Llms Can Now Solve Challenging Math Problems With Minimal Data: Researchers From Uc Berkeley And Ai2 Unveil A Fine-tuning Recipe That Unlocks Mathematical Reasoning Across Difficulty Levels

Language models person made important strides successful tackling reasoning tasks, pinch moreover small-scale supervised fine-tuning (SFT) approaches specified arsenic LIMO and s1 demonstrating singular improvements successful mathematical problem-solving capabilities. However, basal questions stay astir these advancements: Do these models genuinely generalise beyond their training data, aliases are they simply overfitting to trial sets? The investigation organization faces challenges successful knowing which capabilities are enhanced done small-scale SFT and which limitations persist contempt these improvements. Despite awesome capacity connected celebrated benchmarks, location is an incomplete knowing of these fine-tuned models’ circumstantial strengths and weaknesses, creating a captious spread successful knowledge astir their existent reasoning abilities and applicable limitations.

Various attempts person been made to understand nan effects of reasoning-based supervised fine-tuning beyond elemental benchmark scores. Researchers person questioned whether SFT simply improves capacity connected antecedently seen problem types aliases genuinely enables models to transportation problem-solving strategies to caller contexts, specified arsenic applying coordinate-based techniques successful geometry. Existing methods attraction connected factors for illustration correctness, solution length, and consequence diversity, which first studies propose play important roles successful exemplary betterment done SFT. However, these approaches deficiency nan granularity needed to find precisely which types of antecedently unsolvable questions go solvable aft fine-tuning, and which problem categories stay resistant to betterment contempt extended training. The investigation organization still struggles to found whether observed improvements bespeak deeper learning aliases simply memorisation of training trajectories, highlighting nan request for much blase study methods.

The researchers from nan University of California, Berkeley and nan Allen Institute for AI propose a gradual study model to analyse really supervised fine-tuning affects reasoning capabilities successful connection models. This attack utilises nan AIME24 dataset, chosen for its complexity and wide usage successful reasoning research, which exhibits a ladder-like building wherever models solving higher-tier questions typically win connected lower-tier ones. By categorising questions into 4 trouble tiers, Easy, Medium, Hard, and Exh, nan study systematically examines nan circumstantial requirements for advancing betwixt tiers. The study reveals that progression from Easy to Medium chiefly requires adopting an R1 reasoning style pinch agelong conclusion context, while Hard-level questions request greater computational stableness during heavy exploration. Exh-level questions coming a fundamentally different challenge, requiring unconventional problem-solving strategies that existent models uniformly struggle with. The investigation besides identifies 4 cardinal insights: nan capacity spread betwixt imaginable and stableness successful small-scale SFT models, minimal benefits from observant dataset curation, diminishing returns from scaling SFT datasets, and imaginable intelligence barriers that whitethorn not beryllium flooded done SFT alone.

The methodology employs a broad gradual study utilizing nan AIME24 dataset arsenic nan superior trial benchmark. This prime stems from 3 cardinal attributes: nan dataset’s hierarchical trouble that challenges moreover state-of-the-art models, its divers sum of mathematical domains, and its attraction connected precocious schoolhouse mathematics that isolates axenic reasoning expertise from domain-specific knowledge. Qwen2.5-32 B-Instruct serves arsenic nan guidelines exemplary owed to its wide take and inherent cognitive behaviours, including verification, backtracking, and subgoal setting. The fine-tuning information consists of question-response pairs from nan Openr1-Math-220k dataset, specifically utilizing CoT trajectories generated by DeepSeek R1 for problems from NuminaMath1.5, pinch incorrect solutions filtered out. The training configuration mirrors anterior studies pinch a learning complaint of 1 × 10−5, weight decay of 1 × 10−4, batch size of 32, and 5 epochs. Performance information employs avg@n (average walk complaint complete aggregate attempts) and cov@n metrics, pinch questions categorised into 4 trouble levels (Easy, Medium, Hard, and Extremely Hard) based connected exemplary capacity patterns.

Research results uncover that effective progression from Easy to Medium-level mathematical problem-solving requires minimal but circumstantial conditions. The study systematically examined aggregate training variables, including foundational knowledge crossed divers mathematical categories, dataset size variations (100-1000 examples per category), trajectory magnitude (short, normal, aliases long), and trajectory style (comparing DeepSeek-R1 pinch Gemini-flash). Through broad ablation studies, researchers isolated nan effect of each magnitude connected exemplary performance, represented arsenic P = f(C, N, L, S), wherever C represents category, N represents nan number of trajectories, L represents length, and S represents style. The findings show that achieving capacity ≥90% connected Medium-level questions minimally requires astatine slightest 500 normal aliases agelong R1-style trajectories, sloppy of nan circumstantial mathematical category. Models consistently neglect to meet capacity thresholds erstwhile trained pinch less trajectories, shorter trajectories, aliases Gemini-style trajectories. This indicates that reasoning trajectory magnitude and amount correspond captious factors successful processing mathematical reasoning capabilities, while nan circumstantial taxable matter of nan trajectories proves little important than their structural characteristics.

The investigation demonstrates that models pinch small-scale supervised fine-tuning tin perchance lick arsenic galore questions arsenic much blase models for illustration Deepseek-R1, though important challenges remain. The superior limitation identified is instability successful mathematical reasoning, alternatively than capability. Experimental results show that geometry-trained models tin execute a sum people of 90, matching R1’s capacity erstwhile fixed aggregate attempts, yet their wide accuracy lags by much than 20%. This capacity spread stems chiefly from instability successful heavy exploration and computational limitations during analyzable problem-solving. While expanding nan SFT dataset size offers 1 solution path, capacity enhancement follows a logarithmic scaling inclination pinch diminishing returns. Notably, nan study challenges caller assertions astir nan value of observant dataset curation, revealing that capacity crossed various mathematical categories remains accordant wrong a constrictive scope of 55±4%, pinch only marginal differences betwixt specifically constructed akin datasets and randomly constructed ones. This conclusion suggests that nan amount and value of reasoning trajectories matter much than subject-specific contented for processing robust mathematical reasoning capabilities.

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