In this article, we tackle the math word problem, namely, automatically answering a mathematical problem according to its textual description. Although recent methods have demonstrated their promising results, most of these methods are based on template-based generation scheme which results in limited generalization capability. To this end, we propose a novel human-like analogical learning method in a recall and learn manner. Our proposed framework is composed of modules of memory, representation, analogy, and reasoning, which are designed to make a new exercise by referring to the exercises learned in the past. Specifically, given a math word problem, the model first retrieves similar questions by a memory module and then encodes the unsolved problem and each retrieved question using a representation module. Moreover, to solve the problem in a way of analogy, an analogy module and a reasoning module with a copy mechanism are proposed to model the interrelationship between the problem and each retrieved question. Extensive experiments on two well-known datasets show the superiority of our proposed algorithm as compared to other state-of-the-art competitors from both overall performance comparison and micro-scope studies.
Standard accuracy metrics have shown that Math Word Problem (MWP) solvers have achieved high performance on benchmark datasets. However, the extent to which existing MWP solvers truly understand language and its relation with numbers is still unclear. In this paper, we generate adversarial attacks to evaluate the robustness of state-of-the-art MWP solvers. We propose two methods Question Reordering and Sentence Paraphrasing to generate adversarial attacks. We conduct experiments across three neural MWP solvers over two benchmark datasets. On average, our attack method is able to reduce the accuracy of MWP solvers by over 40 percentage points on these datasets. Our results demonstrate that existing MWP solvers are sensitive to linguistic variations in the problem text. We verify the validity and quality of generated adversarial examples through human evaluation.
Developing automatic Math Word Problem (MWP) solvers has been an interest of NLP researchers since the 1960s. Over the last few years, there are a growing number of datasets and deep learning-based methods proposed for effectively solving MWPs. However, most existing methods are benchmarked soly on one or two datasets, varying in different configurations, which leads to a lack of unified, standardized, fair, and comprehensive comparison between methods. This paper presents MWPToolkit, the first open-source framework for solving MWPs. In MWPToolkit, we decompose the procedure of existing MWP solvers into multiple core components and decouple their models into highly reusable modules. We also provide a hyper-parameter search function to boost the performance. In total, we implement and compare 17 MWP solvers on 4 widely-used single equation generation benchmarks and 2 multiple equations generation benchmarks. These features enable our MWPToolkit to be suitable for researchers to reproduce advanced baseline models and develop new MWP solvers quickly. Code and documents are available at https://github.com/LYH-YF/MWPToolkit.
Obtaining human-like performance in NLP is often argued to require compositional generalisation. Whether neural networks exhibit this ability is usually studied by training models on highly compositional synthetic data. However, compositionality in natural language is much more complex than the rigid, arithmetic-like version such data adheres to, and artificial compositionality tests thus do not allow us to determine how neural models deal with more realistic forms of compositionality. In this work, we re-instantiate three compositionality tests from the literature and reformulate them for neural machine translation (NMT). Our results highlight that: i) unfavourably, models trained on more data are more compositional; ii) models are sometimes less compositional than expected, but sometimes more, exemplifying that different levels of compositionality are required, and models are not always able to modulate between them correctly; iii) some of the non-compositional behaviours are mistakes, whereas others reflect the natural variation in data. Apart from an empirical study, our work is a call to action: we should rethink the evaluation of compositionality in neural networks and develop benchmarks using real data to evaluate compositionality on natural language, where composing meaning is not as straightforward as doing the math.
Math word problem (MWP) solving faces a dilemma in number representation learning. In order to avoid the number representation issue and reduce the search space of feasible solutions, existing works striving for MWP solving usually replace real numbers with symbolic placeholders to focus on logic reasoning. However, different from common symbolic reasoning tasks like program synthesis and knowledge graph reasoning, MWP solving has extra requirements in numerical reasoning. In other words, instead of the number value itself, it is the reusable numerical property that matters more in numerical reasoning. Therefore, we argue that injecting numerical properties into symbolic placeholders with contextualized representation learning schema can provide a way out of the dilemma in the number representation issue here. In this work, we introduce this idea to the popular pre-training language model (PLM) techniques and build MWP-BERT, an effective contextual number representation PLM. We demonstrate the effectiveness of our MWP-BERT on MWP solving and several MWP-specific understanding tasks on both English and Chinese benchmarks.
We study the dynamics of membrane vesicle motor transport into dendritic spines, which are bulbous intracellular compartments in neurons that play a key role in transmitting signals between neurons. We consider the stochastic analog of the vesicle transport model in [Park and Fai, The Dynamics of Vesicles Driven Into Closed Constrictions by Molecular Motors. Bull. Math. Biol. 82, 141 (2020)]. The stochastic version, which may be considered as an agent-based model, relies mostly on the action of individual myosin motors to produce vesicle motion. To aid in our analysis, we coarse-grain this agent-based model using a master equation combined with a partial differential equation describing the probability of local motor positions. We confirm through convergence studies that the coarse-graining captures the essential features of bistability in velocity (observed in experiments) and waiting-time distributions to switch between steady-state velocities. Interestingly, these results allow us to reformulate the translocation problem in terms of the mean first passage time for a run-and-tumble particle moving on a finite domain with absorbing boundaries at the two ends. We conclude by presenting numerical and analytical calculations of vesicle translocation.
Previous math word problem solvers following the encoder-decoder paradigm fail to explicitly incorporate essential math symbolic constraints, leading to unexplainable and unreasonable predictions. Herein, we propose Neural-Symbolic Solver (NS-Solver) to explicitly and seamlessly incorporate different levels of symbolic constraints by auxiliary tasks. Our NS-Solver consists of a problem reader to encode problems, a programmer to generate symbolic equations, and a symbolic executor to obtain answers. Along with target expression supervision, our solver is also optimized via 4 new auxiliary objectives to enforce different symbolic reasoning: a) self-supervised number prediction task predicting both number quantity and number locations; b) commonsense constant prediction task predicting what prior knowledge (e.g. how many legs a chicken has) is required; c) program consistency checker computing the semantic loss between predicted equation and target equation to ensure reasonable equation mapping; d) duality exploiting task exploiting the quasi duality between symbolic equation generation and problem's part-of-speech generation to enhance the understanding ability of a solver. Besides, to provide a more realistic and challenging benchmark for developing a universal and scalable solver, we also construct a new large-scale MWP benchmark CM17K consisting of 4 kinds of MWPs (arithmetic, one-unknown linear, one-unknown non-linear, equation set) with more than 17K samples. Extensive experiments on Math23K and our CM17k demonstrate the superiority of our NS-Solver compared to state-of-the-art methods.
We present ASDiv (Academia Sinica Diverse MWP Dataset), a diverse (in terms of both language patterns and problem types) English math word problem (MWP) corpus for evaluating the capability of various MWP solvers. Existing MWP corpora for studying AI progress remain limited either in language usage patterns or in problem types. We thus present a new English MWP corpus with 2,305 MWPs that cover more text patterns and most problem types taught in elementary school. Each MWP is annotated with its problem type and grade level (for indicating the level of difficulty). Furthermore, we propose a metric to measure the lexicon usage diversity of a given MWP corpus, and demonstrate that ASDiv is more diverse than existing corpora. Experiments show that our proposed corpus reflects the true capability of MWP solvers more faithfully.
This open-source book represents our attempt to make deep learning approachable, teaching readers the concepts, the context, and the code. The entire book is drafted in Jupyter notebooks, seamlessly integrating exposition figures, math, and interactive examples with self-contained code. Our goal is to offer a resource that could (i) be freely available for everyone; (ii) offer sufficient technical depth to provide a starting point on the path to actually becoming an applied machine learning scientist; (iii) include runnable code, showing readers how to solve problems in practice; (iv) allow for rapid updates, both by us and also by the community at large; (v) be complemented by a forum for interactive discussion of technical details and to answer questions.
Since the introduction of the original BERT (i.e., BASE BERT), researchers have developed various customized BERT models with improved performance for specific domains and tasks by exploiting the benefits of transfer learning. Due to the nature of mathematical texts, which often use domain specific vocabulary along with equations and math symbols, we posit that the development of a new BERT model for mathematics would be useful for many mathematical downstream tasks. In this resource paper, we introduce our multi-institutional effort (i.e., two learning platforms and three academic institutions in the US) toward this need: MathBERT, a model created by pre-training the BASE BERT model on a large mathematical corpus ranging from pre-kindergarten (pre-k), to high-school, to college graduate level mathematical content. In addition, we select three general NLP tasks that are often used in mathematics education: prediction of knowledge component, auto-grading open-ended Q&A, and knowledge tracing, to demonstrate the superiority of MathBERT over BASE BERT. Our experiments show that MathBERT outperforms prior best methods by 1.2-22% and BASE BERT by 2-8% on these tasks. In addition, we build a mathematics specific vocabulary 'mathVocab' to train with MathBERT. We discover that MathBERT pre-trained with 'mathVocab' outperforms MathBERT trained with the BASE BERT vocabulary (i.e., 'origVocab'). MathBERT is currently being adopted at the participated leaning platforms: Stride, Inc, a commercial educational resource provider, and ASSISTments.org, a free online educational platform. We release MathBERT for public usage at: https://github.com/tbs17/MathBERT.
In conversation, uptake happens when a speaker builds on the contribution of their interlocutor by, for example, acknowledging, repeating or reformulating what they have said. In education, teachers' uptake of student contributions has been linked to higher student achievement. Yet measuring and improving teachers' uptake at scale is challenging, as existing methods require expensive annotation by experts. We propose a framework for computationally measuring uptake, by (1) releasing a dataset of student-teacher exchanges extracted from US math classroom transcripts annotated for uptake by experts; (2) formalizing uptake as pointwise Jensen-Shannon Divergence (pJSD), estimated via next utterance classification; (3) conducting a linguistically-motivated comparison of different unsupervised measures and (4) correlating these measures with educational outcomes. We find that although repetition captures a significant part of uptake, pJSD outperforms repetition-based baselines, as it is capable of identifying a wider range of uptake phenomena like question answering and reformulation. We apply our uptake measure to three different educational datasets with outcome indicators. Unlike baseline measures, pJSD correlates significantly with instruction quality in all three, providing evidence for its generalizability and for its potential to serve as an automated professional development tool for teachers.
With the recent advancements in deep learning, neural solvers have gained promising results in solving math word problems. However, these SOTA solvers only generate binary expression trees that contain basic arithmetic operators and do not explicitly use the math formulas. As a result, the expression trees they produce are lengthy and uninterpretable because they need to use multiple operators and constants to represent one single formula. In this paper, we propose sequence-to-general tree (S2G) that learns to generate interpretable and executable operation trees where the nodes can be formulas with an arbitrary number of arguments. With nodes now allowed to be formulas, S2G can learn to incorporate mathematical domain knowledge into problem-solving, making the results more interpretable. Experiments show that S2G can achieve a better performance against strong baselines on problems that require domain knowledge.
Automatic math problem solving has recently attracted increasing attention as a long-standing AI benchmark. In this paper, we focus on solving geometric problems, which requires a comprehensive understanding of textual descriptions, visual diagrams, and theorem knowledge. However, the existing methods were highly dependent on handcraft rules and were merely evaluated on small-scale datasets. Therefore, we propose a Geometric Question Answering dataset GeoQA, containing 4,998 geometric problems with corresponding annotated programs, which illustrate the solving process of the given problems. Compared with another publicly available dataset GeoS, GeoQA is 25 times larger, in which the program annotations can provide a practical testbed for future research on explicit and explainable numerical reasoning. Moreover, we introduce a Neural Geometric Solver (NGS) to address geometric problems by comprehensively parsing multimodal information and generating interpretable programs. We further add multiple self-supervised auxiliary tasks on NGS to enhance cross-modal semantic representation. Extensive experiments on GeoQA validate the effectiveness of our proposed NGS and auxiliary tasks. However, the results are still significantly lower than human performance, which leaves large room for future research. Our benchmark and code are released at https://github.com/chen-judge/GeoQA .
Kids have an amazing capacity to use modern electronic devices such as tablets, smartphones, etc. This has been incredibly boosted by the ease of access of these devices given the expansion of such devices through the world, reaching even third world countries. Also, it is well known that children tend to have difficulty learning some subjects at pre-school. We as a society focus extensively on alphabetization, but in the end, children end up having differences in another essential area: Mathematics. With this work, we create the basis for an intuitive application that could join the fact that children have a lot of ease when using such technological applications, trying to shrink the gap between a fun and enjoyable activity with something that will improve the children knowledge and ability to understand concepts when in a low age, by using a novel convolutional neural network to achieve so, named FINNger.
A longstanding question in cognitive science concerns the learning mechanisms underlying compositionality in human cognition. Humans can infer the structured relationships (e.g., grammatical rules) implicit in their sensory observations (e.g., auditory speech), and use this knowledge to guide the composition of simpler meanings into complex wholes. Recent progress in artificial neural networks has shown that when large models are trained on enough linguistic data, grammatical structure emerges in their representations. We extend this work to the domain of mathematical reasoning, where it is possible to formulate precise hypotheses about how meanings (e.g., the quantities corresponding to numerals) should be composed according to structured rules (e.g., order of operations). Our work shows that neural networks are not only able to infer something about the structured relationships implicit in their training data, but can also deploy this knowledge to guide the composition of individual meanings into composite wholes.
In this paper, we revisit math word problems~(MWPs) from the cross-lingual and multilingual perspective. We construct our MWP solvers over pretrained multilingual language models using sequence-to-sequence model with copy mechanism. We compare how the MWP solvers perform in cross-lingual and multilingual scenarios. To facilitate the comparison of cross-lingual performance, we first adapt the large-scale English dataset MathQA as a counterpart of the Chinese dataset Math23K. Then we extend several English datasets to bilingual datasets through machine translation plus human annotation. Our experiments show that the MWP solvers may not be transferred to a different language even if the target expressions have the same operator set and constants. But for both cross-lingual and multilingual cases, it can be better generalized if problem types exist on both source language and target language.
This report summarizes work performed as part of an internship at INRIA, in partial requirement for the completion of a master degree in math and informatics. The goal of the internship was to develop a software environment to simulate electricity transmission in a power grid and actions performed by operators to maintain this grid in security. Our environment lends itself to automate the control of the power grid with reinforcement learning agents, assisting human operators. It is amenable to organizing benchmarks, including a challenge in machine learning planned by INRIA and RTE for 2019. Our framework, built on top of open-source libraries, is available at https://github.com/MarvinLer/pypownet. In this report we present intermediary results and its usage in the context of a reinforcement learning game.
The problem of designing NLP solvers for math word problems (MWP) has seen sustained research activity and steady gains in the test accuracy. Since existing solvers achieve high performance on the benchmark datasets for elementary level MWPs containing one-unknown arithmetic word problems, such problems are often considered "solved" with the bulk of research attention moving to more complex MWPs. In this paper, we restrict our attention to English MWPs taught in grades four and lower. We provide strong evidence that the existing MWP solvers rely on shallow heuristics to achieve high performance on the benchmark datasets. To this end, we show that MWP solvers that do not have access to the question asked in the MWP can still solve a large fraction of MWPs. Similarly, models that treat MWPs as bag-of-words can also achieve surprisingly high accuracy. Further, we introduce a challenge dataset, SVAMP, created by applying carefully chosen variations over examples sampled from existing datasets. The best accuracy achieved by state-of-the-art models is substantially lower on SVAMP, thus showing that much remains to be done even for the simplest of the MWPs.
Many intellectual endeavors require mathematical problem solving, but this skill remains beyond the capabilities of computers. To measure this ability in machine learning models, we introduce MATH, a new dataset of 12,500 challenging competition mathematics problems. Each problem in MATH has a full step-by-step solution which can be used to teach models to generate answer derivations and explanations. To facilitate future research and increase accuracy on MATH, we also contribute a large auxiliary pretraining dataset which helps teach models the fundamentals of mathematics. Even though we are able to increase accuracy on MATH, our results show that accuracy remains relatively low, even with enormous Transformer models. Moreover, we find that simply increasing budgets and model parameter counts will be impractical for achieving strong mathematical reasoning if scaling trends continue. While scaling Transformers is automatically solving most other text-based tasks, scaling is not currently solving MATH. To have more traction on mathematical problem solving we will likely need new algorithmic advancements from the broader research community.
Gulisashvili et al. [Quant. Finance, 2018, 18(10), 1753-1765] provide a small-time asymptotics for the mass at zero under the uncorrelated stochastic-alpha-beta-rho (SABR) model by approximating the integrated variance with a moment-matched lognormal distribution. We improve the accuracy of the numerical integration by using the Gauss--Hermite quadrature. We further obtain the option price by integrating the constant elasticity of variance (CEV) option prices in the same manner without resorting to the small-strike volatility smile asymptotics of De Marco et al. [SIAM J. Financ. Math., 2017, 8(1), 709-737]. For the uncorrelated SABR model, the new option pricing method is accurate and arbitrage-free across all strike prices.