Large language models (LLMs) are displaying emergent abilities for math reasoning tasks,and there is a growing attention on enhancing the ability of open-source LLMs through supervised fine-tuning (SFT).In this paper, we aim to explore a general data strategy for supervised data to help optimize and expand math reasoning ability.Firstly, we determine the ability boundary of reasoning paths augmentation by identifying these paths' minimal optimal set.Secondly, we validate that different abilities of the model can be cumulatively enhanced by Mix of Minimal Optimal Sets of corresponding types of data, while our models MMOS achieve SOTA performance on series base models under much lower construction costs.Besides, we point out GSM-HARD is not really hard and today's LLMs no longer lack numerical robustness.Also, we provide an Auto Problem Generator for robustness testing and educational applications.Our code and data are publicly available at https://github.com/cyzhh/MMOS.

A causal falling rule list (CFRL) is a sequence of if-then rules that specifies heterogeneous treatment effects, where (i) the order of rules determines the treatment effect subgroup a subject belongs to, and (ii) the treatment effect decreases monotonically down the list. A given CFRL parameterizes a hierarchical bayesian regression model in which the treatment effects are incorporated as parameters, and assumed constant within model-specific subgroups. We formulate the search for the CFRL best supported by the data as a Bayesian model selection problem, where we perform a search over the space of CFRL models, and approximate the evidence for a given CFRL model using standard variational techniques. We apply CFRL to a census wage dataset to identify subgroups of differing wage inequalities between men and women.

Harry K. sits at his desk in Vancouver, Canada, scanning sepia-tinted swirls, loops and blobs on his computer screen. Every second or so, he jabs at his mouse and adds a fluorescent dot to the image. After a minute, a new image pops up in front of him. Harry is tagging images of cells removed from breast cancers. It's a painstaking job but not a difficult one, he says: "It's like playing Etch A Sketch or a video game where you color in certain dots." Harry found the gig on Crowdflower, a crowdworking platform. Usually that cell-tagging task would be the job of pathologists, who typically start their careers with annual salaries of around 200,000 -- an hourly wage of about 80. Harry, on the other hand, earns just four cents for annotating a batch of five images, which takes him between two to eight minutes. His hourly wage is about 60 cents. Granted, Harry can't perform most of the tasks in a pathologist's repertoire.