arXiv subjects

Zeyu Gan

Publications and source records attributed to Zeyu Gan.

8 recordsLinked to original sources

arXiv2026

Normalize-Then-Precondition: A Hierarchical Approach to Marginal Scale and Interaction Geometry for LLM Training

Matrix optimizers have emerged as a promising direction, with Muon standing out as a prominent design. Revisiting Muon through its full-Gram representation, we observe that it jointly processes marginal-scale and interaction information. This opens an alternative way to organize geometric information hierarchically, motivating the Normalize-Then-Precondition framework. Specifically, it first uses diagonal-Gram information to construct a marginally normalized update, then applies spectral preconditioning to its directional interaction geometry. Building on this framework, we develop NormPre with NormPre-G and NormPre-L adopting global and localized spectral preconditioning, grounded in spectral-norm steepest descent and a regularized formulation followed by leading mode selection, respectively. To enable large-scale training, NormPre-G uses Newton-Schulz iterations and NormPre-L employs randomized sketching to approximate the leading interaction eigenspace. Theoretically, we establish $\mathcal{O}(T^{-1/2})$ convergence guarantees for simplified versions of NormPre. Across extensive pretraining experiments on GPT-2 Small, LLaMA and Qwen3, both variants consistently outperform AdamW, Muon and MANO under matched training budgets. Further efficiency and spectral analyses reveal the complementary strengths of two variants and characterize their performance-efficiency trade-off. We open-source our code through a GitHub repository at https://github.com/zx-gong/NormPre.

cs.LG↗

arXiv2026

Harness Evolution as Learning: Approximation, Generalization, and Optimization Limits of Self-Improving Personal Agents

As the capabilities of large language models (LLMs) continue to advance, increasing attention is turning to how to translate their abilities into useful behavior. Personal agents bring this question into everyday settings, where models are expected to serve individual users and continually adapt to their preferences. With the underlying model held fixed, such adaptation relies on harness engineering: designing and evolving the surrounding layer that manages context, memory, tools, and execution. Despite rapid progress, the factors governing effective harness evolution remain insufficiently understood. To narrow this gap, we investigate three central questions concerning harness architecture, harness scale, and self-evolution algorithms through complementary empirical and theoretical analyses. Empirically, we introduce a preference-oriented benchmark and systematically characterize the capabilities and limitations of personal agents associated with these three dimensions. Theoretically, we formulate harness evolution as a learning problem and explain these phenomena through approximation, generalization, and optimization errors. Analyses of reachable policies, capacity under finite interaction evidence, and biased update dynamics provide theoretical accounts of the observed phenomena. Together, these results offer a unified perspective on the limits of personalization through harness evolution and inform future harness design.

cs.AI↗

arXiv2026

Statistical Priors for Implicit Preferences: Decoupling Skill Selection as a Local Harness in Personal Agents

As Large Language Model (LLM) capabilities advance, locally deployed personal agents relying on API-based remote models and external skills have emerged as a novel paradigm. With the rapid expansion of available skills, enabling personal agents to learn and adapt to implicit user preferences becomes a critical challenge. However, local deployment constraints preclude complex centralized selection algorithms, creating an urgent need for a lightweight local preference harness. This paper explores the implementation of such a harness through a novel architecture that strictly decouples statistical preference learning from semantic intent parsing. Specifically, we leverage localized statistical results to influence and modulate the selection decisions of the remote LLM. Extensive evaluations demonstrate that our decoupled approach achieves the lowest cumulative regret and highest test accuracy, significantly outperforming traditional memory-augmented agents.

cs.AI↗

arXiv2026

Why Does Reasoning Length Converge? Unveiling the Underfitting-Overfitting Trade-off in Chain-of-Thought

Test-time scaling, primarily manifested through multi-step Chain-of-Thought (CoT) reasoning via Reinforcement Learning (RL), has emerged as a pivotal paradigm for enhancing the reasoning capabilities of Large Language Models (LLMs). However, a significant theoretical gap persists: traditional token-level analysis fails to capture the macroscopic dynamics of reasoning-level scaling. To address this, we introduce CoT-Space, a novel theoretical framework that recasts the reasoning process from a discrete token-prediction task to an optimization process within a continuous, reasoning-level semantic space. By modeling the reasoning trajectory from both noise and risk perspectives and revitalizing foundational principles from classical learning theory, we demonstrate that the observed convergence to an optimal CoT length is a natural consequence of the fundamental trade-off between underfitting and overfitting. We further utilize RL as a tool to elicit and verify these results in our experiments. Our findings provide a mechanistic explanation for the internal test-time scaling via RL, offering a principled theoretical foundation to optimize reasoning trajectories in modern LLMs.

cs.AI↗

arXiv2026

Information Gain-based Policy Optimization: A Simple and Effective Approach for Multi-Turn Search Agents

Large language model (LLM)-based agents are increasingly trained with reinforcement learning (RL) to enhance their ability to interact with external environments through tool use, particularly in search-based settings that require multi-turn reasoning and knowledge acquisition. However, existing approaches typically rely on outcome-based rewards that are only provided exclusively upon generating the final answer. This reward sparsity becomes particularly problematic in multi-turn settings, where long trajectories exacerbate three critical issues: (i) advantage collapse, where all rollouts receive identical rewards and provide no useful learning signals; (ii) lack of fine-grained credit assignment, where the correctness of intermediate turns is obscured, especially in long-horizon tasks; and (iii) poor sample efficiency, where each rollout yields only a single outcome signal, leading to low data utilization. In this paper, we propose Information Gain-based Policy Optimization (IGPO), a simple yet effective RL framework that provides dense and intrinsic supervision for multi-turn agent training. IGPO models each interaction turn as an incremental process of acquiring information about the ground truth, and defines turn-level rewards as the marginal increase in the policy's probability of producing the correct answer. Unlike prior process-level reward approaches that depend on external reward models or costly Monte Carlo estimation, IGPO derives intrinsic rewards directly from the model's own belief updates. These intrinsic turn-level rewards are combined with outcome-level supervision to form dense reward signals. Extensive experiments on both in-domain and out-of-domain benchmarks demonstrate that IGPO consistently outperforms strong baselines in multi-turn scenarios, achieving higher accuracy and improved data efficiency. Our code is available at https://github.com/GuoqingWang1/IGPO.

cs.CL↗

arXiv2026

Beyond the Black Box: A Survey on the Theory and Mechanism of Large Language Models

The rapid emergence of Large Language Models (LLMs) has precipitated a profound paradigm shift in Artificial Intelligence, delivering monumental engineering successes that increasingly impact modern society. However, a critical paradox persists within the current field: despite the empirical efficacy, our theoretical understanding of LLMs remains disproportionately nascent, forcing these systems to be treated largely as ``black boxes''. To address this theoretical fragmentation, this survey proposes a unified lifecycle-based taxonomy that organizes the research landscape into six distinct stages: Data Preparation, Model Preparation, Training, Alignment, Inference, and Evaluation. Within this framework, we provide a systematic review of the foundational theories and internal mechanisms driving LLM performance. Specifically, we analyze core theoretical issues such as the mathematical justification for data mixtures, the representational limits of various architectures, and the optimization dynamics of alignment algorithms. Moving beyond current best practices, we identify critical frontier challenges, including the theoretical limits of synthetic data self-improvement, the mathematical bounds of safety guarantees, and the mechanistic origins of emergent intelligence. By connecting empirical observations with rigorous scientific inquiry, this work provides a structured roadmap for transitioning LLM development from engineering heuristics toward a principled scientific discipline.

cs.CL↗

arXiv2025

Rethinking External Slow-Thinking: From Snowball Errors to Probability of Correct Reasoning

Test-time scaling, which is also often referred to as slow-thinking, has been demonstrated to enhance multi-step reasoning in large language models (LLMs). However, despite its widespread utilization, the mechanisms underlying slow-thinking methods remain poorly understood. This paper explores the mechanisms of external slow-thinking from a theoretical standpoint. We begin by examining the snowball error effect within the LLM reasoning process and connect it to the likelihood of correct reasoning using information theory. Building on this, we show that external slow-thinking methods can be interpreted as strategies to mitigate the error probability. We further provide a comparative analysis of popular external slow-thinking approaches, ranging from simple to complex, highlighting their differences and interrelationships. Our findings suggest that the efficacy of these methods is not primarily determined by the specific framework employed, and that expanding the search scope or the model's internal reasoning capacity may yield more sustained improvements in the long term. We open-source our code at https://github.com/ZyGan1999/Snowball-Errors-and-Probability.

cs.AI↗

arXiv2025

Towards a Theoretical Understanding of Synthetic Data in LLM Post-Training: A Reverse-Bottleneck Perspective

Synthetic data has become a pivotal resource in post-training tasks for large language models (LLMs) due to the scarcity of high-quality, specific data. While various methods have been developed to generate synthetic data, there remains a discernible gap between the practical effects of synthetic data and our theoretical comprehension. To address this challenge, we commence by presenting a detailed modeling of the prevalent synthetic data generation process. Building upon this modeling, we demonstrate that the generalization capability of the post-trained model is critically determined by the information gain derived from the generative model, as analyzed from a novel reverse-bottleneck perspective. Moreover, we introduce the concept of Generalization Gain via Mutual Information (GGMI) and elucidate the relationship between generalization gain and information gain. This analysis serves as a theoretical foundation for synthetic data generation and further highlights its connection with the generalization capability of post-trained models, offering an understanding about the design of synthetic data generation techniques and the optimization of the post-training process. We open-source our code at https://github.com/ZyGan1999/Towards-a-Theoretical-Understanding-of-Synthetic-Data-in-LLM-Post-Training.

cs.AI↗