Iterative preference optimization is essential for aligning Large Language Models on mathematical reasoning tasks, yet its efficiency is often throttled by signal scarcity: as the model improves, static problem sets become increasingly mismatched to the model’s evolving competence, producing rollouts that are either too easy or too hard and therefore non-informative, which leads to a scarcity of valid preference pairs. We propose DIAG, a Diagnostic Iterative Alignment and Generation framework that adaptively reshapes the practice distribution to increase informative supervision and focus training near the student’s current competence boundary. DIAG consists of two phases: (1) diagnosing valid preference-pair yield to calibrate the exploration-exploitation trade-off and allocate topic quotas via an Empirical Bayes shrinkage estimator, thereby prioritizing high-yield concepts; and (2) generating targeted practice, where a teacher synthesizes variants from the student’s failure traces. We further provide a theoretical view interpreting DIAG as a teacher-mediated approximation to KL-regularized reweighting of the practice distribution toward the student’s competence boundary, where valid preference-pair yield is maximized. Experiments show that DIAG boosts yield across iterations and delivers stronger reasoning performance under an iso-effective training budget, demonstrating that it can distill more informative preference supervision for mathematical reasoning.

Iterative Direct Preference Optimization (DPO) has emerged as a widely used paradigm for aligning Large Language Models on reasoning tasks. Existing approaches typically rely on Best-of-N sampling (N≥8) to mine positive trajectories from the distribution tail. In this work, we show that in mathematical reasoning, increasing N yields diminishing returns while increasing verifier-induced false-positive risk and the distribution shift required for policy updates. To address this, we introduce PACE (Proximal Alignment via Corrective Exploration), a generation-based corrective framework that replaces exhaustive mining with low-budget exploration (2≤ N≤3). Rather than searching for increasingly rare positive samples, PACE synthesizes high-fidelity preference pairs from failed explorations through corrective hindsight refinement and verification-guided filtering. Empirically, PACE matches or exceeds the performance of DPO-R1 (N=16) while using about 1/5 of the compute, and remains robust under 20% label corruption, where high-N baselines exhibit substantially higher noise exploitation.

In mathematical reasoning, data selection strategies predominantly rely on static, externally defined metrics, which fail to adapt to the evolving capabilities of models during training. This misalignment limits the efficiency of Supervised Fine-Tuning and Reinforcement Learning. To bridge this gap, we introduce SAI-DPO (Self-Aware Iterative Data Persistent Optimization), a dynamic sampling framework that aligns training data with the model’s intrinsic competence. SAI-DPO operationalizes two novel metrics: Knowledge Semantic Alignment for targeting domain weaknesses, and Self-Aware Difficulty, derived from pass rates and reasoning path characteristics, to gauge instance complexity relative to the model’s current state. By iteratively recalibrating the data distribution based on real-time feedback, SAI-DPO dynamically aligns training samples with the model’s evolving competence, ensuring the data remains strictly relevant to the model’s current capability level. Extensive experiments on eight benchmarks (including AIME24 and AMC23) demonstrate that SAI-DPO outperforms static baselines at most nearly 6 points, achieving state-of-the-art efficiency with significantly less data.

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.