Mathematical word problems (MWPs) involve the task of converting textual descriptions into mathematical equations. This poses a significant challenge in natural language processing, particularly for low-resource languages such as Bengali. This paper addresses this challenge by developing an innovative approach to solving Bengali MWPs using transformer-based models, including Basic Transformer, mT5, BanglaT5, and mBART50. To support this effort, the "PatiGonit" dataset was introduced, containing 10,000 Bengali math problems, and these models were fine-tuned to translate the word problems into equations accurately. The evaluation revealed that the mT5 model achieved the highest accuracy of 97.30%, demonstrating the effectiveness of transformer models in this domain. This research marks a significant step forward in Bengali natural language processing, offering valuable methodologies and resources for educational AI tools. By improving math education, it also supports the development of advanced problem-solving skills for Bengali-speaking students.
Cutting-edge large language models (LLMs) demonstrate promising performance in solving complex math problems with a divide-and-conquer pipeline and the assistance of in-context learning (ICL) examples. However, their potential for improvement is limited by two critical problems within their ICL examples: granularity-mismatch and the ensuing negative-effect noise problem. Specifically, the LLMs are capable of the dividing process yet mostly failed by inaccurate reasoning within a few conquer steps, while the ICL examples retrieved in question-grained sometimes lack relevant steps for a specific challenging reasoning step. Further, this disconnect may hinder the correct reasoning due to its irrelevance. To this end, we focus on improving the reasoning quality within each step and present BoostStep. BoostStep aligns the granularity between the retrieving and reasoning on step grained, and provides highly related ICL examples for each reasoning step with a novel `first-try' strategy. BoostStep provides more relevant examples than the coarse question-grained strategy, enhancing the model reasoning quality within each step steadily. BoostStep is a general and robust reasoning-enhancing method that not only improves standalone reasoning performance but also integrates seamlessly with Monte Carlo Tree Search methods (MCTS) to refine both candidate generation and decision-making. Quantitatively, it improves GPT-4o and Qwen2.5-Math-72B by 3.6\% and 2.0\% respectively on various mathematical benchmarks, and 7.5\% gain combined with MCTS.
Large language models have achieved significant advancements in complex mathematical reasoning benchmarks, such as MATH. However, their substantial computational requirements present challenges for practical deployment. Model quantization has emerged as an effective strategy to reduce memory usage and computational costs by employing lower precision and bit-width representations. In this study, we systematically evaluate the impact of quantization on mathematical reasoning tasks. We introduce a multidimensional evaluation framework that qualitatively assesses specific capability dimensions and conduct quantitative analyses on the step-by-step outputs of various quantization methods. Our results demonstrate that quantization differentially affects numerical computation and reasoning planning abilities, identifying key areas where quantized models experience performance degradation.
Enhancing the reasoning capabilities of Large Language Models remains a critical challenge in artificial intelligence. We introduce RDoLT, Recursive Decomposition of Logical Thought prompting, a novel framework that significantly boosts LLM reasoning performance. RDoLT is built on three key innovations: (1) recursively breaking down complex reasoning tasks into sub-tasks of progressive complexity; (2) employing an advanced selection and scoring mechanism to identify the most promising reasoning thoughts; and (3) integrating a knowledge propagation module that mimics human learning by keeping track of strong and weak thoughts for information propagation. Our approach was evaluated across multiple benchmarks, including GSM8K, SVAMP, MultiArith, LastLetterConcatenation, and Gaokao2023 Math. The results demonstrate that RDoLT consistently outperforms existing state-of-the-art techniques, achieving a 90.98 percent accuracy on GSM8K with ChatGPT-4, surpassing state-of-the-art techniques by 6.28 percent. Similar improvements were observed on other benchmarks, with accuracy gains ranging from 5.5 percent to 6.75 percent. These findings highlight RDoLT's potential to advance prompt engineering, offering a more effective and generalizable approach to complex reasoning tasks.
Current inference scaling methods, such as Self-consistency and Best-of-N, have proven effective in improving the accuracy of LLMs on complex reasoning tasks. However, these methods rely heavily on the quality of candidate responses and are unable to produce correct answers when all candidates are incorrect. In this paper, we propose a novel inference scaling strategy, CoT-based Synthesizer, which leverages CoT reasoning to synthesize superior answers by analyzing complementary information from multiple candidate responses, even when all candidate responses are flawed. To enable a lightweight and cost-effective implementation, we introduce an automated data generation pipeline that creates diverse training data. This allows smaller LLMs trained on this data to improve the inference accuracy of larger models, including API-based LLMs. Experimental results across four benchmark datasets with seven policy models demonstrate that our method significantly enhances performance, with gains of 11.8% for Llama3-8B and 10.3% for GPT-4o on the MATH dataset. The corresponding training data and code are publicly available on https://github.com/RUCKBReasoning/CoT-based-Synthesizer.
In this work, we present a probabilistic model for directed graphs where nodes have attributes and labels. This model serves as a generative classifier capable of predicting the labels of unseen nodes using either maximum likelihood or maximum a posteriori estimations. The predictions made by this model are highly interpretable, contrasting with some common methods for node classification, such as graph neural networks. We applied the model to two datasets, demonstrating predictive performance that is competitive with, and even superior to, state-of-the-art methods. One of the datasets considered is adapted from the Math Genealogy Project, which has not previously been utilized for this purpose. Consequently, we evaluated several classification algorithms on this dataset to compare the performance of our model and provide benchmarks for this new resource.
In this paper, we rethink delay Doppler channels (also called doubly selective channels). We prove that no modulation schemes (including the current active VOFDM/OTFS) can compensate a non-trivial Doppler spread well. We then discuss some of the existing methods to deal with time-varying channels, in particular time-frequency (TF) coding in an OFDM system. TF coding is equivalent to space-time coding in the math part. We also summarize state of the art on space-time coding that was an active research topic over a decade ago.
Recent advances in large language models (LLMs) have demonstrated the effectiveness of Iterative Self-Improvement (ISI) techniques. However, continuous training on self-generated data leads to reduced output diversity, a limitation particularly critical in reasoning tasks where diverse solution paths are essential. We present DIVE (Diversified Iterative Self-Improvement), a novel framework that addresses this challenge through two key components: Sample Pool Expansion for broader solution exploration, and Data Selection for balancing diversity and quality in preference pairs. Experiments on MATH and GSM8k datasets show that DIVE achieves a 10% to 45% relative increase in output diversity metrics while maintaining performance quality compared to vanilla ISI. Our ablation studies confirm both components' significance in achieving these improvements. Code is available at https://github.com/qinyiwei/DIVE.
Since the launch of ChatGPT in late 2022, the capacities of Large Language Models and their evaluation have been in constant discussion and evaluation both in academic research and in the industry. Scenarios and benchmarks have been developed in several areas such as law, medicine and math (Bommasani et al., 2023) and there is continuous evaluation of model variants. One area that has not received sufficient scenario development attention is journalism, and in particular journalistic sourcing and ethics. Journalism is a crucial truth-determination function in democracy (Vincent, 2023), and sourcing is a crucial pillar to all original journalistic output. Evaluating the capacities of LLMs to annotate stories for the different signals of sourcing and how reporters justify them is a crucial scenario that warrants a benchmark approach. It offers potential to build automated systems to contrast more transparent and ethically rigorous forms of journalism with everyday fare. In this paper we lay out a scenario to evaluate LLM performance on identifying and annotating sourcing in news stories on a five-category schema inspired from journalism studies (Gans, 2004). We offer the use case, our dataset and metrics and as the first step towards systematic benchmarking. Our accuracy findings indicate LLM-based approaches have more catching to do in identifying all the sourced statements in a story, and equally, in matching the type of sources. An even harder task is spotting source justifications.
Partial differential equation (PDE) is an important math tool in science and engineering. This paper experimentally demonstrates an optical neural PDE solver by leveraging the back-propagation-free on-photonic-chip training of physics-informed neural networks.
Recently, "visual o1" began to enter people's vision, with expectations that this slow-thinking design can solve visual reasoning tasks, especially geometric math problems. However, the reality is that current LVLMs (Large Vision Language Models) can hardly even accurately copy a geometric figure, let alone truly understand the complex inherent logic and spatial relationships within geometric shapes. We believe accurate copying (strong perception) is the first step to visual o1. Accordingly, we introduce the concept of "slow perception" (SP), which guides the model to gradually perceive basic point-line combinations, as our humans, reconstruct complex geometric structures progressively. There are two-fold stages in SP: a) perception decomposition. Perception is not instantaneous. In this stage, complex geometric figures are broken down into basic simple units to unify geometry representation. b) perception flow, which acknowledges that accurately tracing a line is not an easy task. This stage aims to avoid "long visual jumps" in regressing line segments by using a proposed "perceptual ruler" to trace each line stroke-by-stroke. Surprisingly, such a human-like perception manner enjoys an inference time scaling law -- the slower, the better. Researchers strive to speed up the model's perception in the past, but we slow it down again, allowing the model to read the image step-by-step and carefully.
We present Dynamic Skill Adaptation (DSA), an adaptive and dynamic framework to adapt novel and complex skills to Large Language Models (LLMs). Compared with previous work which learns from human-curated and static data in random orders, we propose to first automatically generate and organize the training data by mimicking the learning pathways of human and then dynamically tailor the training data based on the training dynamics. Specifically, inspired by the learning structures and teaching strategies in the human education system, we first construct a skill graph by decomposing complex skills into sub-skills and arranging them based on their dependencies in human syllables. For every skill, we utilize LLMs to generate both textbook-like data which contains detailed descriptions of skills for pre-training and exercise-like data which targets at explicitly utilizing the skills to solve problems for instruction-tuning. Furthermore, during the instruction-tuning, we dynamically update the training data which down-weight easy-to-learn examples, generate more complex examples, and filter out data with errors. Experiments on large language models such as LLAMA and Mistral demonstrate the effectiveness of our proposed methods in adapting math reasoning skills and social study skills.
Large language models (LLMs) have been routinely used to solve various tasks using step-by-step reasoning. However, the structure of intermediate reasoning steps, or thoughts, is rigid and unidirectional, such as chains, trees, or acyclic-directed graphs. Consequently, the resulting inflexible and forward-only reasoning may not address challenging tasks and fail when the LLM frequently gives false responses, i.e., ``hallucinations''. This paper proposes a new reasoning framework, called Thought Rollback (TR), allowing LLMs to adaptively build thought structure while maintaining effective reasoning toward problem-solving under ``hallucinations''. The core mechanism of TR is rolling back thoughts, which allows LLMs to perform error analysis on thoughts, and thus roll back to any previously mistaken thought for revision. Subsequently, by including such trial-and-error in the prompt to guide the LLM, each rollback leads to one more reliable reasoning path. Therefore, starting with a simple prompt without human annotations, LLM with TR adaptively and gradually explores thoughts for a correct solution. Comprehensive experiments on mathematical problems and multi-task reasoning demonstrate the state-of-the-art performance of TR in terms of problem-solving rate and interaction cost. For instance, the solving rate of GPT-4 with TR outperforms the current best by $9\%$ on the MATH dataset.
The rapid development of large language models (LLMs) necessitates robust, unbiased, and scalable methods for evaluating their capabilities. However, human annotations are expensive to scale, model-based evaluations are prone to biases in answer style, while target-answer-based benchmarks are vulnerable to data contamination and cheating. To address these limitations, we propose StructTest, a novel benchmark that evaluates LLMs on their ability to produce compositionally specified structured outputs as an unbiased, cheap-to-run and difficult-to-cheat measure. The evaluation is done deterministically by a rule-based evaluator, which can be easily extended to new tasks. By testing structured outputs across diverse task domains -- including Summarization, Code, HTML and Math -- we demonstrate that StructTest serves as a good proxy for general reasoning abilities, as producing structured outputs often requires internal logical reasoning. We believe that StructTest offers a critical, complementary approach to objective and robust model evaluation.
Causal reasoning capabilities are essential for large language models (LLMs) in a wide range of applications, such as education and healthcare. But there is still a lack of benchmarks for a better understanding of such capabilities. Current LLM benchmarks are mainly based on conversational tasks, academic math tests, and coding tests. Such benchmarks evaluate LLMs in well-regularized settings, but they are limited in assessing the skills and abilities to solve real-world problems. In this work, we provide a benchmark, named by CARL-GT, which evaluates CAusal Reasoning capabilities of large Language models using Graphs and Tabular data. The benchmark has a diverse range of tasks for evaluating LLMs from causal graph reasoning, knowledge discovery, and decision-making aspects. In addition, effective zero-shot learning prompts are developed for the tasks. In our experiments, we leverage the benchmark for evaluating open-source LLMs and provide a detailed comparison of LLMs for causal reasoning abilities. We found that LLMs are still weak in casual reasoning, especially with tabular data to discover new insights. Furthermore, we investigate and discuss the relationships of different benchmark tasks by analyzing the performance of LLMs. The experimental results show that LLMs have different strength over different tasks and that their performance on tasks in different categories, i.e., causal graph reasoning, knowledge discovery, and decision-making, shows stronger correlation than tasks in the same category.
We present a novel approach to generating mathematical conjectures using Large Language Models (LLMs). Focusing on the solubilizer, a relatively recent construct in group theory, we demonstrate how LLMs such as ChatGPT, Gemini, and Claude can be leveraged to generate conjectures. These conjectures are pruned by allowing the LLMs to generate counterexamples. Our results indicate that LLMs are capable of producing original conjectures that, while not groundbreaking, are either plausible or falsifiable via counterexamples, though they exhibit limitations in code execution.
We use modifications of the Adams method and very fast and accurate sinh-acceleration method of the Fourier inversion (iFT) (S.Boyarchenko and Levendorski\u{i}, IJTAF 2019, v.22) to evaluate prices of vanilla options; for options of moderate and long maturities and strikes not very far from the spot, thousands of prices can be calculated in several msec. with relative errors of the order of 0.5\% and smaller running Matlab on a Mac with moderate characteristics. We demonstrate that for the calibrated set of parameters in Euch and Rosenbaum, Math. Finance 2019, v. 29, the correct implied volatility surface is significantly flatter and fits the data very poorly, hence, the calibration results in op.cit. is an example of the {\em ghost calibration} (M.Boyarchenko and Levendorki\u{i}, Quantitative Finance 2015, v. 15): the errors of the model and numerical method almost cancel one another. We explain how calibration errors of this sort are generated by each of popular versions of numerical realizations of iFT (Carr-Madan, Lipton-Lewis and COS methods) with prefixed parameters of a numerical method, resulting in spurious volatility smiles and skews. We suggest a general {\em Conformal Bootstrap principle} which allows one to avoid ghost calibration errors. We outline schemes of application of Conformal Bootstrap principle and the method of the paper to the design of accurate and fast calibration procedures.
Large Language Models still encounter substantial challenges in reasoning tasks, especially for smaller models, which many users may be restricted to due to resource constraints (e.g. GPU memory restrictions). Inference-time methods to boost LLM performance, such as prompting methods to invoke certain reasoning pathways in responses, have been shown effective in past works, though they largely rely on sequential queries. The ensemble method, which consists of multiple constituent models running in parallel, is a promising approach to achieving better inference-time performance, especially given recent developments that enabled significant speed-ups in LLM batch inference. In this work, we propose a novel, training-free LLM ensemble framework where a single LLM model is fed an optimized, diverse set of prompts in parallel, effectively producing an ensemble at inference time to achieve performance improvement in reasoning tasks. We empirically demonstrate that our method leads to significant gains on math reasoning tasks, e.g., on MATH, where our ensemble consisting of a few small models (e.g., three Qwen2-MATH-1.5B-it models) can outperform a larger model (e.g., Qwen2-MATH-7B-it).
The rise of large language models (LLMs) offers new opportunities for automatic error detection in education, particularly for math word problems (MWPs). While prior studies demonstrate the promise of LLMs as error detectors, they overlook the presence of multiple valid solutions for a single MWP. Our preliminary analysis reveals a significant performance gap between conventional and alternative solutions in MWPs, a phenomenon we term conformity bias in this work. To mitigate this bias, we introduce the Ask-Before-Detect (AskBD) framework, which generates adaptive reference solutions using LLMs to enhance error detection. Experiments on 200 examples of GSM8K show that AskBD effectively mitigates bias and improves performance, especially when combined with reasoning-enhancing techniques like chain-of-thought prompting.
With current state-of-the-art approaches aimed at enhancing the reasoning capabilities of Large Language Models(LLMs) through iterative preference learning inspired by AlphaZero, we propose to further enhance the step-wise reasoning capabilities through intrinsic self-correction to some extent. Our work leverages step-wise preference learning to enhance self-verification via reinforcement learning. We initially conduct our work through a two-stage training procedure. At the first stage, the self-correction reasoning ability of an LLM is enhanced through its own predictions, relying entirely on self-generated data within the intrinsic self-correction to some extent. At the second stage, the baseline step-wise preference learning is leveraged via the application of the enhanced self-correct policy achieved at the first stage. In the evaluation of arithmetic reasoning tasks, our approach outperforms OpenMath2-Llama3.1-8B, dart-math-mistral-7b-uniform on MATH with increases in accuracy to 71.34%(+4.18%) and 48.06%(+4.94%) and LLama-3.1-8B-Instruct, Mistral-7B-Instruct-v0.1 on GSM8K with increases in accuracy to 86.76%(+2.00%) and 38.06%(+2.28%).