In reinforcement learning, different reward functions can be equivalent in terms of the optimal policies they induce. A particularly well-known and important example is potential shaping, a class of functions that can be added to any reward function without changing the optimal policy set under arbitrary transition dynamics. Potential shaping is conceptually similar to potentials, conservative vector fields and gauge transformations in math and physics, but this connection has not previously been formally explored. We develop a formalism for discrete calculus on graphs that abstract a Markov Decision Process, and show how potential shaping can be formally interpreted as a gradient within this framework. This allows us to strengthen results from Ng et al. (1999) describing conditions under which potential shaping is the only additive reward transformation to always preserve optimal policies. As an additional application of our formalism, we define a rule for picking a single unique reward function from each potential shaping equivalence class.
The data revolution has generated a huge demand for data-driven solutions. This demand propels a growing number of easy-to-use tools and training for aspiring data scientists that enable the rapid building of predictive models. Today, weapons of math destruction can be easily built and deployed without detailed planning and validation. This rapidly extends the list of AI failures, i.e. deployments that lead to financial losses or even violate democratic values such as equality, freedom and justice. The lack of planning, rules and standards around the model development leads to the ,,anarchisation of AI". This problem is reported under different names such as validation debt, reproducibility crisis, and lack of explainability. Post-mortem analysis of AI failures often reveals mistakes made in the early phase of model development or data acquisition. Thus, instead of curing the consequences of deploying harmful models, we shall prevent them as early as possible by putting more attention to the initial planning stage. In this paper, we propose a quick and simple framework to support planning of AI solutions. The POCA framework is based on four pillars: Performance, Opaqueness, Consequences, and Assumptions. It helps to set the expectations and plan the constraints for the AI solution before any model is built and any data is collected. With the help of the POCA method, preliminary requirements can be defined for the model-building process, so that costly model misspecification errors can be identified as soon as possible or even avoided. AI researchers, product owners and business analysts can use this framework in the initial stages of building AI solutions.
Drori et al. (2022) report that "A neural network solves, explains, and generates university math problems by program synthesis and few-shot learning at human level ... [It] automatically answers 81\% of university-level mathematics problems." The system they describe is indeed impressive; however, the above description is very much overstated. The work of solving the problems is done, not by a neural network, but by the symbolic algebra package Sympy. Problems of various formats are excluded from consideration. The so-called "explanations" are just rewordings of lines of code. Answers are marked as correct that are not in the form specified in the problem. Most seriously, it seems that in many cases the system uses the correct answer given in the test corpus to guide its path to solving the problem.
This paper describes the design and implementation of a new machine learning model for online learning systems. We aim at improving the intelligent level of the systems by enabling an automated math word problem solver which can support a wide range of functions such as homework correction, difficulty estimation, and priority recommendation. We originally planned to employ existing models but realized that they processed a math word problem as a sequence or a homogeneous graph of tokens. Relationships between the multiple types of tokens such as entity, unit, rate, and number were ignored. We decided to design and implement a novel model to use such relational data to bridge the information gap between human-readable language and machine-understandable logical form. We propose a heterogeneous line graph transformer (HLGT) model that constructs a heterogeneous line graph via semantic role labeling on math word problems and then perform node representation learning aware of edge types. We add numerical comparison as an auxiliary task to improve model training for real-world use. Experimental results show that the proposed model achieves a better performance than existing models and suggest that it is still far below human performance. Information utilization and knowledge discovery is continuously needed to improve the online learning systems.
We introduce CLEVR-Math, a multi-modal math word problems dataset consisting of simple math word problems involving addition/subtraction, represented partly by a textual description and partly by an image illustrating the scenario. The text describes actions performed on the scene that is depicted in the image. Since the question posed may not be about the scene in the image, but about the state of the scene before or after the actions are applied, the solver envision or imagine the state changes due to these actions. Solving these word problems requires a combination of language, visual and mathematical reasoning. We apply state-of-the-art neural and neuro-symbolic models for visual question answering on CLEVR-Math and empirically evaluate their performances. Our results show how neither method generalise to chains of operations. We discuss the limitations of the two in addressing the task of multi-modal word problem solving.
Responsive teaching is a highly effective strategy that promotes student learning. In math classrooms, teachers might "funnel" students towards a normative answer or "focus" students to reflect on their own thinking, deepening their understanding of math concepts. When teachers focus, they treat students' contributions as resources for collective sensemaking, and thereby significantly improve students' achievement and confidence in mathematics. We propose the task of computationally detecting funneling and focusing questions in classroom discourse. We do so by creating and releasing an annotated dataset of 2,348 teacher utterances labeled for funneling and focusing questions, or neither. We introduce supervised and unsupervised approaches to differentiating these questions. Our best model, a supervised RoBERTa model fine-tuned on our dataset, has a strong linear correlation of .76 with human expert labels and with positive educational outcomes, including math instruction quality and student achievement, showing the model's potential for use in automated teacher feedback tools. Our unsupervised measures show significant but weaker correlations with human labels and outcomes, and they highlight interesting linguistic patterns of funneling and focusing questions. The high performance of the supervised measure indicates its promise for supporting teachers in their instruction.
Systems biology seeks to create math models of biological systems to reduce inherent biological complexity and provide predictions for applications such as therapeutic development. However, it remains a challenge to determine which math model is correct and how to arrive optimally at the answer. We present an algorithm for automated biological model selection using mathematical models of systems biology and likelihood free inference methods. Our algorithm shows improved performance in arriving at correct models without a priori information over conventional heuristics used in experimental biology and random search. This method shows promise to accelerate biological basic science and drug discovery.
In this paper, we investigate the opportunities of automating the judgment process in online one-on-one math classes. We build a Wide & Deep framework to learn fine-grained predictive representations from a limited amount of noisy classroom conversation data that perform better student judgments. We conducted experiments on the task of predicting students' levels of mastery of example questions and the results demonstrate the superiority and availability of our model in terms of various evaluation metrics.
Numerical reasoning is required when solving most problems in our life, but it has been neglected in previous artificial intelligence researches. FinQA challenge has been organized to strengthen the study on numerical reasoning where the participants are asked to predict the numerical reasoning program to solve financial question. The result of FinQA will be evaluated by both execution accuracy and program accuracy. In this paper, we present our approach to tackle the task objective by developing models with different specialized capabilities and fusing their strength. Overall, our approach achieves the 1st place in FinQA challenge, with 71.93% execution accuracy and 67.03% program accuracy.
Learning disabilities, which primarily interfere with the basic learning skills such as reading, writing and math, are known to affect around 10% of children in the world. The poor motor skills and motor coordination as part of the neurodevelopmental disorder can become a causative factor for the difficulty in learning to write (dysgraphia), hindering the academic track of an individual. The signs and symptoms of dysgraphia include but are not limited to irregular handwriting, improper handling of writing medium, slow or labored writing, unusual hand position, etc. The widely accepted assessment criterion for all the types of learning disabilities is the examination performed by medical experts. The few available artificial intelligence-powered screening systems for dysgraphia relies on the distinctive features of handwriting from the corresponding images.This work presents a review of the existing automated dysgraphia diagnosis systems for children in the literature. The main focus of the work is to review artificial intelligence-based systems for dysgraphia diagnosis in children. This work discusses the data collection method, important handwriting features, machine learning algorithms employed in the literature for the diagnosis of dysgraphia. Apart from that, this article discusses some of the non-artificial intelligence-based automated systems also. Furthermore, this article discusses the drawbacks of existing systems and proposes a novel framework for dysgraphia diagnosis.
This paper aims to advance the mathematical intelligence of machines by presenting the first Chinese mathematical pre-trained language model~(PLM) for effectively understanding and representing mathematical problems. Unlike other standard NLP tasks, mathematical texts are difficult to understand, since they involve mathematical terminology, symbols and formulas in the problem statement. Typically, it requires complex mathematical logic and background knowledge for solving mathematical problems. Considering the complex nature of mathematical texts, we design a novel curriculum pre-training approach for improving the learning of mathematical PLMs, consisting of both basic and advanced courses. Specially, we first perform token-level pre-training based on a position-biased masking strategy, and then design logic-based pre-training tasks that aim to recover the shuffled sentences and formulas, respectively. Finally, we introduce a more difficult pre-training task that enforces the PLM to detect and correct the errors in its generated solutions. We conduct extensive experiments on offline evaluation (including nine math-related tasks) and online $A/B$ test. Experimental results demonstrate the effectiveness of our approach compared with a number of competitive baselines. Our code is available at: \textcolor{blue}{\url{https://github.com/RUCAIBox/JiuZhang}}.
Language models demonstrate both quantitative improvement and new qualitative capabilities with increasing scale. Despite their potentially transformative impact, these new capabilities are as yet poorly characterized. In order to inform future research, prepare for disruptive new model capabilities, and ameliorate socially harmful effects, it is vital that we understand the present and near-future capabilities and limitations of language models. To address this challenge, we introduce the Beyond the Imitation Game benchmark (BIG-bench). BIG-bench currently consists of 204 tasks, contributed by 450 authors across 132 institutions. Task topics are diverse, drawing problems from linguistics, childhood development, math, common-sense reasoning, biology, physics, social bias, software development, and beyond. BIG-bench focuses on tasks that are believed to be beyond the capabilities of current language models. We evaluate the behavior of OpenAI's GPT models, Google-internal dense transformer architectures, and Switch-style sparse transformers on BIG-bench, across model sizes spanning millions to hundreds of billions of parameters. In addition, a team of human expert raters performed all tasks in order to provide a strong baseline. Findings include: model performance and calibration both improve with scale, but are poor in absolute terms (and when compared with rater performance); performance is remarkably similar across model classes, though with benefits from sparsity; tasks that improve gradually and predictably commonly involve a large knowledge or memorization component, whereas tasks that exhibit "breakthrough" behavior at a critical scale often involve multiple steps or components, or brittle metrics; social bias typically increases with scale in settings with ambiguous context, but this can be improved with prompting.
Recently, quite a few novel neural architectures were derived to solve math word problems by predicting expression trees. These architectures varied from seq2seq models, including encoders leveraging graph relationships combined with tree decoders. These models achieve good performance on various MWPs datasets but perform poorly when applied to an adversarial challenge dataset, SVAMP. We present a novel model MMTM that leverages multi-tasking and multi-decoder during pre-training. It creates variant tasks by deriving labels using pre-order, in-order and post-order traversal of expression trees, and uses task-specific decoders in a multi-tasking framework. We leverage transformer architectures with lower dimensionality and initialize weights from RoBERTa model. MMTM model achieves better mathematical reasoning ability and generalisability, which we demonstrate by outperforming the best state of the art baseline models from Seq2Seq, GTS, and Graph2Tree with a relative improvement of 19.4% on an adversarial challenge dataset SVAMP.
From the latter half of the last decade, there has been a growing interest in developing algorithms for automatically solving mathematical word problems (MWP). It is a challenging and unique task that demands blending surface level text pattern recognition with mathematical reasoning. In spite of extensive research, we are still miles away from building robust representations of elementary math word problems and effective solutions for the general task. In this paper, we critically examine the various models that have been developed for solving word problems, their pros and cons and the challenges ahead. In the last two years, a lot of deep learning models have recorded competing results on benchmark datasets, making a critical and conceptual analysis of literature highly useful at this juncture. We take a step back and analyse why, in spite of this abundance in scholarly interest, the predominantly used experiment and dataset designs continue to be a stumbling block. From the vantage point of having analyzed the literature closely, we also endeavour to provide a road-map for future math word problem research.
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.
Intelligent systems designed for play-based interactions should be contextually aware of the users and their surroundings. Spoken Dialogue Systems (SDS) are critical for these interactive agents to carry out effective goal-oriented communication with users in real-time. For the real-world (i.e., in-the-wild) deployment of such conversational agents, improving the Natural Language Understanding (NLU) module of the goal-oriented SDS pipeline is crucial, especially with limited task-specific datasets. This study explores the potential benefits of a recently proposed transformer-based multi-task NLU architecture, mainly to perform Intent Recognition on small-size domain-specific educational game datasets. The evaluation datasets were collected from children practicing basic math concepts via play-based interactions in game-based learning settings. We investigate the NLU performances on the initial proof-of-concept game datasets versus the real-world deployment datasets and observe anticipated performance drops in-the-wild. We have shown that compared to the more straightforward baseline approaches, Dual Intent and Entity Transformer (DIET) architecture is robust enough to handle real-world data to a large extent for the Intent Recognition task on these domain-specific in-the-wild game datasets.
Transformer based language models (LMs) demonstrate increasing performance with scale across a wide variety of tasks. Scale alone however cannot enable models to solve tasks that require access to ephemeral, changing, or private data that was unavailable at training time. Many useful tasks may also benefit from LMs being able to access APIs that read or modify state. In this work, we present Tool Augmented Language Models (TALM), combining a text-only approach to augment language models with non-differentiable tools, and an iterative "self-play" technique to bootstrap performance starting from few tool demonstrations. TALM exhibits strong performance on both a knowledge-heavy QA task and a reasoning oriented math task with simple tools. At a given model scale, TALM significantly outperforms non-augmented LMs. We further demonstrate that TALM successfully performs out-of-distribution inferences on both QA and math tasks, where non-augmented LMs fail. Our results suggest that Tool Augmented Language Models are a promising direction to enrich LMs' capabilities, with less dependence on scale.
Chain-of-thought prompting has demonstrated remarkable performance on various natural language reasoning tasks. However, it tends to perform poorly on tasks which requires solving problems harder than the exemplars shown in the prompts. To overcome this challenge of easy-to-hard generalization, we propose a novel prompting strategy, least-to-most prompting. The key idea in this strategy is to break down a complex problem into a series of simpler subproblems and then solve them in sequence. Solving each subproblem is facilitated by the answers to previously solved subproblems. Our experimental results on tasks related to symbolic manipulation, compositional generalization, and math reasoning reveal that least-to-most prompting is capable of generalizing to more difficult problems than those seen in the prompts. A notable finding is that when the GPT-3 code-davinci-002 model is used with least-to-most prompting, it can solve the compositional generalization benchmark SCAN in any split (including length split) with an accuracy of at least 99% using just 14 exemplars, compared to only 16% accuracy with chain-of-thought prompting. This is particularly noteworthy because neural-symbolic models in the literature that specialize in solving SCAN are trained on the entire training set containing over 15,000 examples. We have included prompts for all the tasks in the Appendix.
Math Word Problems (MWP) is an important task that requires the ability of understanding and reasoning over mathematical text. Existing approaches mostly formalize it as a generation task by adopting Seq2Seq or Seq2Tree models to encode an input math problem in natural language as a global representation and generate the output mathematical expression. Such approaches only learn shallow heuristics and fail to capture fine-grained variations in inputs. In this paper, we propose to model a math word problem in a fine-to-coarse manner to capture both the local fine-grained information and the global logical structure of it. Instead of generating a complete equation sequence or expression tree from the global features, we iteratively combine low-level operands to predict a higher-level operator, abstracting the problem and reasoning about the solving operators from bottom to up. Our model is naturally more sensitive to local variations and can better generalize to unseen problem types. Extensive evaluations on Math23k and SVAMP datasets demonstrate the accuracy and robustness of our method.