Math word problems (MWPs) is a task that automatically derives solution expression from a giving math problems in text. The previous studies suffer from spurious correlations between input text and output expression. To mitigate this issue, we propose a self-consistent reasoning framework called SCR, which attempts to adopt a pruning strategy to correct the output distribution shift so as to implicitly fix those spurious correlative samples. Specifically, we firstly obtain a sub-network by pruning a roberta2tree model, for the sake to use the gap on output distribution between the original roberta2tree model and the pruned sub-network to expose spurious correlative samples. Then, we calibrate the output distribution shift by applying symmetric Kullback-Leibler divergence to alleviate spurious correlations. In addition, SCR generates equivalent expressions, thereby, capturing the original text's logic rather than relying on hints from original text. Extensive experiments on two large-scale benchmarks demonstrate that our model substantially outperforms the strong baseline methods.
We present a smoothly broken power law functional form (referred to by us as a Broken Neural Scaling Law (BNSL)) that accurately models and extrapolates the scaling behaviors of deep neural networks (i.e. how the evaluation metric of interest varies as the amount of compute used for training, number of model parameters, training dataset size, model input size, number of training steps, or upstream performance varies) for various architectures and for each of various tasks within a large and diverse set of upstream and downstream tasks, in zero-shot, prompted, and fine-tuned settings. This set includes large-scale vision, language, audio, video, diffusion, generative modeling, multimodal learning, contrastive learning, AI alignment, robotics, out-of-distribution (OOD) generalization, continual learning, transfer learning, uncertainty estimation / calibration, out-of-distribution detection, adversarial robustness, distillation, sparsity, retrieval, quantization, pruning, molecules, computer programming/coding, math word problems, "emergent" "phase transitions / changes", arithmetic, unsupervised/self-supervised learning, and reinforcement learning (single agent and multi-agent). When compared to other functional forms for neural scaling behavior, this functional form yields extrapolations of scaling behavior that are considerably more accurate on this set. Moreover, this functional form accurately models and extrapolates scaling behavior that other functional forms are incapable of expressing such as the non-monotonic transitions present in the scaling behavior of phenomena such as double descent and the delayed, sharp inflection points present in the scaling behavior of tasks such as arithmetic. Lastly, we use this functional form to glean insights about the limit of the predictability of scaling behavior. Code is available at https://github.com/ethancaballero/broken_neural_scaling_laws
As one of the challenging NLP tasks, designing math word problem (MWP) solvers has attracted increasing research attention for the past few years. In previous work, models designed by taking into account the properties of the binary tree structure of mathematical expressions at the output side have achieved better performance. However, the expressions corresponding to a MWP are often diverse (e.g., $n_1+n_2 \times n_3-n_4$, $n_3\times n_2-n_4+n_1$, etc.), and so are the corresponding binary trees, which creates difficulties in model learning due to the non-deterministic output space. In this paper, we propose the Structure-Unified M-Tree Coding Solver (SUMC-Solver), which applies a tree with any M branches (M-tree) to unify the output structures. To learn the M-tree, we use a mapping to convert the M-tree into the M-tree codes, where codes store the information of the paths from tree root to leaf nodes and the information of leaf nodes themselves, and then devise a Sequence-to-Code (seq2code) model to generate the codes. Experimental results on the widely used MAWPS and Math23K datasets have demonstrated that SUMC-Solver not only outperforms several state-of-the-art models under similar experimental settings but also performs much better under low-resource conditions.
We have recently witnessed a number of impressive results on hard mathematical reasoning problems with language models. At the same time, the robustness of these models has also been called into question; recent works have shown that models can rely on shallow patterns in the problem description when generating a solution. Building on the idea of behavioral testing, we propose a novel framework, which pins down the causal effect of various factors in the input, e.g., the surface form of the problem text, the operands, and math operators on the output solution. By grounding the behavioral analysis in a causal graph describing an intuitive reasoning process, we study the behavior of language models in terms of robustness and sensitivity to direct interventions in the input space. We apply our framework on a test bed of math word problems. Our analysis shows that robustness does not appear to continuously improve as a function of size, but the GPT-3 Davinci models (175B) achieve a dramatic improvement in both robustness and sensitivity compared to all other GPT variants.
Math word problem solver requires both precise relation reasoning about quantities in the text and reliable generation for the diverse equation. Current sequence-to-tree or relation extraction methods regard this only from a fixed view, struggling to simultaneously handle complex semantics and diverse equations. However, human solving naturally involves two consistent reasoning views: top-down and bottom-up, just as math equations also can be expressed in multiple equivalent forms: pre-order and post-order. We propose a multi-view consistent contrastive learning for a more complete semantics-to-equation mapping. The entire process is decoupled into two independent but consistent views: top-down decomposition and bottom-up construction, and the two reasoning views are aligned in multi-granularity for consistency, enhancing global generation and precise reasoning. Experiments on multiple datasets across two languages show our approach significantly outperforms the existing baselines, especially on complex problems. We also show after consistent alignment, multi-view can absorb the merits of both views and generate more diverse results consistent with the mathematical laws.
Large pre-trained models exhibit distinct and complementary capabilities dependent on the data they are trained on. Language models such as GPT-3 are capable of textual reasoning but cannot understand visual information, while vision models such as DALL-E can generate photorealistic photos but fail to understand complex language descriptions. In this work, we propose a unified framework for composing ensembles of different pre-trained models -- combining the strengths of each individual model to solve various multimodal problems in a zero-shot manner. We use pre-trained models as "generators" or "scorers" and compose them via closed-loop iterative consensus optimization. The generator constructs proposals and the scorers iteratively provide feedback to refine the generated result. Such closed-loop communication enables models to correct errors caused by other models, significantly boosting performance on downstream tasks, e.g. improving accuracy on grade school math problems by 7.5%, without requiring any model finetuning. We demonstrate that consensus achieved by an ensemble of scorers outperforms the feedback of a single scorer, by leveraging the strengths of each expert model. Results show that the proposed method can be used as a general purpose framework for a wide range of zero-shot multimodal tasks, such as image generation, video question answering, mathematical reasoning, and robotic manipulation. Project page: https://energy-based-model.github.io/composing-pretrained-models.
Resource constrained project scheduling is an important combinatorial optimisation problem with many practical applications. With complex requirements such as precedence constraints, limited resources, and finance-based objectives, finding optimal solutions for large problem instances is very challenging even with well-customised meta-heuristics and matheuristics. To address this challenge, we propose a new math-heuristic algorithm based on Merge Search and parallel computing to solve the resource constrained project scheduling with the aim of maximising the net present value. This paper presents a novel matheuristic framework designed for resource constrained project scheduling, Merge search, which is a variable partitioning and merging mechanism to formulate restricted mixed integer programs with the aim of improving an existing pool of solutions. The solution pool is obtained via a customised parallel ant colony optimisation algorithm, which is also capable of generating high quality solutions on its own. The experimental results show that the proposed method outperforms the current state-of-the-art algorithms on known benchmark problem instances. Further analyses also demonstrate that the proposed algorithm is substantially more efficient compared to its counterparts in respect to its convergence properties when considering multiple cores.
The following work is a preprint collection of formal proofs regarding the convergence properties of the AdaBoost machine learning algorithm's classifier and margins. Various math and computer science papers have been written regarding conjectures and special cases of these convergence properties. Furthermore, the margins of AdaBoost feature prominently in the research surrounding the algorithm. At the zenith of this paper we present how AdaBoost's classifier and margins converge on a value that agrees with decades of research. After this, we show how various quantities associated with the combined classifier converge.
Previous studies have introduced a weakly-supervised paradigm for solving math word problems requiring only the answer value annotation. While these methods search for correct value equation candidates as pseudo labels, they search among a narrow sub-space of the enormous equation space. To address this problem, we propose a novel search algorithm with combinatorial strategy \textbf{ComSearch}, which can compress the search space by excluding mathematically equivalent equations. The compression allows the searching algorithm to enumerate all possible equations and obtain high-quality data. We investigate the noise in the pseudo labels that hold wrong mathematical logic, which we refer to as the \textit{false-matching} problem, and propose a ranking model to denoise the pseudo labels. Our approach holds a flexible framework to utilize two existing supervised math word problem solvers to train pseudo labels, and both achieve state-of-the-art performance in the weak supervision task.
One unsolved sub-task of document analysis is mathematical formula detection (MFD). Research by ourselves and others has shown that existing MFD datasets with inline and display formula labels are small and have insufficient labeling quality. There is therefore an urgent need for datasets with better quality labeling for future research in the MFD field, as they have a high impact on the performance of the models trained on them. We present an advanced labeling pipeline and a new dataset called FormulaNet in this paper. At over 45k pages, we believe that FormulaNet is the largest MFD dataset with inline formula labels. Our experiments demonstrate substantially improved labeling quality for inline and display formulae detection over existing datasets. Additionally, we provide a math formula detection baseline for FormulaNet with an mAP of 0.754. Our dataset is intended to help address the MFD task and may enable the development of new applications, such as making mathematical formulae accessible in PDFs for visually impaired screen reader users.
We evaluate the reasoning abilities of large language models in multilingual settings. We introduce the Multilingual Grade School Math (MGSM) benchmark, by manually translating 250 grade-school math problems from the GSM8K dataset (Cobbe et al., 2021) into ten typologically diverse languages. We find that the ability to solve MGSM problems via chain-of-thought prompting emerges with increasing model scale, and that models have strikingly strong multilingual reasoning abilities, even in underrepresented languages such as Bengali and Swahili. Finally, we show that the multilingual reasoning abilities of language models extend to other tasks such as commonsense reasoning and word-in-context semantic judgment. The MGSM benchmark is publicly available at https://github.com/google-research/url-nlp.
We study the task of prompting large-scale language models to perform multi-step reasoning. Existing work shows that when prompted with a chain of thoughts (CoT), sequences of short sentences describing intermediate reasoning steps towards a final answer, large language models can generate new reasoning chains and predict answers for new inputs. A central question is which reasoning examples make the most effective prompts. In this work, we propose complexity-based prompting, a simple and effective example selection scheme for multi-step reasoning. We show that prompts with higher reasoning complexity, i.e., chains with more reasoning steps, achieve substantially better performance on multi-step reasoning tasks over strong baselines. We further extend our complexity-based criteria from prompting (selecting inputs) to decoding (selecting outputs), where we sample multiple reasoning chains from the model, then choose the majority of generated answers from complex reasoning chains (over simple chains). When used to prompt GPT-3 and Codex, our approach substantially improves multi-step reasoning accuracy and achieves new state-of-the-art (SOTA) performance on three math benchmarks (GSM8K, MultiArith, and MathQA) and two BigBenchHard tasks (Date Understanding and Penguins), with an average +5.3 and up to +18 accuracy improvements. Compared with existing example selection schemes like manual tuning or retrieval-based selection, selection based on reasoning complexity is intuitive, easy to implement, and annotation-efficient. Further results demonstrate the robustness of performance gains from complex prompts under format perturbation and distribution shift.
Mathematical reasoning, a core ability of human intelligence, presents unique challenges for machines in abstract thinking and logical reasoning. Recent large pre-trained language models such as GPT-3 have achieved remarkable progress on mathematical reasoning tasks written in text form, such as math word problems (MWP). However, it is unknown if the models can handle more complex problems that involve math reasoning over heterogeneous information, such as tabular data. To fill the gap, we present Tabular Math Word Problems (TabMWP), a new dataset containing 38,431 open-domain grade-level problems that require mathematical reasoning on both textual and tabular data. Each question in TabMWP is aligned with a tabular context, which is presented as an image, semi-structured text, and a structured table. There are two types of questions: free-text and multi-choice, and each problem is annotated with gold solutions to reveal the multi-step reasoning process. We evaluate different pre-trained models on TabMWP, including the GPT-3 model in a few-shot setting. As earlier studies suggest, since few-shot GPT-3 relies on the selection of in-context examples, its performance is unstable and can degrade to near chance. The unstable issue is more severe when handling complex problems like TabMWP. To mitigate this, we further propose a novel approach, PromptPG, which utilizes policy gradient to learn to select in-context examples from a small amount of training data and then constructs the corresponding prompt for the test example. Experimental results show that our method outperforms the best baseline by 5.31% on the accuracy metric and reduces the prediction variance significantly compared to random selection, which verifies its effectiveness in selecting in-context examples.
Efficient and automated design of optimizers plays a crucial role in full-stack AutoML systems. However, prior methods in optimizer search are often limited by their scalability, generability, or sample efficiency. With the goal of democratizing research and application of optimizer search, we present the first efficient, scalable and generalizable framework that can directly search on the tasks of interest. We first observe that optimizer updates are fundamentally mathematical expressions applied to the gradient. Inspired by the innate tree structure of the underlying math expressions, we re-arrange the space of optimizers into a super-tree, where each path encodes an optimizer. This way, optimizer search can be naturally formulated as a path-finding problem, allowing a variety of well-established tree traversal methods to be used as the search algorithm. We adopt an adaptation of the Monte Carlo method to tree search, equipped with rejection sampling and equivalent- form detection that leverage the characteristics of optimizer update rules to further boost the sample efficiency. We provide a diverse set of tasks to benchmark our algorithm and demonstrate that, with only 128 evaluations, the proposed framework can discover optimizers that surpass both human-designed counterparts and prior optimizer search methods.
To solve Math Word Problems, human students leverage diverse reasoning logic that reaches different possible equation solutions. However, the mainstream sequence-to-sequence approach of automatic solvers aims to decode a fixed solution equation supervised by human annotation. In this paper, we propose a controlled equation generation solver by leveraging a set of control codes to guide the model to consider certain reasoning logic and decode the corresponding equations expressions transformed from the human reference. The empirical results suggest that our method universally improves the performance on single-unknown (Math23K) and multiple-unknown (DRAW1K, HMWP) benchmarks, with substantial improvements up to 13.2% accuracy on the challenging multiple-unknown datasets.
Compositional generalization refers to a model's capability to generalize to newly composed input data based on the data components observed during training. It has triggered a series of compositional generalization analysis on different tasks as generalization is an important aspect of language and problem solving skills. However, the similar discussion on math word problems (MWPs) is limited. In this manuscript, we study compositional generalization in MWP solving. Specifically, we first introduce a data splitting method to create compositional splits from existing MWP datasets. Meanwhile, we synthesize data to isolate the effect of compositions. To improve the compositional generalization in MWP solving, we propose an iterative data augmentation method that includes diverse compositional variation into training data and could collaborate with MWP methods. During the evaluation, we examine a set of methods and find all of them encounter severe performance loss on the evaluated datasets. We also find our data augmentation method could significantly improve the compositional generalization of general MWP methods. Code is available at https://github.com/demoleiwang/CGMWP.
Given the increase of publications, search for relevant papers becomes tedious. In particular, search across disciplines or schools of thinking is not supported. This is mainly due to the retrieval with keyword queries: technical terms differ in different sciences or at different times. Relevant articles might better be identified by their mathematical problem descriptions. Just looking at the equations in a paper already gives a hint to whether the paper is relevant. Hence, we propose a new approach for retrieval of mathematical expressions based on machine learning. We design an unsupervised representation learning task that combines embedding learning with self-supervised learning. Using graph convolutional neural networks we embed mathematical expression into low-dimensional vector spaces that allow efficient nearest neighbor queries. To train our models, we collect a huge dataset with over 29 million mathematical expressions from over 900,000 publications published on arXiv.org. The math is converted into an XML format, which we view as graph data. Our empirical evaluations involving a new dataset of manually annotated search queries show the benefits of using embedding models for mathematical retrieval. This work was originally published at KDD 2020.
Fukasawa introduced in [Fukasawa, Math Financ, 2012] two necessary conditions for no butterfly arbitrage which require that the $d_1$ and $d_2$ functions of the Black-Scholes formula have to be decreasing. In this article we characterize the set of smiles satisfying these conditions, using the parametrization of the smile in delta. We obtain a parametrization of the set via one real number and three positive functions. We also show that such smiles and their symmetric smiles can be transformed into smiles in the strike space by a bijection. Our result motivates the study of the challenging question of characterizing the subset of butterfly arbitrage-free smiles.
The paper is dedicated to the use of LodView for navigation over the multilingual Linguistic Linked Open Data cloud. First, we define the class of Pubby-like tools, that LodView belongs to, and clarify the relation of this class to the classes of URI dereferenciation tools, RDF browsers and LOD visualization tools. Second, we reveal several limitations of LodView that impede its use for the designated purpose, and propose improvements to be made for fixing these limitations. These improvements are: 1) resolution of Cyrillic URIs; 2) decoding Cyrillic URIs in Turtle representations of resources; 3) support of Cyrillic literals; 4) user-friendly URLs for RDF representations of resources; 5) support of hash URIs; 6) expanding nested resources; 7) support of RDF collections; 8) pagination of resource property values; and 9) support of $\LaTeX$ math notation. Third, we partially implement several of the proposed improvements.
Automatic tagging of knowledge points for practice problems is the basis for managing question bases and improving the automation and intelligence of education. Therefore, it is of great practical significance to study the automatic tagging technology for practice problems. However, there are few studies on the automatic tagging of knowledge points for math problems. Math texts have more complex structures and semantics compared with general texts because they contain unique elements such as symbols and formulas. Therefore, it is difficult to meet the accuracy requirement of knowledge point prediction by directly applying the text classification techniques in general domains. In this paper, K12 math problems taken as the research object, the LABS model based on label-semantic attention and multi-label smoothing combining textual features is proposed to improve the automatic tagging of knowledge points for math problems. The model combines the text classification techniques in general domains and the unique features of math texts. The results show that the models using label-semantic attention or multi-label smoothing perform better on precision, recall, and F1-score metrics than the traditional BiLSTM model, while the LABS model using both performs best. It can be seen that label information can guide the neural networks to extract meaningful information from the problem text, which improves the text classification performance of the model. Moreover, multi-label smoothing combining textual features can fully explore the relationship between text and labels, improve the model's prediction ability for new data and improve the model's classification accuracy.