinyuelou/SaLAD.

Goal inference with Rao-Blackwellized Particle Filters

Inferring the eventual goal of a mobile agent from noisy observations of its trajectory is a fundamental estimation problem. We initiate the study of such intent inference using a variant of a Rao-Blackwellized Particle Filter (RBPF), subject to the assumption that the agent's intent manifests through closed-loop behavior with a state-of-the-art provable practical stability property. Leveraging the assumed closed-form agent dynamics, the RBPF analytically marginalizes the linear-Gaussian substructure and updates particle weights only, improving sample efficiency over a standard particle filter. Two difference estimators are introduced: a Gaussian mixture model using the RBPF weights and a reduced version confining the mixture to the effective sample. We quantify how well the adversary can recover the agent's intent using information-theoretic leakage metrics and provide computable lower bounds on the Kullback-Leibler (KL) divergence between the true intent distribution and RBPF estimates via Gaussian-mixture KL bounds. We also provide a bound on the difference in performance between the two estimators, highlighting the fact that the reduced estimator performs almost as well as the complete one. Experiments illustrate fast and accurate intent recovery for compliant agents, motivating future work on designing intent-obfuscating controllers.

Seer Self-Consistency: Advance Budget Estimation for Adaptive Test-Time Scaling

Test-time scaling improves the inference performance of Large Language Models (LLMs) but also incurs substantial computational costs. Although recent studies have reduced token consumption through dynamic self-consistency, they remain constrained by the high latency of sequential requests. In this paper, we propose SeerSC, a dynamic self-consistency framework that simultaneously improves token efficiency and latency by integrating System 1 and System 2 reasoning. Specifically, we utilize the rapid System 1 to compute the answer entropy for given queries. This score is then used to evaluate the potential of samples for scaling, enabling dynamic self-consistency under System 2. Benefiting from the advance and accurate estimation provided by System 1, the proposed method can reduce token usage while simultaneously achieving a significant decrease in latency through parallel generation. It outperforms existing methods, achieving up to a 47% reduction in token consumption and a 43% reduction in inference latency without significant performance loss.

Initialization Schemes for Kolmogorov-Arnold Networks: An Empirical Study

Kolmogorov-Arnold Networks (KANs) are a recently introduced neural architecture that replace fixed nonlinearities with trainable activation functions, offering enhanced flexibility and interpretability. While KANs have been applied successfully across scientific and machine learning tasks, their initialization strategies remain largely unexplored. In this work, we study initialization schemes for spline-based KANs, proposing two theory-driven approaches inspired by LeCun and Glorot, as well as an empirical power-law family with tunable exponents. Our evaluation combines large-scale grid searches on function fitting and forward PDE benchmarks, an analysis of training dynamics through the lens of the Neural Tangent Kernel, and evaluations on a subset of the Feynman dataset. Our findings indicate that the Glorot-inspired initialization significantly outperforms the baseline in parameter-rich models, while power-law initialization achieves the strongest performance overall, both across tasks and for architectures of varying size. All code and data accompanying this manuscript are publicly available at https://github.com/srigas/KAN_Initialization_Schemes.

High precision PINNs in unbounded domains: application to singularity formulation in PDEs

We investigate the high-precision training of Physics-Informed Neural Networks (PINNs) in unbounded domains, with a special focus on applications to singularity formulation in PDEs. We propose a modularized approach and study the choices of neural network ansatz, sampling strategy, and optimization algorithm. When combined with rigorous computer-assisted proofs and PDE analysis, the numerical solutions identified by PINNs, provided they are of high precision, can serve as a powerful tool for studying singularities in PDEs. For 1D Burgers equation, our framework can lead to a solution with very high precision, and for the 2D Boussinesq equation, which is directly related to the singularity formulation in 3D Euler and Navier-Stokes equations, we obtain a solution whose loss is $4$ digits smaller than that obtained in \cite{wang2023asymptotic} with fewer training steps. We also discuss potential directions for pushing towards machine precision for higher-dimensional problems.

Tag-Evol: Achieving Efficient Instruction Evolving via Tag Injection

Evol-Instruct has made significant improvements as a data synthesis method in several areas. Existing methods typically rely on a fixed set of strategies to evolve, which require manual design and are monolithic in form. In addition, iterative evolution also makes the acquisition of hard samples expensive. In view of this, we propose the Tag-Evol framework, a more diverse and efficient instruction evolving method. Specifically, Tag-Evol uses diverse and specific knowledge tags as strategies to achieve controlled evolution by injecting different combinations of tags into the original instructions. Experiments with multiple backbones in diverse domain benchmarks show that the proposed method generates significantly better evolved data than other methods. Furthermore, we conduct a thorough analysis of the evolved data, demonstrating that Tag-Evol is not only efficient but also generates more diverse and challenging data.

Pretraining Language Models to Ponder in Continuous Space

Humans ponder before articulating complex sentence elements, enabling deeper cognitive processing through focused effort. In this work, we introduce this pondering process into language models by repeatedly invoking the forward process within a single token generation step. During pondering, instead of generating an actual token sampled from the prediction distribution, the model ponders by yielding a weighted sum of all token embeddings according to the predicted token distribution. The generated embedding is then fed back as input for another forward pass. We show that the model can learn to ponder in this way through self-supervised learning, without any human annotations. Our method is straightforward and can be seamlessly integrated with various existing language models. Experiments across three widely used open-source architectures-GPT-2, Pythia, and LLaMA-and extensive downstream task evaluations demonstrate the effectiveness and generality of our method. For language modeling tasks, pondering language models achieve performance comparable to vanilla models with twice the number of parameters. On 9 downstream benchmarks, our pondering-enhanced Pythia models significantly outperform the official Pythia models. Notably, pondering-enhanced Pythia-1B is comparable to TinyLlama-1.1B, which is trained on 10 times more data. The code is available at https://github.com/LUMIA-Group/PonderingLM.

Think Before You Accept: Semantic Reflective Verification for Faster Speculative Decoding

Large language models (LLMs) suffer from high inference latency due to the auto-regressive decoding process. Speculative decoding accelerates inference by generating multiple draft tokens using a lightweight model and verifying them in parallel. However, existing verification methods rely heavily on distributional consistency while overlooking semantic correctness, thereby limiting the potential speedup of speculative decoding. While some methods employ additional models for relaxed verification of draft tokens, they often fail to generalize effectively to more diverse or open-domain settings. In this work, we propose Reflective Verification, a training-free and semantics-aware approach that achieves a better trade-off between correctness and efficiency. Specifically, we leverage the inherent reflective capacity of LLMs to semantically assess the correctness of draft tokens in parallel during verification. Using prompt-based probing, we obtain both the original and reflective distributions of draft tokens in a single forward pass. The fusion of these distributions enables semantic-level verification of draft tokens that incorporates both consistency and correctness. Experiments across multiple domain benchmarks and model scales demonstrate that our method significantly increases the acceptance length of draft tokens without compromising model performance. Furthermore, we find that the proposed Reflective Verification is orthogonal to existing statistical verification methods, and their combination yields additional 5$\sim$15\% improvements in decoding speed.