Despite their success in many natural language tasks, solving math problems remains a significant challenge for large language models (LLMs). A large gap exists between LLMs' pass-at-one and pass-at-N performance in solving math problems, suggesting LLMs might be close to finding correct solutions, motivating our exploration of fine-tuning methods to unlock LLMs' performance. Using the challenging MATH dataset, we investigate three fine-tuning strategies: (1) solution fine-tuning, where we fine-tune to generate a detailed solution for a given math problem; (2) solution-cluster re-ranking, where the LLM is fine-tuned as a solution verifier/evaluator to choose among generated candidate solution clusters; (3) multi-task sequential fine-tuning, which integrates both solution generation and evaluation tasks together efficiently to enhance the LLM performance. With these methods, we present a thorough empirical study on a series of PaLM 2 models and find: (1) The quality and style of the step-by-step solutions used for fine-tuning can make a significant impact on the model performance; (2) While solution re-ranking and majority voting are both effective for improving the model performance when used separately, they can also be used together for an even greater performance boost; (3) Multi-task fine-tuning that sequentially separates the solution generation and evaluation tasks can offer improved performance compared with the solution fine-tuning baseline. Guided by these insights, we design a fine-tuning recipe that yields approximately 58.8% accuracy on the MATH dataset with fine-tuned PaLM 2-L models, an 11.2% accuracy improvement over the few-shot performance of pre-trained PaLM 2-L model with majority voting.

We show that an important and computationally challenging solution space feature of the graph coloring problem (COL), namely the number of clusters of solutions, can be accurately estimated by a technique very similar to one for counting the number of solutions. This cluster counting approach can be naturally written in terms of a new factor graph derived from the factor graph representing the COL instance. Using a variant of the Belief Propagation inference framework, we can efficiently approximate cluster counts in random COL problems over a large range of graph densities. We illustrate the algorithm on instances with up to 100, 000 vertices. Moreover, we supply a methodology for computing the number of clus- ters exactly using advanced techniques from the knowledge compilation literature.

This work proposes new approximate (and exact) inference methods for reasoning about an important and hard-to-compute property of the solution space of combinatorial problems, namely clusters of solutions. We introduce an approximate method that first reformulates the constraint satisfaction problem (CSP) as a "factor graph" over an extended set of variable domains, approximates the number of clusters using an exponential size expression defined over this factor graph, and then estimates the value of this expression using message passing techniques, specifically an extension of the belief propagation (BP) algorithm. We provide formal exactness results as well as an empirical evaluation attesting to the accuracy of our method in counting the number of solution clusters.

We show that an important and computationally challenging solution space feature of the graph coloring problem (COL), namely the number of clusters of solutions, can be accurately estimated by a technique very similar to one for counting the number of solutions. This cluster counting approach can be naturally written in terms of a new factor graph derived from the factor graph representing the COL instance. Using a variant of the Belief Propagation inference framework, we can efficiently approximate cluster counts in random COL problems over a large range of graph densities. We illustrate the algorithm on instances with up to 100, 000 vertices. Moreover, we supply a methodology for computing the number of clusters exactlyusing advanced techniques from the knowledge compilation literature.

We show that an important and computationally challenging solution space feature of the graph coloring problem (COL), namely the number of clusters of solutions, can be accurately estimated by a technique very similar to one for counting the number of solutions. This cluster counting approach can be naturally written in terms of a new factor graph derived from the factor graph representing the COL instance. Using a variant of the Belief Propagation inference framework, we can efficiently approximate cluster counts in random COL problems over a large range of graph densities. We illustrate the algorithm on instances with up to 100, 000 vertices. Moreover, we supply a methodology for computing the number of clusters exactly using advanced techniques from the knowledge compilation literature.

We show that an important and computationally challenging solution space feature of the graph coloring problem (COL), namely the number of clusters of solutions, can be accurately estimated by a technique very similar to one for counting the number of solutions. This cluster counting approach can be naturally written in terms of a new factor graph derived from the factor graph representing the COL instance. Using a variant of the Belief Propagation inference framework, we can efficiently approximate cluster counts in random COL problems over a large range of graph densities. We illustrate the algorithm on instances with up to 100, 000 vertices. Moreover, we supply a methodology for computing the number of clusters exactly using advanced techniques from the knowledge compilation literature.