Guhan Chen

Jun Rao, Zixiong Yu, Xuebo Liu et al.

Iterative Direct Preference Optimization has emerged as the state-of-the-art paradigm for aligning Large Language Models on reasoning tasks. Standard implementations (DPO-R1) rely on Best-of-N sampling (e.g., $N \ge 8$) to mine golden trajectories from the distribution tail. In this paper, we challenge this scaling hypothesis and reveal a counter-intuitive phenomenon: in mathematical reasoning, aggressive exploration yields diminishing returns and even catastrophic policy collapse. We theoretically demonstrate that scaling $N$ amplifies verifier noise and induces detrimental distribution shifts. To resolve this, we introduce \textbf{PACE} (Proximal Alignment via Corrective Exploration), which replaces brute-force mining with a generation-based corrective strategy. Operating with a minimal budget ($2<N<3$), PACE synthesizes high-fidelity preference pairs from failed explorations. Empirical evaluations show that PACE outperforms DPO-R1 $(N=16)$ while using only about $1/5$ of the compute, demonstrating superior robustness against reward hacking and label noise.

Zixiong Yu, Jun Rao, Guhan Chen et al.

Synthesizing high-quality mathematical reasoning data without human priors remains a significant challenge. Current approaches typically rely on seed data mutation or simple prompt engineering, often suffering from mode collapse and limited logical complexity. This paper proposes a hierarchical synthesis framework that formulates data synthesis as an unsupervised optimization problem over a constraint graph followed by semantic instantiation, rather than treating it as a direct text generation task. We introduce a Legislator-Executor paradigm: The Legislator adversarially evolves structured generation blueprints encoding the constraints of the problem, while the Executor instantiates these specifications into diverse natural language scenarios. This decoupling of skeleton design from linguistic realization enables a prioritized focus on constructing complex and diverse logical structures, thereby guiding high-quality data synthesis. Experiments conducted on a total of 10 models across the Qwen, Llama, Mistral, and Gemma series demonstrate that our method achieves notable results: models fine-tuned on 1K synthesized samples outperform widely-used datasets of comparable scale (LIMO, s1K) across eight mathematical benchmarks, exhibiting superior out-of-distribution generalization.

Wenxun Wu, Yuanyang Li, Guhan Chen et al.

Recent advances in large language models (LLMs) have popularized test-time scaling, where models generate additional reasoning tokens before producing final answers. These approaches have demonstrated significant performance improvements on benchmarks involving mathematical reasoning. However, language models relying solely on direct inference still struggle with tasks demanding up-to-date knowledge or computational tools such as calculators and code interpreters for complex arithmetic operations. To overcome these limitations, we propose Tool-Augmented Policy Optimization (TAPO), a novel reinforcement learning framework that systematically integrates multi-hop reasoning with adaptive tool-calling capabilities. Our approach employs a modified version of Dynamic Sampling Policy Optimization (DAPO), a recently developed RL paradigm, which we adapt specifically for tool invocation scenarios, enabling models to dynamically interleave complex reasoning with on-demand tool usage (including search APIs and Python interpreters). To support this research, we introduce two new datasets: TAPO-easy-60K and TAPO-hard-18K, specifically designed to train and evaluate both fact-based reasoning and mathematical calculation capabilities. Our experiments on Qwen2.5-3B and Qwen2.5-7B models demonstrate the effectiveness of our approach, with both models achieving state-of-the-art performance on tasks requiring external knowledge and mathematical computation among methods with comparable parameters. Notably, TAPO achieves more efficient tool utilization than baseline methods while preventing excessive calls caused by reward hacking. These results highlight the significant potential of combining advanced reasoning with tool usage to enhance model performance in knowledge-intensive and computationally demanding tasks.

Zixiong Yu, Songtao Tian, Guhan Chen

This paper demonstrates that in classification problems, fully connected neural networks (FCNs) and residual neural networks (ResNets) cannot be approximated by kernel logistic regression based on the Neural Tangent Kernel (NTK) under overtraining (i.e., when training time approaches infinity). Specifically, when using the cross-entropy loss, regardless of how large the network width is (as long as it is finite), the empirical NTK diverges from the NTK on the training samples as training time increases. To establish this result, we first demonstrate the strictly positive definiteness of the NTKs for multi-layer FCNs and ResNets. Then, we prove that during training, % with the cross-entropy loss, the neural network parameters diverge if the smallest eigenvalue of the empirical NTK matrix (Gram matrix) with respect to training samples is bounded below by a positive constant. This behavior contrasts sharply with the lazy training regime commonly observed in regression problems. Consequently, using a proof by contradiction, we show that the empirical NTK does not uniformly converge to the NTK across all times on the training samples as the network width increases. We validate our theoretical results through experiments on both synthetic data and the MNIST classification task. This finding implies that NTK theory is not applicable in this context, with significant theoretical implications for understanding neural networks in classification problems.

Yicheng Li, Zixiong Yu, Guhan Chen et al.

In this paper, we provide a strategy to determine the eigenvalue decay rate (EDR) of a large class of kernel functions defined on a general domain rather than $\mathbb S^{d}$. This class of kernel functions include but are not limited to the neural tangent kernel associated with neural networks with different depths and various activation functions. After proving that the dynamics of training the wide neural networks uniformly approximated that of the neural tangent kernel regression on general domains, we can further illustrate the minimax optimality of the wide neural network provided that the underground truth function $f\in [\mathcal H_{\mathrm{NTK}}]^{s}$, an interpolation space associated with the RKHS $\mathcal{H}_{\mathrm{NTK}}$ of NTK. We also showed that the overfitted neural network can not generalize well. We believe our approach for determining the EDR of kernels might be also of independent interests.