Despite the recent advancements in language models (LMs), their ability to solve complex problems remains limited. This paper introduces Cumulative Reasoning (CR), a novel approach that utilizes LMs cumulatively and iteratively, mirroring human thought processes for problem-solving. CR decomposes tasks into smaller, manageable components and leverages previous propositions for effective composition, significantly enhancing problem-solving capabilities. We demonstrate CR's superiority through several complex reasoning tasks: it outperforms existing methods in logical inference tasks with up to a 9.3% improvement, achieving 98.04% accuracy on the curated FOLIO wiki dataset. In the Game of 24, it achieves 98% accuracy, marking a 24% improvement over the prior state-of-the-art. Additionally, CR sets new state-of-the-art on the MATH dataset, achieving a 4.2% increase from previous methods and a 43% relative improvement in the most challenging problems. By extending CR to incorporate a code environment without external aids like retrieval or web browsing, we further harness the computational and logical reasoning capabilities of LMs, achieving a remarkable 72.2% accuracy on the MATH dataset and outperforming the PAL/PoT method by 38.8%. Our work not only sets new state-of-the-art but also paves the way toward more sophisticated AI reasoning methods. The code is available at https://github.com/iiis-ai/cumulative-reasoning.
Translate-test is a popular technique to improve the performance of multilingual language models. This approach works by translating the input into English using an external machine translation system, and running inference over the translated input. However, these improvements can be attributed to the use of a separate translation system, which is typically trained on large amounts of parallel data not seen by the language model. In this work, we introduce a new approach called self-translate, which overcomes the need of an external translation system by leveraging the few-shot translation capabilities of multilingual language models. Experiments over 5 tasks show that self-translate consistently outperforms direct inference, demonstrating that language models are unable to leverage their full multilingual potential when prompted in non-English languages. Our code is available at https://github.com/juletx/self-translate.
We investigate how to elicit compositional generalization capabilities in large language models (LLMs). Compositional generalization empowers LLMs to solve complex problems by combining foundational skills, a critical reasoning ability akin to human intelligence. However, even the most advanced LLMs currently struggle with this form of reasoning. We examine this problem within the framework of in-context learning and find that demonstrating both foundational skills and compositional examples grounded in these skills within the same prompt context is crucial. We refer to this prompt structure as skills-in-context (SKiC). With as few as two exemplars, this in-context learning structure enables LLMs to tackle more challenging problems requiring innovative skill combinations, achieving near-perfect systematic generalization across a broad range of tasks. Intriguingly, SKiC also unlocks the latent potential of LLMs, allowing them to more actively utilize pre-existing internal skills acquired during earlier pretraining stages to solve complex reasoning problems. The SKiC structure is robust across different skill constructions and exemplar choices and demonstrates strong transferability to new tasks. Finally, inspired by our in-context learning study, we show that fine-tuning LLMs with SKiC-style data can elicit zero-shot weak-to-strong generalization, enabling the models to solve much harder problems directly with standard prompting.
In this work, we develop and release Llama 2, a collection of pretrained and fine-tuned large language models (LLMs) ranging in scale from 7 billion to 70 billion parameters. Our fine-tuned LLMs, called Llama 2-Chat, are optimized for dialogue use cases. Our models outperform open-source chat models on most benchmarks we tested, and based on our human evaluations for helpfulness and safety, may be a suitable substitute for closed-source models. We provide a detailed description of our approach to fine-tuning and safety improvements of Llama 2-Chat in order to enable the community to build on our work and contribute to the responsible development of LLMs.
We present the Chinese Elementary School Math Word Problems (CMATH) dataset, comprising 1.7k elementary school-level math word problems with detailed annotations, source from actual Chinese workbooks and exams. This dataset aims to provide a benchmark tool for assessing the following question: to what grade level of elementary school math do the abilities of popular large language models (LLMs) correspond? We evaluate a variety of popular LLMs, including both commercial and open-source options, and discover that only GPT-4 achieves success (accuracy $\geq$ 60\%) across all six elementary school grades, while other models falter at different grade levels. Furthermore, we assess the robustness of several top-performing LLMs by augmenting the original problems in the CMATH dataset with distracting information. Our findings reveal that GPT-4 is able to maintains robustness, while other model fail. We anticipate that our study will expose limitations in LLMs' arithmetic and reasoning capabilities, and promote their ongoing development and advancement.
The art of mathematical reasoning stands as a fundamental pillar of intellectual progress and is a central catalyst in cultivating human ingenuity. Researchers have recently published a plethora of works centered around the task of solving Math Word Problems (MWP) $-$ a crucial stride towards general AI. These existing models are susceptible to dependency on shallow heuristics and spurious correlations to derive the solution expressions. In order to ameliorate this issue, in this paper, we propose a framework for MWP solvers based on the generation of linguistic variants of the problem text. The approach involves solving each of the variant problems and electing the predicted expression with the majority of the votes. We use DeBERTa (Decoding-enhanced BERT with disentangled attention) as the encoder to leverage its rich textual representations and enhanced mask decoder to construct the solution expressions. Furthermore, we introduce a challenging dataset, $\mathrm{P\small{ARA}\normalsize{MAWPS}}$, consisting of paraphrased, adversarial, and inverse variants of selectively sampled MWPs from the benchmark $\mathrm{M\small{AWPS}}$ dataset. We extensively experiment on this dataset along with other benchmark datasets using some baseline MWP solver models. We show that training on linguistic variants of problem statements and voting on candidate predictions improve the mathematical reasoning and robustness of the model. We make our code and data publicly available.
Mathematical reasoning is regarded as a necessary ability for Language Models (LMs). Recent works demonstrate large LMs' impressive performance in solving math problems. The success is attributed to their Chain-of-Thought (CoT) reasoning abilities, i.e., the ability to decompose complex questions into step-by-step reasoning chains, but such ability seems only to emerge from models with abundant parameters. This work investigates how to incorporate relatively small LMs with the capabilities of multi-step reasoning. We propose to inject such abilities by continually pre-training LMs on a synthetic dataset MsAT which is composed of Multi-step Arithmetic Tasks. Our experiments on four math word problem datasets show the effectiveness of the proposed method in enhancing LMs' math reasoning abilities.
Employing Large Language Models (LLMs) to address mathematical problems is an intriguing research endeavor, considering the abundance of math problems expressed in natural language across numerous science and engineering fields. LLMs, with their generalized ability, are used as a foundation model to build AI agents for different tasks. In this paper, we study the effectiveness of utilizing LLM agents to solve math problems through conversations. We propose MathChat, a conversational problem-solving framework designed for math problems. MathChat consists of an LLM agent and a user proxy agent which is responsible for tool execution and additional guidance. This synergy facilitates a collaborative problem-solving process, where the agents engage in a dialogue to solve the problems. We perform evaluation on difficult high school competition problems from the MATH dataset. Utilizing Python, we show that MathChat can further improve previous tool-using prompting methods by 6%.
In recent years, large language models have greatly improved in their ability to perform complex multi-step reasoning. However, even state-of-the-art models still regularly produce logical mistakes. To train more reliable models, we can turn either to outcome supervision, which provides feedback for a final result, or process supervision, which provides feedback for each intermediate reasoning step. Given the importance of training reliable models, and given the high cost of human feedback, it is important to carefully compare the both methods. Recent work has already begun this comparison, but many questions still remain. We conduct our own investigation, finding that process supervision significantly outperforms outcome supervision for training models to solve problems from the challenging MATH dataset. Our process-supervised model solves 78% of problems from a representative subset of the MATH test set. Additionally, we show that active learning significantly improves the efficacy of process supervision. To support related research, we also release PRM800K, the complete dataset of 800,000 step-level human feedback labels used to train our best reward model.
Recent developments in large pre-trained language models have enabled unprecedented performance on a variety of downstream tasks. Achieving best performance with these models often leverages in-context learning, where a model performs a (possibly new) task given one or more examples. However, recent work has shown that the choice of examples can have a large impact on task performance and that finding an optimal set of examples is non-trivial. While there are many existing methods for selecting in-context examples, they generally score examples independently, ignoring the dependency between them and the order in which they are provided to the model. In this work, we propose Retrieval for In-Context Learning (RetICL), a learnable method for modeling and optimally selecting examples sequentially for in-context learning. We frame the problem of sequential example selection as a Markov decision process and train an example retriever using reinforcement learning. We evaluate RetICL on math word problem solving and scientific question answering tasks and show that it consistently outperforms or matches heuristic and learnable baselines. We also use case studies to show that RetICL implicitly learns representations of problem solving strategies.
In this paper, we present a novel approach for distilling math word problem solving capabilities from large language models (LLMs) into smaller, more efficient student models. Our approach is designed to consider the student model's weaknesses and foster a tailored learning experience by generating targeted exercises aligned with educational science principles, such as knowledge tracing and personalized learning. Concretely, we let GPT-3 be a math tutor and run two steps iteratively: 1) assessing the student model's current learning status on a GPT-generated exercise book, and 2) improving the student model by training it with tailored exercise samples generated by GPT-3. Experimental results reveal that our approach outperforms LLMs (e.g., GPT-3 and PaLM) in accuracy across three distinct benchmarks while employing significantly fewer parameters. Furthermore, we provide a comprehensive analysis of the various components within our methodology to substantiate their efficacy.
Chain-of-Thought (CoT) and Program-Aided Language Models (PAL) represent two distinct reasoning methods, each with its own strengths. CoT employs natural language, offering flexibility and interpretability, while PAL utilizes programming language, yielding more structured and rigorous logic. We introduce a model selection method to combine the best of both worlds by employing a large language model (LLM) to dynamically select between them. Our theoretical analysis underscores the feasibility of this method, which is further corroborated by empirical results. Our proposed method demonstrates significant performance improvements across eight reasoning datasets with Codex, ChatGPT, and GPT-4. Additionally, our method is complementary to self-consistency; when integrated, it can further enhance performance while significantly reducing computation costs. Moreover, we achieve new state-of-the-art results on GSM8K and SVAMP, with respective accuracies of 96.8% and 93.7%. Our code, data and prompts are available at https://github.com/XuZhao0/Model-Selection-Reasoning
The performance of Large Language Models (LLMs) in reasoning tasks depends heavily on prompt design, with Chain-of-Thought (CoT) and self-consistency being critical methods that enhance this ability. However, these methods do not fully exploit the answers generated by the LLM to guide subsequent responses. This paper proposes a new prompting method, named Progressive-Hint Prompting (PHP), that enables automatic multiple interactions between users and LLMs by using previously generated answers as hints to progressively guide toward the correct answers. PHP is orthogonal to CoT and self-consistency, making it easy to combine with state-of-the-art techniques to further improve performance. We conducted extensive and comprehensive experiments on seven benchmarks. The results show that PHP significantly improves accuracy while remaining highly efficient. For instance, with text-davinci-003, we observed a 4.2% improvement on GSM8K with greedy decoding compared to Complex CoT, and a 46.17% reduction in sample paths with self-consistency. With GPT-4 and PHP, we achieve state-of-the-art performances on SVAMP (89.1% -> 91.9%), GSM8K (92% -> 95.5%), AQuA (76.4% -> 79.9%) and MATH (50.3% -> 53.9%).
Artificial intelligence (AI) researchers have been developing and refining large language models (LLMs) that exhibit remarkable capabilities across a variety of domains and tasks, challenging our understanding of learning and cognition. The latest model developed by OpenAI, GPT-4, was trained using an unprecedented scale of compute and data. In this paper, we report on our investigation of an early version of GPT-4, when it was still in active development by OpenAI. We contend that (this early version of) GPT-4 is part of a new cohort of LLMs (along with ChatGPT and Google's PaLM for example) that exhibit more general intelligence than previous AI models. We discuss the rising capabilities and implications of these models. We demonstrate that, beyond its mastery of language, GPT-4 can solve novel and difficult tasks that span mathematics, coding, vision, medicine, law, psychology and more, without needing any special prompting. Moreover, in all of these tasks, GPT-4's performance is strikingly close to human-level performance, and often vastly surpasses prior models such as ChatGPT. Given the breadth and depth of GPT-4's capabilities, we believe that it could reasonably be viewed as an early (yet still incomplete) version of an artificial general intelligence (AGI) system. In our exploration of GPT-4, we put special emphasis on discovering its limitations, and we discuss the challenges ahead for advancing towards deeper and more comprehensive versions of AGI, including the possible need for pursuing a new paradigm that moves beyond next-word prediction. We conclude with reflections on societal influences of the recent technological leap and future research directions.
We introduce LLaMA, a collection of foundation language models ranging from 7B to 65B parameters. We train our models on trillions of tokens, and show that it is possible to train state-of-the-art models using publicly available datasets exclusively, without resorting to proprietary and inaccessible datasets. In particular, LLaMA-13B outperforms GPT-3 (175B) on most benchmarks, and LLaMA-65B is competitive with the best models, Chinchilla-70B and PaLM-540B. We release all our models to the research community.
Socratic questioning is an educational method that allows students to discover answers to complex problems by asking them a series of thoughtful questions. Generation of didactically sound questions is challenging, requiring understanding of the reasoning process involved in the problem. We hypothesize that such questioning strategy can not only enhance the human performance, but also assist the math word problem (MWP) solvers. In this work, we explore the ability of large language models (LMs) in generating sequential questions for guiding math word problem-solving. We propose various guided question generation schemes based on input conditioning and reinforcement learning. On both automatic and human quality evaluations, we find that LMs constrained with desirable question properties generate superior questions and improve the overall performance of a math word problem solver. We conduct a preliminary user study to examine the potential value of such question generation models in the education domain. Results suggest that the difficulty level of problems plays an important role in determining whether questioning improves or hinders human performance. We discuss the future of using such questioning strategies in education.
Large language models (LLMs) have recently demonstrated an impressive ability to perform arithmetic and symbolic reasoning tasks, when provided with a few examples at test time ("few-shot prompting"). Much of this success can be attributed to prompting methods such as "chain-of-thought'', which employ LLMs for both understanding the problem description by decomposing it into steps, as well as solving each step of the problem. While LLMs seem to be adept at this sort of step-by-step decomposition, LLMs often make logical and arithmetic mistakes in the solution part, even when the problem is decomposed correctly. In this paper, we present Program-Aided Language models (PAL): a novel approach that uses the LLM to read natural language problems and generate programs as the intermediate reasoning steps, but offloads the solution step to a runtime such as a Python interpreter. With PAL, decomposing the natural language problem into runnable steps remains the only learning task for the LLM, while solving is delegated to the interpreter. We demonstrate this synergy between a neural LLM and a symbolic interpreter across 13 mathematical, symbolic, and algorithmic reasoning tasks from BIG-Bench Hard and other benchmarks. In all these natural language reasoning tasks, generating code using an LLM and reasoning using a Python interpreter leads to more accurate results than much larger models. For example, PAL using Codex achieves state-of-the-art few-shot accuracy on the GSM8K benchmark of math word problems, surpassing PaLM-540B which uses chain-of-thought by absolute 15% top-1. Our code and data are publicly available at http://reasonwithpal.com/ .
Information overload is a major obstacle to scientific progress. The explosive growth in scientific literature and data has made it ever harder to discover useful insights in a large mass of information. Today scientific knowledge is accessed through search engines, but they are unable to organize scientific knowledge alone. In this paper we introduce Galactica: a large language model that can store, combine and reason about scientific knowledge. We train on a large scientific corpus of papers, reference material, knowledge bases and many other sources. We outperform existing models on a range of scientific tasks. On technical knowledge probes such as LaTeX equations, Galactica outperforms the latest GPT-3 by 68.2% versus 49.0%. Galactica also performs well on reasoning, outperforming Chinchilla on mathematical MMLU by 41.3% to 35.7%, and PaLM 540B on MATH with a score of 20.4% versus 8.8%. It also sets a new state-of-the-art on downstream tasks such as PubMedQA and MedMCQA dev of 77.6% and 52.9%. And despite not being trained on a general corpus, Galactica outperforms BLOOM and OPT-175B on BIG-bench. We believe these results demonstrate the potential for language models as a new interface for science. We open source the model for the benefit of the scientific community.
Math word problem solver requires both precise relation reasoning about quantities in the text and reliable generation for the diverse equation. Current sequence-to-tree or relation extraction methods regard this only from a fixed view, struggling to simultaneously handle complex semantics and diverse equations. However, human solving naturally involves two consistent reasoning views: top-down and bottom-up, just as math equations also can be expressed in multiple equivalent forms: pre-order and post-order. We propose a multi-view consistent contrastive learning for a more complete semantics-to-equation mapping. The entire process is decoupled into two independent but consistent views: top-down decomposition and bottom-up construction, and the two reasoning views are aligned in multi-granularity for consistency, enhancing global generation and precise reasoning. Experiments on multiple datasets across two languages show our approach significantly outperforms the existing baselines, especially on complex problems. We also show after consistent alignment, multi-view can absorb the merits of both views and generate more diverse results consistent with the mathematical laws.
Numerical reasoning over text is a challenging task of Artificial Intelligence (AI), requiring reading comprehension and numerical reasoning abilities. Previous approaches use numerical reasoning programs to represent the reasoning process. However, most works do not separate the generation of operators and operands, which are key components of a numerical reasoning program, thus limiting their ability to generate such programs for complicated tasks. In this paper, we introduce the numEricaL reASoning with adapTive symbolIc Compiler (ELASTIC) model, which is constituted of the RoBERTa as the Encoder and a Compiler with four modules: Reasoning Manager, Operator Generator, Operands Generator, and Memory Register. ELASTIC is robust when conducting complicated reasoning. Also, it is domain agnostic by supporting the expansion of diverse operators without caring about the number of operands it contains. Experiments show that ELASTIC achieves 68.96 and 65.21 of execution accuracy and program accuracy on the FinQA dataset and 83.00 program accuracy on the MathQA dataset, outperforming previous state-of-the-art models significantly.