MathCoder2: Better Math Reasoning from Continued Pretraining on Model-translated Mathematical Code

Code has been shown to be effective in enhancing the mathematical reasoning abilities of large language models due to its precision and accuracy. Previous works involving continued mathematical pretraining often include code that utilizes math-related packages, which are primarily designed for fields such as engineering, machine learning, signal processing, or module testing, rather than being directly focused on mathematical reasoning. In this paper, we introduce a novel method for generating mathematical code accompanied with corresponding reasoning steps for continued pretraining. Our approach begins with the construction of a high-quality mathematical continued pretraining dataset by incorporating math-related web data, code using mathematical packages, math textbooks, and synthetic data. Next, we construct reasoning steps by extracting LaTeX expressions, the conditions needed for the expressions, and the results of the expressions from the previously collected dataset. Based on this extracted information, we generate corresponding code to accurately capture the mathematical reasoning process. Appending the generated code to each reasoning step results in data consisting of paired natural language reasoning steps and their corresponding code. Combining this data with the original dataset results in a 19.2B-token high-performing mathematical pretraining corpus, which we name MathCode-Pile. Training several popular base models with this corpus significantly improves their mathematical abilities, leading to the creation of the MathCoder2 family of models. All of our data processing and training code is open-sourced, ensuring full transparency and easy reproducibility of the entire data collection and training pipeline. The code is released at https://github.com/mathllm/MathCoder2 .

Step-Controlled DPO: Leveraging Stepwise Error for Enhanced Mathematical Reasoning

Direct Preference Optimization (DPO) has proven effective at improving the performance of large language models (LLMs) on downstream tasks such as reasoning and alignment. In this work, we propose Step-Controlled DPO (SCDPO), a method for automatically providing stepwise error supervision by creating negative samples of mathematical reasoning rationales that start making errors at a specified step. By applying these samples in DPO training, SCDPO can better align the model to understand reasoning errors and output accurate reasoning steps. We apply SCDPO to both code-integrated and chain-of-thought solutions, empirically showing that it consistently improves the performance compared to naive DPO on three different SFT models, including one existing SFT model and two models we finetuned. Qualitative analysis of the credit assignment of SCDPO and DPO demonstrates the effectiveness of SCDPO at identifying errors in mathematical solutions. We then apply SCDPO to an InternLM2-20B model, resulting in a 20B model that achieves high scores of 88.5% on GSM8K and 58.1% on MATH, rivaling all other open-source LLMs, showing the great potential of our method.

ReflectionCoder: Learning from Reflection Sequence for Enhanced One-off Code Generation

Code generation plays a crucial role in various tasks, such as code auto-completion and mathematical reasoning. Previous work has proposed numerous methods to enhance code generation performance, including integrating feedback from the compiler. Inspired by this, we present ReflectionCoder, a novel approach that effectively leverages reflection sequences constructed by integrating compiler feedback to improve one-off code generation performance. Furthermore, we propose reflection self-distillation and dynamically masked distillation to effectively utilize these reflection sequences. Extensive experiments on three benchmarks, i.e., HumanEval (+), MBPP (+), and MultiPl-E, demonstrate that models fine-tuned with our method achieve state-of-the-art performance. Notably, ReflectionCoder-DeepSeek-Coder-33B reaches pass@1 of 82.9 (76.8) on HumanEval (+) and 84.1 (72.0) on MBPP (+), on par with GPT-3.5-Turbo and Claude-3-opus, and surpasses early GPT-4. Beyond the code domain, we believe this approach can benefit other domains that focus on final results and require long reasoning paths. Code and data are available at https://github.com/SenseLLM/ReflectionCoder.

Empowering Character-level Text Infilling by Eliminating Sub-Tokens

In infilling tasks, sub-tokens, representing instances where a complete token is segmented into two parts, often emerge at the boundaries of prefixes, middles, and suffixes. Traditional methods focused on training models at the token level, leading to sub-optimal performance in character-level infilling tasks during the inference stage. Alternately, some approaches considered character-level infilling, but they relied on predicting sub-tokens in inference, yet this strategy diminished ability in character-level infilling tasks due to the large perplexity of the model on sub-tokens. In this paper, we introduce FIM-SE, which stands for Fill-In-the-Middle with both Starting and Ending character constraints. The proposed method addresses character-level infilling tasks by utilizing a line-level format to avoid predicting any sub-token in inference. In addition, we incorporate two special tokens to signify the rest of the incomplete lines, thereby enhancing generation guidance. Extensive experiments demonstrate that our proposed approach surpasses previous methods, offering a significant advantage. Code is available at https://github.com/SenseLLM/FIM-SE.

MathGenie: Generating Synthetic Data with Question Back-translation for Enhancing Mathematical Reasoning of LLMs

Large language models (LLMs) have exhibited great potential in mathematical reasoning. However, there remains a performance gap in this area between existing open-source models and closed-source models such as GPT-4. In this paper, we introduce MathGenie, a novel method for generating diverse and reliable math problems from a small-scale problem-solution dataset (denoted as seed data). We augment the ground-truth solutions of our seed data and train a back-translation model to translate the augmented solutions back into new questions. Subsequently, we generate code-integrated solutions for the new questions. To ensure the correctness of the code-integrated solutions, we employ rationale-based strategy for solution verification. Various pretrained models, ranging from 7B to 70B, are trained on the newly curated data to test the effectiveness of the proposed augmentation technique, resulting in a family of models known as MathGenieLM. These models consistently outperform previous open-source models across five representative mathematical reasoning datasets, achieving state-of-the-art performance. In particular, MathGenieLM-InternLM2 achieves an accuracy of 87.7% on GSM8K and 55.7% on MATH, securing the best overall score among open-source language models.

MathCoder: Seamless Code Integration in LLMs for Enhanced Mathematical Reasoning

The recently released GPT-4 Code Interpreter has demonstrated remarkable proficiency in solving challenging math problems, primarily attributed to its ability to seamlessly reason with natural language, generate code, execute code, and continue reasoning based on the execution output. In this paper, we present a method to fine-tune open-source language models, enabling them to use code for modeling and deriving math equations and, consequently, enhancing their mathematical reasoning abilities. We propose a method of generating novel and high-quality datasets with math problems and their code-based solutions, referred to as MathCodeInstruct. Each solution interleaves natural language, code, and execution results. We also introduce a customized supervised fine-tuning and inference approach. This approach yields the MathCoder models, a family of models capable of generating code-based solutions for solving challenging math problems. Impressively, the MathCoder models achieve state-of-the-art scores among open-source LLMs on the MATH (45.2%) and GSM8K (83.9%) datasets, substantially outperforming other open-source alternatives. Notably, the MathCoder model not only surpasses ChatGPT-3.5 and PaLM-2 on GSM8K and MATH but also outperforms GPT-4 on the competition-level MATH dataset. The dataset and models will be released at https://github.com/mathllm/MathCoder.

Typos-aware Bottlenecked Pre-Training for Robust Dense Retrieval

Current dense retrievers (DRs) are limited in their ability to effectively process misspelled queries, which constitute a significant portion of query traffic in commercial search engines. The main issue is that the pre-trained language model-based encoders used by DRs are typically trained and fine-tuned using clean, well-curated text data. Misspelled queries are typically not found in the data used for training these models, and thus misspelled queries observed at inference time are out-of-distribution compared to the data used for training and fine-tuning. Previous efforts to address this issue have focused on \textit{fine-tuning} strategies, but their effectiveness on misspelled queries remains lower than that of pipelines that employ separate state-of-the-art spell-checking components. To address this challenge, we propose ToRoDer (TypOs-aware bottlenecked pre-training for RObust DEnse Retrieval), a novel \textit{pre-training} strategy for DRs that increases their robustness to misspelled queries while preserving their effectiveness in downstream retrieval tasks. ToRoDer utilizes an encoder-decoder architecture where the encoder takes misspelled text with masked tokens as input and outputs bottlenecked information to the decoder. The decoder then takes as input the bottlenecked embeddings, along with token embeddings of the original text with the misspelled tokens masked out. The pre-training task is to recover the masked tokens for both the encoder and decoder. Our extensive experimental results and detailed ablation studies show that DRs pre-trained with ToRoDer exhibit significantly higher effectiveness on misspelled queries, sensibly closing the gap with pipelines that use a separate, complex spell-checker component, while retaining their effectiveness on correctly spelled queries.

Lexicon-Enhanced Self-Supervised Training for Multilingual Dense Retrieval

Recent multilingual pre-trained models have shown better performance in various multilingual tasks. However, these models perform poorly on multilingual retrieval tasks due to lacking multilingual training data. In this paper, we propose to mine and generate self-supervised training data based on a large-scale unlabeled corpus. We carefully design a mining method which combines the sparse and dense models to mine the relevance of unlabeled queries and passages. And we introduce a query generator to generate more queries in target languages for unlabeled passages. Through extensive experiments on Mr. TYDI dataset and an industrial dataset from a commercial search engine, we demonstrate that our method performs better than baselines based on various pre-trained multilingual models. Our method even achieves on-par performance with the supervised method on the latter dataset.

Empowering Dual-Encoder with Query Generator for Cross-Lingual Dense Retrieval

In monolingual dense retrieval, lots of works focus on how to distill knowledge from cross-encoder re-ranker to dual-encoder retriever and these methods achieve better performance due to the effectiveness of cross-encoder re-ranker. However, we find that the performance of the cross-encoder re-ranker is heavily influenced by the number of training samples and the quality of negative samples, which is hard to obtain in the cross-lingual setting. In this paper, we propose to use a query generator as the teacher in the cross-lingual setting, which is less dependent on enough training samples and high-quality negative samples. In addition to traditional knowledge distillation, we further propose a novel enhancement method, which uses the query generator to help the dual-encoder align queries from different languages, but does not need any additional parallel sentences. The experimental results show that our method outperforms the state-of-the-art methods on two benchmark datasets.

Self-supervised Trajectory Representation Learning with Temporal Regularities and Travel Semantics

Trajectory Representation Learning (TRL) is a powerful tool for spatial-temporal data analysis and management. TRL aims to convert complicated raw trajectories into low-dimensional representation vectors, which can be applied to various downstream tasks, such as trajectory classification, clustering, and similarity computation. Existing TRL works usually treat trajectories as ordinary sequence data, while some important spatial-temporal characteristics, such as temporal regularities and travel semantics, are not fully exploited. To fill this gap, we propose a novel Self-supervised trajectory representation learning framework with TemporAl Regularities and Travel semantics, namely START. The proposed method consists of two stages. The first stage is a Trajectory Pattern-Enhanced Graph Attention Network (TPE-GAT), which converts the road network features and travel semantics into representation vectors of road segments. The second stage is a Time-Aware Trajectory Encoder (TAT-Enc), which encodes representation vectors of road segments in the same trajectory as a trajectory representation vector, meanwhile incorporating temporal regularities with the trajectory representation. Moreover, we also design two self-supervised tasks, i.e., span-masked trajectory recovery and trajectory contrastive learning, to introduce spatial-temporal characteristics of trajectories into the training process of our START framework. The effectiveness of the proposed method is verified by extensive experiments on two large-scale real-world datasets for three downstream tasks. The experiments also demonstrate that our method can be transferred across different cities to adapt heterogeneous trajectory datasets.