Pretrained language models have shown superior performance on many natural language processing tasks, yet they still struggle at multi-step formal reasoning tasks like grade school math problems. One key challenge of finetuning them to solve such math reasoning problems is that many existing datasets only contain one reference solution for each problem, despite the fact that there are often alternative solutions resembling different reasoning paths to the final answer. This way, the finetuned models are biased towards the limited reference solutions, which limits their generalization to unseen examples. To mitigate this issue, we propose to let the model perform sampling during training and learn from both self-sampled fully-correct solutions, which yield the correct answer upon execution, and partially-correct solutions, whose intermediate state matches an intermediate state of a known correct solution. We show that our use of self-sampled correct and partially-correct solutions can benefit learning and help guide the sampling process, leading to more efficient exploration of the solution space. Additionally, we explore various training objectives to support learning from multiple solutions per example and find they greatly affect the performance. Experiments on two math reasoning datasets show the effectiveness of our method compared to learning from a single reference solution with MLE, where we improve PASS@100 from 35.5% to 44.5% for GSM8K, and 27.6% to 36.2% PASS@80 for MathQA. Such improvements are also consistent across different model sizes. Our code is available at https://github.com/microsoft/TraceCodegen.
Automatic short answer grading is an important research direction in the exploration of how to use artificial intelligence (AI)-based tools to improve education. Current state-of-the-art approaches use neural language models to create vectorized representations of students responses, followed by classifiers to predict the score. However, these approaches have several key limitations, including i) they use pre-trained language models that are not well-adapted to educational subject domains and/or student-generated text and ii) they almost always train one model per question, ignoring the linkage across a question and result in a significant model storage problem due to the size of advanced language models. In this paper, we study the problem of automatic short answer grading for students' responses to math questions and propose a novel framework for this task. First, we use MathBERT, a variant of the popular language model BERT adapted to mathematical content, as our base model and fine-tune it for the downstream task of student response grading. Second, we use an in-context learning approach that provides scoring examples as input to the language model to provide additional context information and promote generalization to previously unseen questions. We evaluate our framework on a real-world dataset of student responses to open-ended math questions and show that our framework (often significantly) outperforms existing approaches, especially for new questions that are not seen during training.
Recently, deep learning models have made great progress in MWP solving on answer accuracy. However, they are uninterpretable since they mainly rely on shallow heuristics to achieve high performance without understanding and reasoning the grounded math logic. To address this issue and make a step towards interpretable MWP solving, we first construct a high-quality MWP dataset named InterMWP which consists of 11,495 MWPs and annotates interpretable logical formulas based on algebraic knowledge as the grounded linguistic logic of each solution equation. Different from existing MWP datasets, our InterMWP benchmark asks for a solver to not only output the solution expressions but also predict the corresponding logical formulas. We further propose a novel approach with logical prompt and interpretation generation, called LogicSolver. For each MWP, our LogicSolver first retrieves some highly-correlated algebraic knowledge and then passes them to the backbone model as prompts to improve the semantic representations of MWPs. With these improved semantic representations, our LogicSolver generates corresponding solution expressions and interpretable knowledge formulas in accord with the generated solution expressions, simultaneously. Experimental results show that our LogicSolver has stronger logical formula-based interpretability than baselines while achieving higher answer accuracy with the help of logical prompts, simultaneously. The source code and dataset is available at https://github.com/yangzhch6/InterMWP.
In this paper, we revisit the solving bias when evaluating models on current Math Word Problem (MWP) benchmarks. However, current solvers exist solving bias which consists of data bias and learning bias due to biased dataset and improper training strategy. Our experiments verify MWP solvers are easy to be biased by the biased training datasets which do not cover diverse questions for each problem narrative of all MWPs, thus a solver can only learn shallow heuristics rather than deep semantics for understanding problems. Besides, an MWP can be naturally solved by multiple equivalent equations while current datasets take only one of the equivalent equations as ground truth, forcing the model to match the labeled ground truth and ignoring other equivalent equations. Here, we first introduce a novel MWP dataset named UnbiasedMWP which is constructed by varying the grounded expressions in our collected data and annotating them with corresponding multiple new questions manually. Then, to further mitigate learning bias, we propose a Dynamic Target Selection (DTS) Strategy to dynamically select more suitable target expressions according to the longest prefix match between the current model output and candidate equivalent equations which are obtained by applying commutative law during training. The results show that our UnbiasedMWP has significantly fewer biases than its original data and other datasets, posing a promising benchmark for fairly evaluating the solvers' reasoning skills rather than matching nearest neighbors. And the solvers trained with our DTS achieve higher accuracies on multiple MWP benchmarks. The source code is available at https://github.com/yangzhch6/UnbiasedMWP.
Recently, the explosion of online education platforms makes a success in encouraging us to easily access online education resources. However, most of them ignore the integration of massive unstructured information, which inevitably brings the problem of \textit{information overload} and \textit{knowledge trek}. In this paper, we proposed a mathematical knowledge graph named Math-KG, which automatically constructed by the pipeline method with the natural language processing technology to integrate the resources of the mathematics. It is built from the corpora of Baidu Baike, Wikipedia. We implement a simple application system to validate the proposed Math-KG can make contributions on a series of scenes, including faults analysis and semantic search. The system is publicly available at GitHub \footnote{\url{https://github.com/wjn1996/Mathematical-Knowledge-Entity-Recognition}.}.
Existing Math Word Problem (MWP) solvers have achieved high accuracy on benchmark datasets. However, prior works have shown that such solvers do not generalize well and rely on superficial cues to achieve high performance. In this paper, we first conduct experiments to showcase that this behaviour is mainly associated with the limited size and diversity present in existing MWP datasets. Next, we propose several data augmentation techniques broadly categorized into Substitution and Paraphrasing based methods. By deploying these methods we increase the size of existing datasets by five folds. Extensive experiments on two benchmark datasets across three state-of-the-art MWP solvers show that proposed methods increase the generalization and robustness of existing solvers. On average, proposed methods significantly increase the state-of-the-art results by over five percentage points on benchmark datasets. Further, the solvers trained on the augmented dataset perform comparatively better on the challenge test set. We also show the effectiveness of proposed techniques through ablation studies and verify the quality of augmented samples through human evaluation.
Transcripts of teaching episodes can be effective tools to understand discourse patterns in classroom instruction. According to most educational experts, sustained classroom discourse is a critical component of equitable, engaging, and rich learning environments for students. This paper describes the TalkMoves dataset, composed of 567 human-annotated K-12 mathematics lesson transcripts (including entire lessons or portions of lessons) derived from video recordings. The set of transcripts primarily includes in-person lessons with whole-class discussions and/or small group work, as well as some online lessons. All of the transcripts are human-transcribed, segmented by the speaker (teacher or student), and annotated at the sentence level for ten discursive moves based on accountable talk theory. In addition, the transcripts include utterance-level information in the form of dialogue act labels based on the Switchboard Dialog Act Corpus. The dataset can be used by educators, policymakers, and researchers to understand the nature of teacher and student discourse in K-12 math classrooms. Portions of this dataset have been used to develop the TalkMoves application, which provides teachers with automated, immediate, and actionable feedback about their mathematics instruction.
We present SymForce, a library for fast symbolic computation, code generation, and nonlinear optimization for robotics applications like computer vision, motion planning, and controls. SymForce combines the development speed and flexibility of symbolic math with the performance of autogenerated, highly optimized code in C++ or any target runtime language. SymForce provides geometry and camera types, Lie group operations, and branchless singularity handling for creating and analyzing complex symbolic expressions in Python, built on top of SymPy. Generated functions can be integrated as factors into our tangent-space nonlinear optimizer, which is highly optimized for real-time production use. We introduce novel methods to automatically compute tangent-space Jacobians, eliminating the need for bug-prone handwritten derivatives. This workflow enables faster runtime code, faster development time, and fewer lines of handwritten code versus the state-of-the-art. Our experiments demonstrate that our approach can yield order of magnitude speedups on computational tasks core to robotics. Code is available at https://github.com/symforce-org/symforce.
Current approaches to generic segmentation start by creating a hierarchy of nested image partitions and then specifying a segmentation from it. Our first contribution is to describe several ways, most of them new, for specifying segmentations using the hierarchy elements. Then, we consider the best hierarchy-induced segmentation specified by a limited number of hierarchy elements. We focus on a common quality measure for binary segmentations, the Jaccard index (also known as IoU). Optimizing the Jaccard index is highly non-trivial, and yet we propose an efficient approach for doing exactly that. This way we get algorithm-independent upper bounds on the quality of any segmentation created from the hierarchy. We found that the obtainable segmentation quality varies significantly depending on the way that the segments are specified by the hierarchy elements, and that representing a segmentation with only a few hierarchy elements is often possible. (Code is available).
Solving math word problems requires deductive reasoning over the quantities in the text. Various recent research efforts mostly relied on sequence-to-sequence or sequence-to-tree models to generate mathematical expressions without explicitly performing relational reasoning between quantities in the given context. While empirically effective, such approaches typically do not provide explanations for the generated expressions. In this work, we view the task as a complex relation extraction problem, proposing a novel approach that presents explainable deductive reasoning steps to iteratively construct target expressions, where each step involves a primitive operation over two quantities defining their relation. Through extensive experiments on four benchmark datasets, we show that the proposed model significantly outperforms existing strong baselines. We further demonstrate that the deductive procedure not only presents more explainable steps but also enables us to make more accurate predictions on questions that require more complex reasoning.
With the recent success of dense retrieval methods based on bi-encoders, studies have applied this approach to various interesting downstream retrieval tasks with good efficiency and in-domain effectiveness. Recently, we have also seen the presence of dense retrieval models in Math Information Retrieval (MIR) tasks, but the most effective systems remain classic retrieval methods that consider hand-crafted structure features. In this work, we try to combine the best of both worlds:\ a well-defined structure search method for effective formula search and efficient bi-encoder dense retrieval models to capture contextual similarities. Specifically, we have evaluated two representative bi-encoder models for token-level and passage-level dense retrieval on recent MIR tasks. Our results show that bi-encoder models are highly complementary to existing structure search methods, and we are able to advance the state-of-the-art on MIR datasets.
Language models typically need to be trained or finetuned in order to acquire new knowledge, which involves updating their weights. We instead envision language models that can simply read and memorize new data at inference time, thus acquiring new knowledge immediately. In this work, we extend language models with the ability to memorize the internal representations of past inputs. We demonstrate that an approximate kNN lookup into a non-differentiable memory of recent (key, value) pairs improves language modeling across various benchmarks and tasks, including generic webtext (C4), math papers (arXiv), books (PG-19), code (Github), as well as formal theorems (Isabelle). We show that the performance steadily improves when we increase the size of memory up to 262K tokens. On benchmarks including code and mathematics, we find that the model is capable of making use of newly defined functions and theorems during test time.
In this paper, we present an end-to-end joint entity and relation extraction approach based on transformer-based language models. We apply the model to the task of linking mathematical symbols to their descriptions in LaTeX documents. In contrast to existing approaches, which perform entity and relation extraction in sequence, our system incorporates information from relation extraction into entity extraction. This means that the system can be trained even on data sets where only a subset of all valid entity spans is annotated. We provide an extensive evaluation of the proposed system and its strengths and weaknesses. Our approach, which can be scaled dynamically in computational complexity at inference time, produces predictions with high precision and reaches 3rd place in the leaderboard of SemEval-2022 Task 12. For inputs in the domain of physics and math, it achieves high relation extraction macro F1 scores of 95.43% and 79.17%, respectively. The code used for training and evaluating our models is available at: https://github.com/nicpopovic/RE1st
We study the feasibility of a Data Science assistant powered by a sequence-to-sequence transformer by training a new model JuPyT5 on all publicly available Jupyter Notebook GitHub repositories and developing a new metric: Data Science Problems (DSP). DSP is a collection of 1119 problems curated from 306 pedagogical notebooks with 92 dataset dependencies, natural language and Markdown problem descriptions, and assert-based unit tests. These notebooks were designed to test university students' mastery of various Python implementations of Math and Data Science, and we now leverage them to study the ability of JuPyT5 to understand and pass the tests. We analyze the content of DSP, validate its quality, and we find that given 100 sampling attempts JuPyT5 is able to solve 77.5\% of the DSP problems. We further present various ablation and statistical analyses and compare DSP to other recent natural language to code benchmarks.
Reasoning over natural language is a long-standing goal for the research community. However, studies have shown that existing language models are inadequate in reasoning. To address the issue, we present POET, a novel reasoning pre-training paradigm. Through pre-training language models with programs and their execution results, POET empowers language models to harvest the reasoning knowledge possessed by program executors via a data-driven approach. POET is conceptually simple and can be instantiated by different kinds of program executors. In this paper, we showcase two simple instances POET-Math and POET-Logic, in addition to a complex instance, POET-SQL. Experimental results on six benchmarks demonstrate that POET can significantly boost model performance in natural language reasoning, such as numerical reasoning, logical reasoning, and multi-hop reasoning. POET opens a new gate on reasoning-enhancement pre-training, and we hope our analysis would shed light on the future research of reasoning like program executors.
We demonstrate that a neural network pre-trained on text and fine-tuned on code solves mathematics course problems, explains solutions, and generates new questions at a human level. We automatically synthesize programs using few-shot learning and OpenAI's Codex transformer and execute them to solve course problems at 81% automatic accuracy. We curate a new dataset of questions from MIT's largest mathematics courses (Single Variable and Multivariable Calculus, Differential Equations, Introduction to Probability and Statistics, Linear Algebra, and Mathematics for Computer Science) and Columbia University's Computational Linear Algebra. We solve questions from a MATH dataset (on Prealgebra, Algebra, Counting and Probability, Intermediate Algebra, Number Theory, and Precalculus), the latest benchmark of advanced mathematics problems designed to assess mathematical reasoning. We randomly sample questions and generate solutions with multiple modalities, including numbers, equations, and plots. The latest GPT-3 language model pre-trained on text automatically solves only 18.8% of these university questions using zero-shot learning and 30.8% using few-shot learning and the most recent chain of thought prompting. In contrast, program synthesis with few-shot learning using Codex fine-tuned on code generates programs that automatically solve 81% of these questions. Our approach improves the previous state-of-the-art automatic solution accuracy on the benchmark topics from 8.8% to 81.1%. We perform a survey to evaluate the quality and difficulty of generated questions. This work is the first to automatically solve university-level mathematics course questions at a human level and the first work to explain and generate university-level mathematics course questions at scale, a milestone for higher education.
May and Leonard (SIAM J. Appl. Math 1975) introduced a three-species Lotka-Volterra type population model that exhibits heteroclinic cycling. Rather than producing a periodic limit cycle, the trajectory takes longer and longer to complete each "cycle", passing closer and closer to unstable fixed points in which one population dominates and the others approach zero. Aperiodic heteroclinic dynamics have subsequently been studied in ecological systems (side-blotched lizards; colicinogenic E. coli), in the immune system, in neural information processing models ("winnerless competition"), and in models of neural central pattern generators. Yet as May and Leonard observed "Biologically, the behavior (produced by the model) is nonsense. Once it is conceded that the variables represent animals, and therefore cannot fall below unity, it is clear that the system will, after a few cycles, converge on some single population, extinguishing the other two." Here, we explore different ways of introducing discrete stochastic dynamics based on May and Leonard's ODE model, with application to ecological population dynamics, and to a neuromotor central pattern generator system. We study examples of several quantitatively distinct asymptotic behaviors, including total extinction of all species, extinction to a single species, and persistent cyclic dominance with finite mean cycle length.
State-of-the-art language models can match human performance on many tasks, but they still struggle to robustly perform multi-step mathematical reasoning. To diagnose the failures of current models and support research, we introduce GSM8K, a dataset of 8.5K high quality linguistically diverse grade school math word problems. We find that even the largest transformer models fail to achieve high test performance, despite the conceptual simplicity of this problem distribution. To increase performance, we propose training verifiers to judge the correctness of model completions. At test time, we generate many candidate solutions and select the one ranked highest by the verifier. We demonstrate that verification significantly improves performance on GSM8K, and we provide strong empirical evidence that verification scales more effectively with increased data than a finetuning baseline.
Math Word Problem (MWP) solving needs to discover the quantitative relationships over natural language narratives. Recent work shows that existing models memorize procedures from context and rely on shallow heuristics to solve MWPs. In this paper, we look at this issue and argue that the cause is a lack of overall understanding of MWP patterns. We first investigate how a neural network understands patterns only from semantics, and observe that, if the prototype equations are the same, most problems get closer representations and those representations apart from them or close to other prototypes tend to produce wrong solutions. Inspired by it, we propose a contrastive learning approach, where the neural network perceives the divergence of patterns. We collect contrastive examples by converting the prototype equation into a tree and seeking similar tree structures. The solving model is trained with an auxiliary objective on the collected examples, resulting in the representations of problems with similar prototypes being pulled closer. We conduct experiments on the Chinese dataset Math23k and the English dataset MathQA. Our method greatly improves the performance in monolingual and multilingual settings.
Solving symbolic mathematics has always been of in the arena of human ingenuity that needs compositional reasoning and recurrence. However, recent studies have shown that large-scale language models such as transformers are universal and surprisingly can be trained as a sequence-to-sequence task to solve complex mathematical equations. These large transformer models need humongous amounts of training data to generalize to unseen symbolic mathematics problems. In this paper, we present a sample efficient way of solving the symbolic tasks by first pretraining the transformer model with language translation and then fine-tuning the pretrained transformer model to solve the downstream task of symbolic mathematics. We achieve comparable accuracy on the integration task with our pretrained model while using around $1.5$ orders of magnitude less number of training samples with respect to the state-of-the-art deep learning for symbolic mathematics. The test accuracy on differential equation tasks is considerably lower comparing with integration as they need higher order recursions that are not present in language translations. We propose the generalizability of our pretrained language model from Anna Karenina Principle (AKP). We pretrain our model with different pairs of language translations. Our results show language bias in solving symbolic mathematics tasks. Finally, we study the robustness of the fine-tuned model on symbolic math tasks against distribution shift, and our approach generalizes better in distribution shift scenarios for the function integration.