Previous neural solvers of math word problems (MWPs) are learned with full supervision and fail to generate diverse solutions. In this paper, we address this issue by introducing a \textit{weakly-supervised} paradigm for learning MWPs. Our method only requires the annotations of the final answers and can generate various solutions for a single problem. To boost weakly-supervised learning, we propose a novel \textit{learning-by-fixing} (LBF) framework, which corrects the misperceptions of the neural network via symbolic reasoning. Specifically, for an incorrect solution tree generated by the neural network, the \textit{fixing} mechanism propagates the error from the root node to the leaf nodes and infers the most probable fix that can be executed to get the desired answer. To generate more diverse solutions, \textit{tree regularization} is applied to guide the efficient shrinkage and exploration of the solution space, and a \textit{memory buffer} is designed to track and save the discovered various fixes for each problem. Experimental results on the Math23K dataset show the proposed LBF framework significantly outperforms reinforcement learning baselines in weakly-supervised learning. Furthermore, it achieves comparable top-1 and much better top-3/5 answer accuracies than fully-supervised methods, demonstrating its strength in producing diverse solutions.
Recent work has proven the existence of extreme inbreeding in a European ancestry sample taken from the contemporary UK population \cite{nature_01}. This result brings our attention again to a math problem related to inbreeding family trees and diversity. Groups with a finite number of individuals could give a variety of genetic relationships. { In previous works \cite{PhysRevE.92.052132, PhysRevE.90.022125, JARNE20191}, we have addressed the issue of building inbreeding trees for biparental reproduction using Markovian models. Here, we extend these studies by presenting an algorithm to generate and represent inbreeding trees with no overlapping generations. We explicitly assume a two-gender reproductory scheme, and we pay particular attention to the links between nodes. We show that even for a simple case with a relatively small number of nodes in the tree, there are a large number of possible ways to rearrange the links between generations. We present an open-source python code to generate the tree graph, the adjacency matrix, and the histogram of the links for each different tree representation. We show how this mapping reflects the difference between tree realizations, and how valuable information may be extracted upon inspection of these matrices. The algorithm includes a feature to average several tree realizations, obtain the connectivity distribution, and calculate the average and mean value. We used this feature to compare trees with a different number of generations and nodes. The code presented here, available in Git-Hub, may be easily modified to be applied to other areas of interest involving connections between individuals, extend the study to add more characteristics of the different nodes, etc.
Although the scientific digital library is growing at a rapid pace, scholars/students often find reading Science, Technology, Engineering, and Mathematics (STEM) literature daunting, especially for the math-content/formula. In this paper, we propose a novel problem, ``mathematics content understanding'', for cyberlearning and cyberreading. To address this problem, we create a Formula Evolution Map (FEM) offline and implement a novel online learning/reading environment, PDF Reader with Math-Assistant (PRMA), which incorporates innovative math-scaffolding methods. The proposed algorithm/system can auto-characterize student emerging math-information need while reading a paper and enable students to readily explore the formula evolution trajectory in FEM. Based on a math-information need, PRMA utilizes innovative joint embedding, formula evolution mining, and heterogeneous graph mining algorithms to recommend high quality Open Educational Resources (OERs), e.g., video, Wikipedia page, or slides, to help students better understand the math-content in the paper. Evaluation and exit surveys show that the PRMA system and the proposed formula understanding algorithm can effectively assist master and PhD students better understand the complex math-content in the class readings.
A practical automatic textual math word problems (MWPs) solver should be able to solve various textual MWPs while most existing works only focused on one-unknown linear MWPs. Herein, we propose a simple but efficient method called Universal Expression Tree (UET) to make the first attempt to represent the equations of various MWPs uniformly. Then a semantically-aligned universal tree-structured solver (SAU-Solver) based on an encoder-decoder framework is proposed to resolve multiple types of MWPs in a unified model, benefiting from our UET representation. Our SAU-Solver generates a universal expression tree explicitly by deciding which symbol to generate according to the generated symbols' semantic meanings like human solving MWPs. Besides, our SAU-Solver also includes a novel subtree-level semanticallyaligned regularization to further enforce the semantic constraints and rationality of the generated expression tree by aligning with the contextual information. Finally, to validate the universality of our solver and extend the research boundary of MWPs, we introduce a new challenging Hybrid Math Word Problems dataset (HMWP), consisting of three types of MWPs. Experimental results on several MWPs datasets show that our model can solve universal types of MWPs and outperforms several state-of-the-art models.
Automatically solving math word problems is a critical task in the field of natural language processing. Recent models have reached their performance bottleneck and require more high-quality data for training. We propose a novel data augmentation method that reverses the mathematical logic of math word problems to produce new high-quality math problems and introduce new knowledge points that can benefit learning the mathematical reasoning logic. We apply the augmented data on two SOTA math word problem solving models and compare our results with a strong data augmentation baseline. Experimental results show the effectiveness of our approach. We release our code and data at https://github.com/yiyunya/RODA.
ML decision-aid systems are increasingly common on the web, but their successful integration relies on people trusting them appropriately: they should use the system to fill in gaps in their ability, but recognize signals that the system might be incorrect. We measured how people's trust in ML recommendations differs by expertise and with more system information through a task-based study of 175 adults. We used two tasks that are difficult for humans: comparing large crowd sizes and identifying similar-looking animals. Our results provide three key insights: (1) People trust incorrect ML recommendations for tasks that they perform correctly the majority of the time, even if they have high prior knowledge about ML or are given information indicating the system is not confident in its prediction; (2) Four different types of system information all increased people's trust in recommendations; and (3) Math and logic skills may be as important as ML for decision-makers working with ML recommendations.
With the success of deep learning in various fields and the advent of numerous Internet of Things (IoT) devices, it is essential to lighten models suitable for low-power devices. In keeping with this trend, MicroNet Challenge, which is the challenge to build efficient models from the view of both storage and computation, was hosted at NeurIPS 2019. To develop efficient models through this challenge, we propose a framework, coined as SIPA, consisting of four stages: Searching, Improving, Pruning, and Accelerating. With the proposed framework, our team, OSI AI, compressed 334x the parameter storage and 357x the math operation compared to WideResNet-28-10 and took 4th place in the CIFAR-100 track at MicroNet Challenge 2019 with the top 10% highly efficient computation. Our source code is available from https://github.com/Lee-Gihun/MicroNet_OSI-AI.
The celebrated Seq2Seq technique and its numerous variants achieve excellent performance on many tasks such as neural machine translation, semantic parsing, and math word problem solving. However, these models either only consider input objects as sequences while ignoring the important structural information for encoding, or they simply treat output objects as sequence outputs instead of structural objects for decoding. In this paper, we present a novel Graph-to-Tree Neural Networks, namely Graph2Tree consisting of a graph encoder and a hierarchical tree decoder, that encodes an augmented graph-structured input and decodes a tree-structured output. In particular, we investigated our model for solving two problems, neural semantic parsing and math word problem. Our extensive experiments demonstrate that our Graph2Tree model outperforms or matches the performance of other state-of-the-art models on these tasks.
There are two famous function decomposition methods in math: Taylor Series and Fourier Series. Fourier series developed into Fourier spectrum, which was applied to signal decomposition\analysis. However, because the Taylor series whose function without a definite functional expression cannot be solved, Taylor Series has rarely been used in engineering. Here, we developed Taylor series by our Dendrite Net, constructed a relation spectrum, and applied it to model or system decomposition\analysis. Specific engineering: the knowledge of the intuitive link between muscle activity and the finger movement is vital for the design of commercial prosthetic hands that do not need user pre-training. However, this link has yet to be understood due to the complexity of human hand. In this study, the relation spectrum was applied to analyze the muscle-finger system. One single muscle actuates multiple fingers, or multiple muscles actuate one single finger simultaneously. Thus, the research was in muscle synergy and muscle coupling for hand. This paper has two main contributions. (1) The findings of hand contribute to designing prosthetic hands. (2) The relation spectrum makes the online model human-readable, which unifies online performance and offline results. Code (novel tool for most fields) is available at https://github.com/liugang1234567/Gang-neuron.
Quantization techniques can reduce the size of Deep Neural Networks and improve inference latency and throughput by taking advantage of high throughput integer instructions. In this paper we review the mathematical aspects of quantization parameters and evaluate their choices on a wide range of neural network models for different application domains, including vision, speech, and language. We focus on quantization techniques that are amenable to acceleration by processors with high-throughput integer math pipelines. We also present a workflow for 8-bit quantization that is able to maintain accuracy within 1% of the floating-point baseline on all networks studied, including models that are more difficult to quantize, such as MobileNets and BERT-large.
A stereotype is an over-generalized belief about a particular group of people, e.g., Asians are good at math or Asians are bad drivers. Such beliefs (biases) are known to hurt target groups. Since pretrained language models are trained on large real world data, they are known to capture stereotypical biases. In order to assess the adverse effects of these models, it is important to quantify the bias captured in them. Existing literature on quantifying bias evaluates pretrained language models on a small set of artificially constructed bias-assessing sentences. We present StereoSet, a large-scale natural dataset in English to measure stereotypical biases in four domains: gender, profession, race, and religion. We evaluate popular models like BERT, GPT-2, RoBERTa, and XLNet on our dataset and show that these models exhibit strong stereotypical biases. We also present a leaderboard with a hidden test set to track the bias of future language models at https://stereoset.mit.edu
Large pre-trained language models (LMs) are known to encode substantial amounts of linguistic information. However, high-level reasoning skills, such as numerical reasoning, are difficult to learn from a language-modeling objective only. Consequently, existing models for numerical reasoning have used specialized architectures with limited flexibility. In this work, we show that numerical reasoning is amenable to automatic data generation, and thus one can inject this skill into pre-trained LMs, by generating large amounts of data, and training in a multi-task setup. We show that pre-training our model, GenBERT, on this data, dramatically improves performance on DROP (49.3 $\rightarrow$ 72.3 F1), reaching performance that matches state-of-the-art models of comparable size, while using a simple and general-purpose encoder-decoder architecture. Moreover, GenBERT generalizes well to math word problem datasets, while maintaining high performance on standard RC tasks. Our approach provides a general recipe for injecting skills into large pre-trained LMs, whenever the skill is amenable to automatic data augmentation.
We introduce the Scanning Single Shot Detector (ScanSSD) for locating math formulas offset from text and embedded in textlines. ScanSSD uses only visual features for detection: no formatting or typesetting information such as layout, font, or character labels are employed. Given a 600 dpi document page image, a Single Shot Detector (SSD) locates formulas at multiple scales using sliding windows, after which candidate detections are pooled to obtain page-level results. For our experiments we use the TFD-ICDAR2019v2 dataset, a modification of the GTDB scanned math article collection. ScanSSD detects characters in formulas with high accuracy, obtaining a 0.926 f-score, and detects formulas with high recall overall. Detection errors are largely minor, such as splitting formulas at large whitespace gaps (e.g., for variable constraints) and merging formulas on adjacent textlines. Formula detection f-scores of 0.796 (IOU $\geq0.5$) and 0.733 (IOU $\ge 0.75$) are obtained. Our data, evaluation tools, and code are publicly available.
Mathematical notation, i.e., the writing system used to communicate concepts in mathematics, encodes valuable information for a variety of information search and retrieval systems. Yet, mathematical notations remain mostly unutilized by today's systems. In this paper, we present the first in-depth study on the distributions of mathematical notation in two large scientific corpora: the open access arXiv (2.5B mathematical objects) and the mathematical reviewing service for pure and applied mathematics zbMATH (61M mathematical objects). Our study lays a foundation for future research projects on mathematical information retrieval for large scientific corpora. Further, we demonstrate the relevance of our results to a variety of use-cases. For example, to assist semantic extraction systems, to improve scientific search engines, and to facilitate specialized math recommendation systems. The contributions of our presented research are as follows: (1) we present the first distributional analysis of mathematical formulae on arXiv and zbMATH; (2) we retrieve relevant mathematical objects for given textual search queries (e.g., linking $P_{n}^{(\alpha, \beta)}\!\left(x\right)$ with `Jacobi polynomial'); (3) we extend zbMATH's search engine by providing relevant mathematical formulae; and (4) we exemplify the applicability of the results by presenting auto-completion for math inputs as the first contribution to math recommendation systems. To expedite future research projects, we have made available our source code and data.
Constructing accurate and automatic solvers of math word problems has proven to be quite challenging. Prior attempts using machine learning have been trained on corpora specific to math word problems to produce arithmetic expressions in infix notation before answer computation. We find that custom-built neural networks have struggled to generalize well. This paper outlines the use of Transformer networks trained to translate math word problems to equivalent arithmetic expressions in infix, prefix, and postfix notations. In addition to training directly on domain-specific corpora, we use an approach that pre-trains on a general text corpus to provide foundational language abilities to explore if it improves performance. We compare results produced by a large number of neural configurations and find that most configurations outperform previously reported approaches on three of four datasets with significant increases in accuracy of over 20 percentage points. The best neural approaches boost accuracy by almost 10% on average when compared to the previous state of the art.
Mathematical equations are an important part of dissemination and communication of scientific information. Students, however, often feel challenged in reading and understanding math content and equations. With the development of the Web, students are posting their math questions online. Nevertheless, constructing a concise math headline that gives a good description of the posted detailed math question is nontrivial. In this study, we explore a novel summarization task denoted as geNerating A concise Math hEadline from a detailed math question (NAME). Compared to conventional summarization tasks, this task has two extra and essential constraints: 1) Detailed math questions consist of text and math equations which require a unified framework to jointly model textual and mathematical information; 2) Unlike text, math equations contain semantic and structural features, and both of them should be captured together. To address these issues, we propose MathSum, a novel summarization model which utilizes a pointer mechanism combined with a multi-head attention mechanism for mathematical representation augmentation. The pointer mechanism can either copy textual tokens or math tokens from source questions in order to generate math headlines. The multi-head attention mechanism is designed to enrich the representation of math equations by modeling and integrating both its semantic and structural features. For evaluation, we collect and make available two sets of real-world detailed math questions along with human-written math headlines, namely EXEQ-300k and OFEQ-10k. Experimental results demonstrate that our model (MathSum) significantly outperforms state-of-the-art models for both the EXEQ-300k and OFEQ-10k datasets.
We incorporate Tensor-Product Representations within the Transformer in order to better support the explicit representation of relation structure. Our Tensor-Product Transformer (TP-Transformer) sets a new state of the art on the recently-introduced Mathematics Dataset containing 56 categories of free-form math word-problems. The essential component of the model is a novel attention mechanism, called TP-Attention, which explicitly encodes the relations between each Transformer cell and the other cells from which values have been retrieved by attention. TP-Attention goes beyond linear combination of retrieved values, strengthening representation-building and resolving ambiguities introduced by multiple layers of standard attention. The TP-Transformer's attention maps give better insights into how it is capable of solving the Mathematics Dataset's challenging problems. Pretrained models and code will be made available after publication.
We present an open source math-aware Question Answering System based on Ask Platypus. Our system returns as a single mathematical formula for a natural language question in English or Hindi. This formulae originate from the knowledge-base Wikidata. We translate these formulae to computable data by integrating the calculation engine sympy into our system. This way, users can enter numeric values for the variables occurring in the formula. Moreover, the system loads numeric values for constants occurring in the formula from Wikidata. In a user study, our system outperformed a commercial computational mathematical knowledge engine by 13%. However, the performance of our system heavily depends on the size and quality of the formula data available in Wikidata. Since only a few items in Wikidata contained formulae when we started the project, we facilitated the import process by suggesting formula edits to Wikidata editors. With the simple heuristic that the first formula is significant for the article, 80% of the suggestions were correct.
We introduce a new framework for unifying and systematizing the performance analysis of first-order black-box optimization algorithms for unconstrained convex minimization. The low-cost iteration complexity enjoyed by first-order algorithms renders them particularly relevant for applications in machine learning and large-scale data analysis. Relying on sum-of-squares (SOS) optimization, we introduce a hierarchy of semidefinite programs that give increasingly better convergence bounds for higher levels of the hierarchy. Alluding to the power of the SOS hierarchy, we show that the (dual of the) first level corresponds to the Performance Estimation Problem (PEP) introduced by Drori and Teboulle [Math. Program., 145(1):451--482, 2014], a powerful framework for determining convergence rates of first-order optimization algorithms. Consequently, many results obtained within the PEP framework can be reinterpreted as degree-1 SOS proofs, and thus, the SOS framework provides a promising new approach for certifying improved rates of convergence by means of higher-order SOS certificates. To determine analytical rate bounds, in this work we use the first level of the SOS hierarchy and derive new result{s} for noisy gradient descent with inexact line search methods (Armijo, Wolfe, and Goldstein).
Reducing hardware overhead of neural networks for faster or lower power inference and training is an active area of research. Uniform quantization using integer multiply-add has been thoroughly investigated, which requires learning many quantization parameters, fine-tuning training or other prerequisites. Little effort is made to improve floating point relative to this baseline; it remains energy inefficient, and word size reduction yields drastic loss in needed dynamic range. We improve floating point to be more energy efficient than equivalent bit width integer hardware on a 28 nm ASIC process while retaining accuracy in 8 bits with a novel hybrid log multiply/linear add, Kulisch accumulation and tapered encodings from Gustafson's posit format. With no network retraining, and drop-in replacement of all math and float32 parameters via round-to-nearest-even only, this open-sourced 8-bit log float is within 0.9% top-1 and 0.2% top-5 accuracy of the original float32 ResNet-50 CNN model on ImageNet. Unlike int8 quantization, it is still a general purpose floating point arithmetic, interpretable out-of-the-box. Our 8/38-bit log float multiply-add is synthesized and power profiled at 28 nm at 0.96x the power and 1.12x the area of 8/32-bit integer multiply-add. In 16 bits, our log float multiply-add is 0.59x the power and 0.68x the area of IEEE 754 float16 fused multiply-add, maintaining the same signficand precision and dynamic range, proving useful for training ASICs as well.