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.
Contextually aware intelligent agents are often required to understand the users and their surroundings in real-time. Our goal is to build Artificial Intelligence (AI) systems that can assist children in their learning process. Within such complex frameworks, Spoken Dialogue Systems (SDS) are crucial building blocks to handle efficient task-oriented communication with children in game-based learning settings. We are working towards a multimodal dialogue system for younger kids learning basic math concepts. Our focus is on improving the Natural Language Understanding (NLU) module of the task-oriented SDS pipeline with limited datasets. This work explores the potential benefits of data augmentation with paraphrase generation for the NLU models trained on small task-specific datasets. We also investigate the effects of extracting entities for conceivably further data expansion. We have shown that paraphrasing with model-in-the-loop (MITL) strategies using small seed data is a promising approach yielding improved performance results for the Intent Recognition task.
This paper introduces a novel nonlocal partial difference equation (G-PDE) for labeling metric data on graphs. The G-PDE is derived as nonlocal reparametrization of the assignment flow approach that was introduced in \textit{J.~Math.~Imaging \& Vision} 58(2), 2017. Due to this parameterization, solving the G-PDE numerically is shown to be equivalent to computing the Riemannian gradient flow with respect to a nonconvex potential. We devise an entropy-regularized difference-of-convex-functions (DC) decomposition of this potential and show that the basic geometric Euler scheme for integrating the assignment flow is equivalent to solving the G-PDE by an established DC programming scheme. Moreover, the viewpoint of geometric integration reveals a basic way to exploit higher-order information of the vector field that drives the assignment flow, in order to devise a novel accelerated DC programming scheme. A detailed convergence analysis of both numerical schemes is provided and illustrated by numerical experiments.
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.
We provide a short-time large deviation principle (LDP) for stochastic volatility models, where the volatility is expressed as a function of a Volterra process. This LDP does not require strict self-similarity assumptions on the Volterra process. For this reason, we are able to apply such an LDP to two notable examples of non self-similar rough volatility models: models where the volatility is given as a function of a log-modulated fractional Brownian motion [Bayer et al., Log-modulated rough stochastic volatility models. SIAM J. Financ. Math, 2021, 12(3), 1257-1284], and models where it is given as a function of a fractional Ornstein-Uhlenbeck (fOU) process [Gatheral et al., Volatility is rough. Quant. Finance, 2018, 18(6), 933-949]. In both cases we derive consequences for short-maturity European option prices, implied volatility surfaces and implied volatility skew. In the fOU case we also discuss moderate deviations pricing and simulation results.
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).
A math word problem (MWP) is a coherent narrative which reflects the underlying logic of math equations. Successful MWP generation can automate the writing of mathematics questions. Previous methods mainly generate MWP text based on inflexible pre-defined templates. In this paper, we propose a neural model for generating MWP text from math equations. Firstly, we incorporate a matching model conditioned on the domain knowledge to retrieve a MWP instance which is most consistent with the ground-truth, where the domain is a latent variable extracted with a domain summarizer. Secondly, by constructing a Quantity Cell Graph (QCG) from the retrieved MWP instance and reasoning over it, we improve the model's comprehension of real-world scenarios and derive a domain-constrained instance sketch to guide the generation. Besides, the QCG also interacts with the equation encoder to enhance the alignment between math tokens (e.g., quantities and variables) and MWP text. Experiments and empirical analysis on educational MWP set show that our model achieves impressive performance in both automatic evaluation metrics and human evaluation metrics.
Large language models have been shown to achieve remarkable performance across a variety of natural language tasks using few-shot learning, which drastically reduces the number of task-specific training examples needed to adapt the model to a particular application. To further our understanding of the impact of scale on few-shot learning, we trained a 540-billion parameter, densely activated, Transformer language model, which we call Pathways Language Model PaLM. We trained PaLM on 6144 TPU v4 chips using Pathways, a new ML system which enables highly efficient training across multiple TPU Pods. We demonstrate continued benefits of scaling by achieving state-of-the-art few-shot learning results on hundreds of language understanding and generation benchmarks. On a number of these tasks, PaLM 540B achieves breakthrough performance, outperforming the finetuned state-of-the-art on a suite of multi-step reasoning tasks, and outperforming average human performance on the recently released BIG-bench benchmark. A significant number of BIG-bench tasks showed discontinuous improvements from model scale, meaning that performance steeply increased as we scaled to our largest model. PaLM also has strong capabilities in multilingual tasks and source code generation, which we demonstrate on a wide array of benchmarks. We additionally provide a comprehensive analysis on bias and toxicity, and study the extent of training data memorization with respect to model scale. Finally, we discuss the ethical considerations related to large language models and discuss potential mitigation strategies.
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.
Chain-of-thought prompting combined with pre-trained large language models has achieved encouraging results on complex reasoning tasks. In this paper, we propose a new decoding strategy, self-consistency, to replace the naive greedy decoding used in chain-of-thought prompting. It first samples a diverse set of reasoning paths instead of only taking the greedy one, and then selects the most consistent answer by marginalizing out the sampled reasoning paths. Self-consistency leverages the intuition that a complex reasoning problem typically admits multiple different ways of thinking leading to its unique correct answer. Our extensive empirical evaluation shows that self-consistency boosts the performance of chain-of-thought prompting with a striking margin on a range of popular arithmetic and commonsense reasoning benchmarks, including GSM8K (+17.9%), SVAMP (+11.0%), AQuA (+12.2%), StrategyQA (+6.4%) and ARC-challenge (+3.9%).
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
In this paper we investigate the homeostatic mechanism in two biologically motivated models: intracellular copper regulation and self immune recognition. The analysis is based on the notions of infinitesimal homeostasis and near-perfect homeostasis. We introduce a theoretical background that makes it possible to consider points of infinitesimal homeostasis that lie at the boundary of the domain of definition of the input-output function. We show that the two models display near-perfect homeostasis. Moreover, we show that, unlike the examples of [M. Reed, J. Best, M. Golubitsky, I. Stewart, and H. F. Nijhout. Analysis of homeostatic mechanisms in biochemical networks. Bull. Math. Biol., 79(11):2534-2557, 2017], the obstruction of occurrence of infinitesimal homeostasis in both of them is caused by the modeling assumptions that force the point of infinitesimal homeostasis to lie at the boundary of domain of definition of the respective input-output functions.
We review briefly the characteristic topological data of Calabi--Yau threefolds and focus on the question of when two threefolds are equivalent through related topological data. This provides an interesting test case for machine learning methodology in discrete mathematics problems motivated by physics.
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.