Mathematical symbol definition extraction is important for improving scholarly reading interfaces and scholarly information extraction (IE). However, the task poses several challenges: math symbols are difficult to process as they are not composed of natural language morphemes; and scholarly papers often contain sentences that require resolving complex coordinate structures. We present SymDef, an English language dataset of 5,927 sentences from full-text scientific papers where each sentence is annotated with all mathematical symbols linked with their corresponding definitions. This dataset focuses specifically on complex coordination structures such as "respectively" constructions, which often contain overlapping definition spans. We also introduce a new definition extraction method that masks mathematical symbols, creates a copy of each sentence for each symbol, specifies a target symbol, and predicts its corresponding definition spans using slot filling. Our experiments show that our definition extraction model significantly outperforms RoBERTa and other strong IE baseline systems by 10.9 points with a macro F1 score of 84.82. With our dataset and model, we can detect complex definitions in scholarly documents to make scientific writing more readable.
While automatic dialogue tutors hold great potential in making education personalized and more accessible, research on such systems has been hampered by a lack of sufficiently large and high-quality datasets. Collecting such datasets remains challenging, as recording tutoring sessions raises privacy concerns and crowdsourcing leads to insufficient data quality. To address this, we propose a framework to generate such dialogues by pairing human teachers with a Large Language Model (LLM) prompted to represent common student errors. We describe how we use this framework to collect MathDial, a dataset of 3k one-to-one teacher-student tutoring dialogues grounded in multi-step math reasoning problems. While models like GPT-3 are good problem solvers, they fail at tutoring because they generate factually incorrect feedback or are prone to revealing solutions to students too early. To overcome this, we let teachers provide learning opportunities to students by guiding them using various scaffolding questions according to a taxonomy of teacher moves. We demonstrate MathDial and its extensive annotations can be used to finetune models to be more effective tutors (and not just solvers). We confirm this by automatic and human evaluation, notably in an interactive setting that measures the trade-off between student solving success and telling solutions. The dataset is released publicly.
Large language models (LLMs) have shown remarkable reasoning capabilities, especially when prompted to generate intermediate reasoning steps (e.g., Chain-of-Thought, CoT). However, LLMs can still struggle with problems that are easy for humans, such as generating action plans for executing tasks in a given environment, or performing complex math, logical, and commonsense reasoning. The deficiency stems from the key fact that LLMs lack an internal $\textit{world model}$ to predict the world $\textit{state}$ (e.g., environment status, intermediate variable values) and simulate long-term outcomes of actions. This prevents LLMs from performing deliberate planning akin to human brains, which involves exploring alternative reasoning paths, anticipating future states and rewards, and iteratively refining existing reasoning steps. To overcome the limitations, we propose a new LLM reasoning framework, $\underline{R}$easoning vi$\underline{a}$ $\underline{P}$lanning $\textbf{(RAP)}$. RAP repurposes the LLM as both a world model and a reasoning agent, and incorporates a principled planning algorithm (based on Monto Carlo Tree Search) for strategic exploration in the vast reasoning space. During reasoning, the LLM (as agent) incrementally builds a reasoning tree under the guidance of the LLM (as world model) and task-specific rewards, and obtains a high-reward reasoning path efficiently with a proper balance between exploration $\textit{vs.}$ exploitation. We apply RAP to a variety of challenging reasoning problems including plan generation, math reasoning, and logical inference. Empirical results on these tasks demonstrate the superiority of RAP over various strong baselines, including CoT and least-to-most prompting with self-consistency. RAP on LLAMA-33B surpasses CoT on GPT-4 with 33% relative improvement in a plan generation setting.
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 nature, the behaviors of many complex systems can be described by parsimonious math equations. Automatically distilling these equations from limited data is cast as a symbolic regression process which hitherto remains a grand challenge. Keen efforts in recent years have been placed on tackling this issue and demonstrated success in symbolic regression. However, there still exist bottlenecks that current methods struggle to break when the discrete search space tends toward infinity and especially when the underlying math formula is intricate. To this end, we propose a novel Reinforcement Symbolic Regression Machine (RSRM) that masters the capability of uncovering complex math equations from only scarce data. The RSRM model is composed of three key modules: (1) a Monte Carlo tree search (MCTS) agent that explores optimal math expression trees consisting of pre-defined math operators and variables, (2) a Double Q-learning block that helps reduce the feasible search space of MCTS via properly understanding the distribution of reward, and (3) a modulated sub-tree discovery block that heuristically learns and defines new math operators to improve representation ability of math expression trees. Biding of these modules yields the state-of-the-art performance of RSRM in symbolic regression as demonstrated by multiple sets of benchmark examples. The RSRM model shows clear superiority over several representative baseline models.
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.
Although large language models (LLMs) have achieved excellent performance in a variety of evaluation benchmarks, they still struggle in complex reasoning tasks which require specific knowledge and multi-hop reasoning. To improve the reasoning abilities, we propose ChatCoT, a tool-augmented chain-of-thought reasoning framework for chat-based LLMs (e.g., ChatGPT). In ChatCoT, we model the chain-of-thought (CoT) reasoning as multi-turn conversations, to utilize tools in a more natural way through chatting. At each turn, LLMs can either interact with tools or perform the reasoning. Our approach can effectively leverage the multi-turn conversation ability of chat-based LLMs, and integrate the thought chain following and tools manipulation in a unified way. Specially, we initialize the early turns of the conversation by the knowledge about tools, tasks, and reasoning format, and propose an iterative tool-augmented reasoning step to perform step-by-step tool-augmented reasoning. The experiment results on two complex reasoning datasets (MATH and HotpotQA) have shown the effectiveness of ChatCoT on complex reasoning tasks, achieving a 7.9% relative improvement over the state-of-the-art baseline. Our code and data are available at: \url{https://github.com/RUCAIBOX/ChatCoT}.
Large Language Models (LLMs) have made significant progress in utilizing tools, but their ability is limited by API availability and the instability of implicit reasoning, particularly when both planning and execution are involved. To overcome these limitations, we propose CREATOR, a novel framework that enables LLMs to create their own tools using documentation and code realization. CREATOR disentangles abstract tool creation and concrete decision execution, resulting in improved performance. We evaluate CREATOR on MATH and TabMWP benchmarks, respectively consisting of challenging math competition problems and diverse tabular contents. Remarkably, CREATOR outperforms existing chain-of-thought, program-of-thought, and tool-using baselines. Additionally, we introduce the Creation Challenge dataset, featuring 2K diverse questions, to emphasize the necessity and benefits of LLMs' tool creation ability. Further research demonstrates that leveraging LLMs as tool creators facilitates knowledge transfer, and LLMs exhibit varying levels of tool creation abilities, enabling them to adapt to diverse situations. The tool creation ability revolutionizes the LLM's problem-solving paradigm, driving us closer to the next frontier of artificial intelligence. All the codes and data are released.
This demo paper presents the first tool to annotate the reuse of text, images, and mathematical formulae in a document pair -- TEIMMA. Annotating content reuse is particularly useful to develop plagiarism detection algorithms. Real-world content reuse is often obfuscated, which makes it challenging to identify such cases. TEIMMA allows entering the obfuscation type to enable novel classifications for confirmed cases of plagiarism. It enables recording different reuse types for text, images, and mathematical formulae in HTML and supports users by visualizing the content reuse in a document pair using similarity detection methods for text and math.
Large language models (LLMs) such as ChatGPT and GPT-4 have shown impressive performance in complex reasoning tasks. However, it is difficult to know whether the models are reasoning based on deep understandings of truth and logic, or leveraging their memorized patterns in a relatively superficial way. In this work, we explore testing LLMs' reasoning by engaging with them in a debate-like conversation, where given a question, the LLM and the user need to discuss to make the correct decision starting from opposing arguments. Upon mitigating the Clever Hans effect, our task requires the LLM to not only achieve the correct answer on its own, but also be able to hold and defend its belief instead of blindly believing or getting misled by the user's (invalid) arguments and critiques, thus testing in greater depth whether the LLM grasps the essence of the reasoning required to solve the problem. Across a range of complex reasoning benchmarks spanning math, commonsense, logic and BIG-Bench tasks, we find that despite their impressive performance as reported in existing work on generating correct step-by-step solutions in the beginning, LLMs like ChatGPT cannot maintain their beliefs in truth for a significant portion of examples when challenged by oftentimes absurdly invalid arguments. Our work points to danger zones of model alignment, and also suggests more careful treatments and interpretations of the recent findings that LLMs can improve their responses based on feedback.
The recent LLMs like GPT-4 and PaLM-2 have made tremendous progress in solving fundamental math problems like GSM8K by achieving over 90% accuracy. However, their capabilities to solve more challenging math problems which require domain-specific knowledge (i.e. theorem) have yet to be investigated. In this paper, we introduce TheoremQA, the first theorem-driven question-answering dataset designed to evaluate AI models' capabilities to apply theorems to solve challenging science problems. TheoremQA is curated by domain experts containing 800 high-quality questions covering 350 theorems (e.g. Taylor's theorem, Lagrange's theorem, Huffman coding, Quantum Theorem, Elasticity Theorem, etc) from Math, Physics, EE&CS, and Finance. We evaluate a wide spectrum of 16 large language and code models with different prompting strategies like Chain-of-Thoughts and Program-of-Thoughts. We found that GPT-4's capabilities to solve these problems are unparalleled, achieving an accuracy of 51% with Program-of-Thoughts Prompting. All the existing open-sourced models are below 15%, barely surpassing the random-guess baseline. Given the diversity and broad coverage of TheoremQA, we believe it can be used as a better benchmark to evaluate LLMs' capabilities to solve challenging science problems. The data and code are released in https://github.com/wenhuchen/TheoremQA.
Prompting becomes an increasingly important research topic for better utilization of LLMs. Although simple prompting performs well on single-step questions, it cannot permanently activate the correct knowledge path for multi-step reasoning tasks. The chain of thought (CoT), which often contains zero-shot CoT and few-shot CoT, is a recently developed prompting method that can explain the reasoning process to the LLM and outperforms simple prompting in three challenging reasoning tasks, including arithmetic, symbolic, and commonsense reasoning. Inspired by zero-shot CoT, and further extending the zero-shot ability, this paper proposes a novel hint of thought (HoT) prompting with explain-ability and zero-shot generalization. It is decomposed into three steps: explainable sub-questions, logical reasoning, and answering. Such three steps are sequentially ordered in step-by-step hints, which can be easily adjusted and explained to different tasks. Finally, experimental results demonstrate that our HoT prompting has a significant advantage on the zero-shot reasoning task compared to existing zero-shot CoT. We did zero-shot experiments on math tasks like GSM8K, ADDSUB, AQUA, SVAMP, and commonsense tasks such as StrategyQA. In particular, the accuracy of the proposed HoT prompting is improved with GSM8K from 40.50% to 70.65%, with AQUA from 31.9% to 46.4%, with SVAMP from 63.7% to 76.9%, and with ADDSUB from 74.7% to 87.34%, respectively, which even defeats the competitive PoT approach on GSM8k, AQUA, and SVAMP.
Recent years witnessed an increase in the amount of research on the task of Question Difficulty Estimation from Text QDET with Natural Language Processing (NLP) techniques, with the goal of targeting the limitations of traditional approaches to question calibration. However, almost the entirety of previous research focused on single silos, without performing quantitative comparisons between different models or across datasets from different educational domains. In this work, we aim at filling this gap, by quantitatively analyzing several approaches proposed in previous research, and comparing their performance on three publicly available real world datasets containing questions of different types from different educational domains. Specifically, we consider reading comprehension Multiple Choice Questions (MCQs), science MCQs, and math questions. We find that Transformer based models are the best performing across different educational domains, with DistilBERT performing almost as well as BERT, and that they outperform other approaches even on smaller datasets. As for the other models, the hybrid ones often outperform the ones based on a single type of features, the ones based on linguistic features perform well on reading comprehension questions, while frequency based features (TF-IDF) and word embeddings (word2vec) perform better in domain knowledge assessment.
Large language models (LLMs) pretrained on vast source code have achieved prominent progress in code intelligence. However, existing code LLMs have two main limitations in terms of architecture and pretraining tasks. First, they often adopt a specific architecture (encoder-only or decoder-only) or rely on a unified encoder-decoder network for different downstream tasks. The former paradigm is limited by inflexibility in applications while in the latter, the model is treated as a single system for all tasks, leading to suboptimal performance on a subset of tasks. Secondly, they often employ a limited set of pretraining objectives which might not be relevant to some downstream tasks and hence result in substantial performance degrade. To address these limitations, we propose ``CodeT5+'', a family of encoder-decoder LLMs for code in which component modules can be flexibly combined to suit a wide range of downstream code tasks. Such flexibility is enabled by our proposed mixture of pretraining objectives to mitigate the pretrain-finetune discrepancy. These objectives cover span denoising, contrastive learning, text-code matching, and causal LM pretraining tasks, on both unimodal and bimodal multilingual code corpora. Furthermore, we propose to initialize CodeT5+ with frozen off-the-shelf LLMs without training from scratch to efficiently scale up our models, and explore instruction-tuning to align with natural language instructions. We extensively evaluate CodeT5+ on over 20 code-related benchmarks in different settings, including zero-shot, finetuning, and instruction-tuning. We observe state-of-the-art (SoTA) model performance on various code-related tasks, such as code generation and completion, math programming, and text-to-code retrieval tasks. Particularly, our instruction-tuned CodeT5+ 16B achieves new SoTA results on HumanEval code generation task against other open code LLMs.
The field of Math-NLP has witnessed significant growth in recent years, motivated by the desire to expand LLM performance to the learning of non-linguistic notions (numerals, and subsequently, arithmetic reasoning). However, non-linguistic skill injection typically comes at a cost for LLMs: it leads to catastrophic forgetting of core linguistic skills, a consequence that often remains unaddressed in the literature. As Math-NLP has been able to create LLMs that can closely approximate the mathematical skills of a grade-schooler or the arithmetic reasoning skills of a calculator, the practicality of these models fail if they concomitantly shed their linguistic capabilities. In this work, we take a closer look into the phenomena of catastrophic forgetting as it pertains to LLMs and subsequently offer a novel framework for non-linguistic skill injection for LLMs based on information theoretic interventions and skill-specific losses that enable the learning of strict arithmetic reasoning. Our model outperforms the state-of-the-art both on injected non-linguistic skills and on linguistic knowledge retention, and does so with a fraction of the non-linguistic training data (1/4) and zero additional synthetic linguistic training data.
We consider the well-studied Robust $(k, z)$-Clustering problem, which generalizes the classic $k$-Median, $k$-Means, and $k$-Center problems. Given a constant $z\ge 1$, the input to Robust $(k, z)$-Clustering is a set $P$ of $n$ weighted points in a metric space $(M,\delta)$ and a positive integer $k$. Further, each point belongs to one (or more) of the $m$ many different groups $S_1,S_2,\ldots,S_m$. Our goal is to find a set $X$ of $k$ centers such that $\max_{i \in [m]} \sum_{p \in S_i} w(p) \delta(p,X)^z$ is minimized. This problem arises in the domains of robust optimization [Anthony, Goyal, Gupta, Nagarajan, Math. Oper. Res. 2010] and in algorithmic fairness. For polynomial time computation, an approximation factor of $O(\log m/\log\log m)$ is known [Makarychev, Vakilian, COLT $2021$], which is tight under a plausible complexity assumption even in the line metrics. For FPT time, there is a $(3^z+\epsilon)$-approximation algorithm, which is tight under GAP-ETH [Goyal, Jaiswal, Inf. Proc. Letters, 2023]. Motivated by the tight lower bounds for general discrete metrics, we focus on \emph{geometric} spaces such as the (discrete) high-dimensional Euclidean setting and metrics of low doubling dimension, which play an important role in data analysis applications. First, for a universal constant $\eta_0 >0.0006$, we devise a $3^z(1-\eta_{0})$-factor FPT approximation algorithm for discrete high-dimensional Euclidean spaces thereby bypassing the lower bound for general metrics. We complement this result by showing that even the special case of $k$-Center in dimension $\Theta(\log n)$ is $(\sqrt{3/2}- o(1))$-hard to approximate for FPT algorithms. Finally, we complete the FPT approximation landscape by designing an FPT $(1+\epsilon)$-approximation scheme (EPAS) for the metric of sub-logarithmic doubling dimension.
Automated feedback as students answer open-ended math questions has significant potential in improving learning outcomes at large scale. A key part of automated feedback systems is an error classification component, which identifies student errors and enables appropriate, predefined feedback to be deployed. Most existing approaches to error classification use a rule-based method, which has limited capacity to generalize. Existing data-driven methods avoid these limitations but specifically require mathematical expressions in student responses to be parsed into syntax trees. This requirement is itself a limitation, since student responses are not always syntactically valid and cannot be converted into trees. In this work, we introduce a flexible method for error classification using pre-trained large language models. We demonstrate that our method can outperform existing methods in algebra error classification, and is able to classify a larger set of student responses. Additionally, we analyze common classification errors made by our method and discuss limitations of automated error classification.
Large language models (LLMs) have recently been shown to deliver impressive performance in various NLP tasks. To tackle multi-step reasoning tasks, few-shot chain-of-thought (CoT) prompting includes a few manually crafted step-by-step reasoning demonstrations which enable LLMs to explicitly generate reasoning steps and improve their reasoning task accuracy. To eliminate the manual effort, Zero-shot-CoT concatenates the target problem statement with "Let's think step by step" as an input prompt to LLMs. Despite the success of Zero-shot-CoT, it still suffers from three pitfalls: calculation errors, missing-step errors, and semantic misunderstanding errors. To address the missing-step errors, we propose Plan-and-Solve (PS) Prompting. It consists of two components: first, devising a plan to divide the entire task into smaller subtasks, and then carrying out the subtasks according to the plan. To address the calculation errors and improve the quality of generated reasoning steps, we extend PS prompting with more detailed instructions and derive PS+ prompting. We evaluate our proposed prompting strategy on ten datasets across three reasoning problems. The experimental results over GPT-3 show that our proposed zero-shot prompting consistently outperforms Zero-shot-CoT across all datasets by a large margin, is comparable to or exceeds Zero-shot-Program-of-Thought Prompting, and has comparable performance with 8-shot CoT prompting on the math reasoning problem. The code can be found at https://github.com/AGI-Edgerunners/Plan-and-Solve-Prompting.
Existing MWP solvers employ sequence or binary tree to present the solution expression and decode it from given problem description. However, such structures fail to handle the variants that can be derived via mathematical manipulation, e.g., $(a_1+a_2) * a_3$ and $a_1 * a_3+a_2 * a_3$ can both be possible valid solutions for a same problem but formulated as different expression sequences or trees. The multiple solution variants depicting different possible solving procedures for the same input problem would raise two issues: 1) making it hard for the model to learn the mapping function between the input and output spaces effectively, and 2) wrongly indicating \textit{wrong} when evaluating a valid expression variant. To address these issues, we introduce a unified tree structure to present a solution expression, where the elements are permutable and identical for all the expression variants. We propose a novel non-autoregressive solver, named \textit{MWP-NAS}, to parse the problem and deduce the solution expression based on the unified tree. For evaluating the possible expression variants, we design a path-based metric to evaluate the partial accuracy of expressions of a unified tree. The results from extensive experiments conducted on Math23K and MAWPS demonstrate the effectiveness of our proposed MWP-NAS. The codes and checkpoints are available at: \url{https://github.com/mengqunhan/MWP-NAS}.
Large language models have emerged abilities including chain-of-thought to answer math word problems step by step. Solving math word problems not only requires abilities to disassemble problems via chain-of-thought but also needs to calculate arithmetic expressions correctly for each step. To the best of our knowledge, there is no work to focus on evaluating the arithmetic ability of large language models. In this work, we propose an arithmetic dataset MATH 401 to test the latest large language models including GPT-4, ChatGPT, InstrctGPT, Galactica, and LLaMA with various arithmetic expressions and provide a detailed analysis of the ability of large language models. MATH 401 and evaluation codes are released at \url{https://github.com/GanjinZero/math401-llm}.