DeepTheorem: Advancing LLM Reasoning for Theorem Proving Through Natural Language and Reinforcement Learning

Theorem proving serves as a major testbed for evaluating complex reasoning abilities in large language models (LLMs). However, traditional automated theorem proving (ATP) approaches rely heavily on formal proof systems that poorly align with LLMs' strength derived from informal, natural language knowledge acquired during pre-training. In this work, we propose DeepTheorem, a comprehensive informal theorem-proving framework exploiting natural language to enhance LLM mathematical reasoning. DeepTheorem includes a large-scale benchmark dataset consisting of 121K high-quality IMO-level informal theorems and proofs spanning diverse mathematical domains, rigorously annotated for correctness, difficulty, and topic categories, accompanied by systematically constructed verifiable theorem variants. We devise a novel reinforcement learning strategy (RL-Zero) explicitly tailored to informal theorem proving, leveraging the verified theorem variants to incentivize robust mathematical inference. Additionally, we propose comprehensive outcome and process evaluation metrics examining proof correctness and the quality of reasoning steps. Extensive experimental analyses demonstrate DeepTheorem significantly improves LLM theorem-proving performance compared to existing datasets and supervised fine-tuning protocols, achieving state-of-the-art accuracy and reasoning quality. Our findings highlight DeepTheorem's potential to fundamentally advance automated informal theorem proving and mathematical exploration.

Two Experts Are All You Need for Steering Thinking: Reinforcing Cognitive Effort in MoE Reasoning Models Without Additional Training

Mixture-of-Experts (MoE) architectures within Large Reasoning Models (LRMs) have achieved impressive reasoning capabilities by selectively activating experts to facilitate structured cognitive processes. Despite notable advances, existing reasoning models often suffer from cognitive inefficiencies like overthinking and underthinking. To address these limitations, we introduce a novel inference-time steering methodology called Reinforcing Cognitive Experts (RICE), designed to improve reasoning performance without additional training or complex heuristics. Leveraging normalized Pointwise Mutual Information (nPMI), we systematically identify specialized experts, termed ''cognitive experts'' that orchestrate meta-level reasoning operations characterized by tokens like ''<think>''. Empirical evaluations with leading MoE-based LRMs (DeepSeek-R1 and Qwen3-235B) on rigorous quantitative and scientific reasoning benchmarks demonstrate noticeable and consistent improvements in reasoning accuracy, cognitive efficiency, and cross-domain generalization. Crucially, our lightweight approach substantially outperforms prevalent reasoning-steering techniques, such as prompt design and decoding constraints, while preserving the model's general instruction-following skills. These results highlight reinforcing cognitive experts as a promising, practical, and interpretable direction to enhance cognitive efficiency within advanced reasoning models.

Optimal Client Sampling in Federated Learning with Client-Level Heterogeneous Differential Privacy

Federated Learning with client-level differential privacy (DP) provides a promising framework for collaboratively training models while rigorously protecting clients' privacy. However, classic approaches like DP-FedAvg struggle when clients have heterogeneous privacy requirements, as they must uniformly enforce the strictest privacy level across clients, leading to excessive DP noise and significant model utility degradation. Existing methods to improve the model utility in such heterogeneous privacy settings often assume a trusted server and are largely heuristic, resulting in suboptimal performance and lacking strong theoretical underpinnings. In this work, we address these challenges under a practical attack model where both clients and the server are honest-but-curious. We propose GDPFed, which partitions clients into groups based on their privacy budgets and achieves client-level DP within each group to reduce the privacy budget waste and hence improve the model utility. Based on the privacy and convergence analysis of GDPFed, we find that the magnitude of DP noise depends on both model dimensionality and the per-group client sampling ratios. To further improve the performance of GDPFed, we introduce GDPFed$^+$, which integrates model sparsification to eliminate unnecessary noise and optimizes per-group client sampling ratios to minimize convergence error. Extensive empirical evaluations on multiple benchmark datasets demonstrate the effectiveness of GDPFed$^+$, showing substantial performance gains compared with state-of-the-art methods.

Trust, But Verify: A Self-Verification Approach to Reinforcement Learning with Verifiable Rewards

Large Language Models (LLMs) show great promise in complex reasoning, with Reinforcement Learning with Verifiable Rewards (RLVR) being a key enhancement strategy. However, a prevalent issue is ``superficial self-reflection'', where models fail to robustly verify their own outputs. We introduce RISE (Reinforcing Reasoning with Self-Verification), a novel online RL framework designed to tackle this. RISE explicitly and simultaneously trains an LLM to improve both its problem-solving and self-verification abilities within a single, integrated RL process. The core mechanism involves leveraging verifiable rewards from an outcome verifier to provide on-the-fly feedback for both solution generation and self-verification tasks. In each iteration, the model generates solutions, then critiques its own on-policy generated solutions, with both trajectories contributing to the policy update. Extensive experiments on diverse mathematical reasoning benchmarks show that RISE consistently improves model's problem-solving accuracy while concurrently fostering strong self-verification skills. Our analyses highlight the advantages of online verification and the benefits of increased verification compute. Additionally, RISE models exhibit more frequent and accurate self-verification behaviors during reasoning. These advantages reinforce RISE as a flexible and effective path towards developing more robust and self-aware reasoners.

Traceable Black-box Watermarks for Federated Learning

Due to the distributed nature of Federated Learning (FL) systems, each local client has access to the global model, posing a critical risk of model leakage. Existing works have explored injecting watermarks into local models to enable intellectual property protection. However, these methods either focus on non-traceable watermarks or traceable but white-box watermarks. We identify a gap in the literature regarding the formal definition of traceable black-box watermarking and the formulation of the problem of injecting such watermarks into FL systems. In this work, we first formalize the problem of injecting traceable black-box watermarks into FL. Based on the problem, we propose a novel server-side watermarking method, $\mathbf{TraMark}$, which creates a traceable watermarked model for each client, enabling verification of model leakage in black-box settings. To achieve this, $\mathbf{TraMark}$ partitions the model parameter space into two distinct regions: the main task region and the watermarking region. Subsequently, a personalized global model is constructed for each client by aggregating only the main task region while preserving the watermarking region. Each model then learns a unique watermark exclusively within the watermarking region using a distinct watermark dataset before being sent back to the local client. Extensive results across various FL systems demonstrate that $\mathbf{TraMark}$ ensures the traceability of all watermarked models while preserving their main task performance.

DeepMath-103K: A Large-Scale, Challenging, Decontaminated, and Verifiable Mathematical Dataset for Advancing Reasoning

The capacity for complex mathematical reasoning is a key benchmark for artificial intelligence. While reinforcement learning (RL) applied to LLMs shows promise, progress is significantly hindered by the lack of large-scale training data that is sufficiently challenging, possesses verifiable answer formats suitable for RL, and is free from contamination with evaluation benchmarks. To address these limitations, we introduce DeepMath-103K, a new, large-scale dataset comprising approximately 103K mathematical problems, specifically designed to train advanced reasoning models via RL. DeepMath-103K is curated through a rigorous pipeline involving source analysis, stringent decontamination against numerous benchmarks, and filtering for high difficulty (primarily Levels 5-9), significantly exceeding existing open resources in challenge. Each problem includes a verifiable final answer, enabling rule-based RL, and three distinct R1-generated solutions suitable for diverse training paradigms like supervised fine-tuning or distillation. Spanning a wide range of mathematical topics, DeepMath-103K promotes the development of generalizable reasoning. We demonstrate that models trained on DeepMath-103K achieve significant improvements on challenging mathematical benchmarks, validating its effectiveness. We release DeepMath-103K publicly to facilitate community progress in building more capable AI reasoning systems: https://github.com/zwhe99/DeepMath.

Dancing with Critiques: Enhancing LLM Reasoning with Stepwise Natural Language Self-Critique

Enhancing the reasoning capabilities of large language models (LLMs), particularly for complex tasks requiring multi-step logical deductions, remains a significant challenge. Traditional inference time scaling methods utilize scalar reward signals from process reward models to evaluate candidate reasoning steps, but these scalar rewards lack the nuanced qualitative information essential for understanding and justifying each step. In this paper, we propose a novel inference-time scaling approach -- stepwise natural language self-critique (PANEL), which employs self-generated natural language critiques as feedback to guide the step-level search process. By generating rich, human-readable critiques for each candidate reasoning step, PANEL retains essential qualitative information, facilitating better-informed decision-making during inference. This approach bypasses the need for task-specific verifiers and the associated training overhead, making it broadly applicable across diverse tasks. Experimental results on challenging reasoning benchmarks, including AIME and GPQA, demonstrate that PANEL significantly enhances reasoning performance, outperforming traditional scalar reward-based methods. Our code is available at https://github.com/puddingyeah/PANEL to support and encourage future research in this promising field.

RaSA: Rank-Sharing Low-Rank Adaptation

Low-rank adaptation (LoRA) has been prominently employed for parameter-efficient fine-tuning of large language models (LLMs). However, the limited expressive capacity of LoRA, stemming from the low-rank constraint, has been recognized as a bottleneck, particularly in rigorous tasks like code generation and mathematical reasoning. To address this limitation, we introduce Rank-Sharing Low-Rank Adaptation (RaSA), an innovative extension that enhances the expressive capacity of LoRA by leveraging partial rank sharing across layers. By forming a shared rank pool and applying layer-specific weighting, RaSA effectively increases the number of ranks without augmenting parameter overhead. Our theoretically grounded and empirically validated approach demonstrates that RaSA not only maintains the core advantages of LoRA but also significantly boosts performance in challenging code and math tasks. Code, data and scripts are available at: https://github.com/zwhe99/RaSA.

Detecting Backdoor Attacks in Federated Learning via Direction Alignment Inspection

The distributed nature of training makes Federated Learning (FL) vulnerable to backdoor attacks, where malicious model updates aim to compromise the global model's performance on specific tasks. Existing defense methods show limited efficacy as they overlook the inconsistency between benign and malicious model updates regarding both general and fine-grained directions. To fill this gap, we introduce AlignIns, a novel defense method designed to safeguard FL systems against backdoor attacks. AlignIns looks into the direction of each model update through a direction alignment inspection process. Specifically, it examines the alignment of model updates with the overall update direction and analyzes the distribution of the signs of their significant parameters, comparing them with the principle sign across all model updates. Model updates that exhibit an unusual degree of alignment are considered malicious and thus be filtered out. We provide the theoretical analysis of the robustness of AlignIns and its propagation error in FL. Our empirical results on both independent and identically distributed (IID) and non-IID datasets demonstrate that AlignIns achieves higher robustness compared to the state-of-the-art defense methods. The code is available at https://github.com/JiiahaoXU/AlignIns.

The First Few Tokens Are All You Need: An Efficient and Effective Unsupervised Prefix Fine-Tuning Method for Reasoning Models

Improving the reasoning capabilities of large language models (LLMs) typically requires supervised fine-tuning with labeled data or computationally expensive sampling. We introduce Unsupervised Prefix Fine-Tuning (UPFT), which leverages the observation of Prefix Self-Consistency -- the shared initial reasoning steps across diverse solution trajectories -- to enhance LLM reasoning efficiency. By training exclusively on the initial prefix substrings (as few as 8 tokens), UPFT removes the need for labeled data or exhaustive sampling. Experiments on reasoning benchmarks show that UPFT matches the performance of supervised methods such as Rejection Sampling Fine-Tuning, while reducing training time by 75% and sampling cost by 99%. Further analysis reveals that errors tend to appear in later stages of the reasoning process and that prefix-based training preserves the model's structural knowledge. This work demonstrates how minimal unsupervised fine-tuning can unlock substantial reasoning gains in LLMs, offering a scalable and resource-efficient alternative to conventional approaches.