The free energy principle (FEP), as an encompassing framework and a unified brain theory, has been widely applied to account for various problems in fields such as cognitive science, neuroscience, social interaction, and hermeneutics. As a computational model deeply rooted in math and statistics, FEP posits an optimization problem based on variational Bayes, which is solved either by dynamic programming or expectation maximization in practice. However, there seems to be a bottleneck in extending the FEP to machine learning and implementing such models with neural networks. This paper gives a preliminary attempt at bridging FEP and machine learning, via a classical neural network model, the Helmholtz machine. As a variational machine learning model, the Helmholtz machine is optimized by minimizing its free energy, the same objective as FEP. Although the Helmholtz machine is not temporal, it gives an ideal parallel to the vanilla FEP and the hierarchical model of the brain, under which the active inference and predictive coding could be formulated coherently. Besides a detailed theoretical discussion, the paper also presents a preliminary experiment to validate the hypothesis. By fine-tuning the trained neural network through active inference, the model performance is promoted to accuracy above 99\%. In the meantime, the data distribution is continuously deformed to a salience that conforms to the model representation, as a result of active sampling.
This study offers a complete analysis of ChatGPT's mathematics abilities in responding to multiple-choice questions for the Vietnamese National High School Graduation Examination (VNHSGE) on a range of subjects and difficulty levels. The dataset included 250 questions divided into four levels: knowledge (K), comprehension (C), application (A), and high application (H), and it included ten themes that covered diverse mathematical concepts. The outcomes demonstrate that ChatGPT's performance varies depending on the difficulty level and subject. It performed best on questions at Level (K), with an accuracy rate of $83\%$; but, as the difficulty level rose, it scored poorly, with an accuracy rate of $10\%$. The study has also shown that ChatGPT significantly succeeds in providing responses to questions on subjects including exponential and logarithmic functions, geometric progression, and arithmetic progression. The study found that ChatGPT had difficulty correctly answering questions on topics including derivatives and applications, spatial geometry, and Oxyz spatial calculus. Additionally, this study contrasted ChatGPT outcomes with Vietnamese students in VNHSGE and in other math competitions. ChatGPT dominated in the SAT Math competition with a success rate of $70\%$, followed by VNHSGE mathematics ($58.8\%)$. However, its success rates were lower on other exams, such as AP Statistics, the GRE Quantitative, AMC 10, AMC 12, and AP Calculus BC. These results suggest that ChatGPT has the potential to be an effective teaching tool for mathematics, but more work is needed to enhance its handling of graphical data and address the challenges presented by questions that are getting more challenging.
The increasing reliance on Large Language Models (LLMs) across academia and industry necessitates a comprehensive understanding of their robustness to prompts. In response to this vital need, we introduce PromptRobust, a robustness benchmark designed to measure LLMs' resilience to adversarial prompts. This study uses a plethora of adversarial textual attacks targeting prompts across multiple levels: character, word, sentence, and semantic. The adversarial prompts, crafted to mimic plausible user errors like typos or synonyms, aim to evaluate how slight deviations can affect LLM outcomes while maintaining semantic integrity. These prompts are then employed in diverse tasks including sentiment analysis, natural language inference, reading comprehension, machine translation, and math problem-solving. Our study generates 4,788 adversarial prompts, meticulously evaluated over 8 tasks and 13 datasets. Our findings demonstrate that contemporary LLMs are not robust to adversarial prompts. Furthermore, we present a comprehensive analysis to understand the mystery behind prompt robustness and its transferability. We then offer insightful robustness analysis and pragmatic recommendations for prompt composition, beneficial to both researchers and everyday users.
Solving math story problems is a complex task for students and NLP models alike, requiring them to understand the world as described in the story and reason over it to compute an answer. Recent years have seen impressive performance on automatically solving these problems with large pre-trained language models and innovative techniques to prompt them. However, it remains unclear if these models possess accurate representations of mathematical concepts. This leads to lack of interpretability and trustworthiness which impedes their usefulness in various applications. In this paper, we consolidate previous work on categorizing and representing math story problems and develop MathWorld, which is a graph-based semantic formalism specific for the domain of math story problems. With MathWorld, we can assign world models to math story problems which represent the situations and actions introduced in the text and their mathematical relationships. We combine math story problems from several existing datasets and annotate a corpus of 1,019 problems and 3,204 logical forms with MathWorld. Using this data, we demonstrate the following use cases of MathWorld: (1) prompting language models with synthetically generated question-answer pairs to probe their reasoning and world modeling abilities, and (2) generating new problems by using the world models as a design space.
Coaching, which involves classroom observation and expert feedback, is a widespread and fundamental part of teacher training. However, the majority of teachers do not have access to consistent, high quality coaching due to limited resources and access to expertise. We explore whether generative AI could become a cost-effective complement to expert feedback by serving as an automated teacher coach. In doing so, we propose three teacher coaching tasks for generative AI: (A) scoring transcript segments based on classroom observation instruments, (B) identifying highlights and missed opportunities for good instructional strategies, and (C) providing actionable suggestions for eliciting more student reasoning. We recruit expert math teachers to evaluate the zero-shot performance of ChatGPT on each of these tasks for elementary math classroom transcripts. Our results reveal that ChatGPT generates responses that are relevant to improving instruction, but they are often not novel or insightful. For example, 82% of the model's suggestions point to places in the transcript where the teacher is already implementing that suggestion. Our work highlights the challenges of producing insightful, novel and truthful feedback for teachers while paving the way for future research to address these obstacles and improve the capacity of generative AI to coach teachers.
Mathematical reasoning is regarded as a necessary ability for Language Models (LMs). Recent works demonstrate large LMs' impressive performance in solving math problems. The success is attributed to their Chain-of-Thought (CoT) reasoning abilities, i.e., the ability to decompose complex questions into step-by-step reasoning chains, but such ability seems only to emerge from models with abundant parameters. This work investigates how to incorporate relatively small LMs with the capabilities of multi-step reasoning. We propose to inject such abilities by continually pre-training LMs on a synthetic dataset MsAT which is composed of Multi-step Arithmetic Tasks. Our experiments on four math word problem datasets show the effectiveness of the proposed method in enhancing LMs' math reasoning abilities.
Employing Large Language Models (LLMs) to address mathematical problems is an intriguing research endeavor, considering the abundance of math problems expressed in natural language across numerous science and engineering fields. LLMs, with their generalized ability, are used as a foundation model to build AI agents for different tasks. In this paper, we study the effectiveness of utilizing LLM agents to solve math problems through conversations. We propose MathChat, a conversational problem-solving framework designed for math problems. MathChat consists of an LLM agent and a user proxy agent which is responsible for tool execution and additional guidance. This synergy facilitates a collaborative problem-solving process, where the agents engage in a dialogue to solve the problems. We perform evaluation on difficult high school competition problems from the MATH dataset. Utilizing Python, we show that MathChat can further improve previous tool-using prompting methods by 6%.
Liver transplantation continues to be the gold standard for treating patients with end-stage liver diseases. However, despite the huge success of liver transplantation in improving patient outcomes, long term graft survival continues to be a major problem. The current clinical practice in the management of liver transplant patients is centered around immunosuppressive multidrug regimens. Current research has been focusing on phenotypic personalized medicine as a novel approach in the optimization of immunosuppression, a regressional math modeling focusing on individual patient dose and response using specific markers like transaminases. A prospective area of study includes the development of a mechanistic computational math modeling for optimizing immunosuppression to improve patient outcomes and increase long-term graft survival by exploring the intricate immune/drug interactions to help us further our understanding and management of medical problems like transplants, autoimmunity, and cancer therapy. Thus, by increasing long-term graft survival, the need for redo transplants will decrease, which will free up organs and potentially help with the organ shortage problem promoting equity and equal opportunity for transplants, as well as decreasing the medical costs associated with additional testing and hospital admissions. Although long-term graft survival remains challenging, computational and quantitative methods have led to significant improvements. In this article, we review recent advances and remaining opportunities. We focus on the following topics: donor organ availability and allocation with a focus on equity, monitoring of patient and graft health, and optimization of immunosuppression dosing.
Large language models (LLMs) have transformed numerous AI applications. On-device LLM is becoming increasingly important: running LLMs locally on edge devices can reduce the cloud computing cost and protect users' privacy. However, the astronomical model size and the limited hardware resource pose significant deployment challenges. We propose Activation-aware Weight Quantization (AWQ), a hardware-friendly approach for LLM low-bit weight-only quantization. AWQ finds that not all weights in an LLM are equally important. Protecting only 1% salient weights can greatly reduce quantization error. To identify salient weight channels, we should refer to the activation distribution, not weights. To avoid the hardware-inefficient mix-precision quantization, we mathematically derive that scaling up the salient channels can reduce the quantization error. AWQ employs an equivalent transformation to scale the salient weight channels to protect them. The scale is determined by collecting the activation statistics offline. AWQ does not rely on any backpropagation or reconstruction, so it generalizes to different domains and modalities without overfitting the calibration set. AWQ outperforms existing work on various language modeling and domain-specific benchmarks (coding and math). Thanks to better generalization, it achieves excellent quantization performance for instruction-tuned LMs and, for the first time, multi-modal LMs. Alongside AWQ, we implement TinyChat, an efficient and flexible inference framework tailored for 4-bit on-device LLM/VLMs. With kernel fusion and platform-aware weight packing, TinyChat offers more than 3x speedup over the Huggingface FP16 implementation on both desktop and mobile GPUs. It also democratizes the deployment of the 70B Llama-2 model on mobile GPUs.
Enriching the quality of early childhood education with interactive math learning at home systems, empowered by recent advances in conversational AI technologies, is slowly becoming a reality. With this motivation, we implement a multimodal dialogue system to support play-based learning experiences at home, guiding kids to master basic math concepts. This work explores Spoken Language Understanding (SLU) pipeline within a task-oriented dialogue system developed for Kid Space, with cascading Automatic Speech Recognition (ASR) and Natural Language Understanding (NLU) components evaluated on our home deployment data with kids going through gamified math learning activities. We validate the advantages of a multi-task architecture for NLU and experiment with a diverse set of pretrained language representations for Intent Recognition and Entity Extraction tasks in the math learning domain. To recognize kids' speech in realistic home environments, we investigate several ASR systems, including the commercial Google Cloud and the latest open-source Whisper solutions with varying model sizes. We evaluate the SLU pipeline by testing our best-performing NLU models on noisy ASR output to inspect the challenges of understanding children for math learning in authentic homes.
Automated scoring of student responses to open-ended questions, including short-answer questions, has great potential to scale to a large number of responses. Recent approaches for automated scoring rely on supervised learning, i.e., training classifiers or fine-tuning language models on a small number of responses with human-provided score labels. However, since scoring is a subjective process, these human scores are noisy and can be highly variable, depending on the scorer. In this paper, we investigate a collection of models that account for the individual preferences and tendencies of each human scorer in the automated scoring task. We apply these models to a short-answer math response dataset where each response is scored (often differently) by multiple different human scorers. We conduct quantitative experiments to show that our scorer models lead to improved automated scoring accuracy. We also conduct quantitative experiments and case studies to analyze the individual preferences and tendencies of scorers. We found that scorers can be grouped into several obvious clusters, with each cluster having distinct features, and analyzed them in detail.
Solutions to math word problems (MWPs) with step-by-step explanations are valuable, especially in education, to help students better comprehend problem-solving strategies. Most existing approaches only focus on obtaining the final correct answer. A few recent approaches leverage intermediate solution steps to improve final answer correctness but often cannot generate coherent steps with a clear solution strategy. Contrary to existing work, we focus on improving the correctness and coherence of the intermediate solutions steps. We propose a step-by-step planning approach for intermediate solution generation, which strategically plans the generation of the next solution step based on the MWP and the previous solution steps. Our approach first plans the next step by predicting the necessary math operation needed to proceed, given history steps, then generates the next step, token-by-token, by prompting a language model with the predicted math operation. Experiments on the GSM8K dataset demonstrate that our approach improves the accuracy and interpretability of the solution on both automatic metrics and human evaluation.
In recent years, large language models have greatly improved in their ability to perform complex multi-step reasoning. However, even state-of-the-art models still regularly produce logical mistakes. To train more reliable models, we can turn either to outcome supervision, which provides feedback for a final result, or process supervision, which provides feedback for each intermediate reasoning step. Given the importance of training reliable models, and given the high cost of human feedback, it is important to carefully compare the both methods. Recent work has already begun this comparison, but many questions still remain. We conduct our own investigation, finding that process supervision significantly outperforms outcome supervision for training models to solve problems from the challenging MATH dataset. Our process-supervised model solves 78% of problems from a representative subset of the MATH test set. Additionally, we show that active learning significantly improves the efficacy of process supervision. To support related research, we also release PRM800K, the complete dataset of 800,000 step-level human feedback labels used to train our best reward model.
Large language models are becoming increasingly integrated into our lives. Hence, it is important to understand the biases present in their outputs in order to avoid perpetuating harmful stereotypes, which originate in our own flawed ways of thinking. This challenge requires developing new benchmarks and methods for quantifying affective and semantic bias, keeping in mind that LLMs act as psycho-social mirrors that reflect the views and tendencies that are prevalent in society. One such tendency that has harmful negative effects is the global phenomenon of anxiety toward math and STEM subjects. Here, we investigate perceptions of math and STEM fields provided by cutting-edge language models, namely GPT-3, Chat-GPT, and GPT-4, by applying an approach from network science and cognitive psychology. Specifically, we use behavioral forma mentis networks (BFMNs) to understand how these LLMs frame math and STEM disciplines in relation to other concepts. We use data obtained by probing the three LLMs in a language generation task that has previously been applied to humans. Our findings indicate that LLMs have an overall negative perception of math and STEM fields, with math being perceived most negatively. We observe significant differences across the three LLMs. We observe that newer versions (i.e. GPT-4) produce richer, more complex perceptions as well as less negative perceptions compared to older versions and N=159 high-school students. These findings suggest that advances in the architecture of LLMs may lead to increasingly less biased models that could even perhaps someday aid in reducing harmful stereotypes in society rather than perpetuating them.
A comparison between three chatbots which are based on large language models, namely ChatGPT-3.5, ChatGPT-4 and Google Bard is presented, focusing on their ability to give correct answers to mathematics and logic problems. In particular, we check their ability to Understand the problem at hand; Apply appropriate algorithms or methods for its solution; and Generate a coherent response and a correct answer. We use 30 questions that are clear, without any ambiguities, fully described with plain text only, and have a unique, well defined correct answer. The questions are divided into two sets of 15 each. The questions of Set A are 15 "Original" problems that cannot be found online, while Set B contains 15 "Published" problems that one can find online, usually with their solution. Each question is posed three times to each chatbot. The answers are recorded and discussed, highlighting their strengths and weaknesses. It has been found that for straightforward arithmetic, algebraic expressions, or basic logic puzzles, chatbots may provide accurate solutions, although not in every attempt. However, for more complex mathematical problems or advanced logic tasks, their answers, although written in a usually "convincing" way, may not be reliable. Consistency is also an issue, as many times a chatbot will provide conflicting answers when given the same question more than once. A comparative quantitative evaluation of the three chatbots is made through scoring their final answers based on correctness. It was found that ChatGPT-4 outperforms ChatGPT-3.5 in both sets of questions. Bard comes third in the original questions of Set A, behind the other two chatbots, while it has the best performance (first place) in the published questions of Set B. This is probably because Bard has direct access to the internet, in contrast to ChatGPT chatbots which do not have any communication with the outside world.
Chain-of-thought (CoT) prompting with large language models has proven effective in numerous natural language processing tasks, but designing prompts that generalize well to diverse problem types can be challenging, especially in the context of math word problem (MWP) solving. Additionally, it is common to have a large amount of training data that have a better diversity coverage but CoT annotations are not available, which limits the use of supervised learning techniques. To address these issues, we investigate two approaches to leverage the training data in a few-shot prompting scenario: dynamic program prompting and program distillation. Our approach is largely inspired by Gao et al., (2022), where they proposed to replace the CoT with the programs as the intermediate reasoning step. Such a prompting strategy allows us to accurately verify the answer correctness through program execution in MWP solving. Our dynamic program prompting involves annotating the training data by sampling correct programs from a large language model, while program distillation involves adapting a smaller model to the program-annotated training data. Our experiments on three standard MWP datasets demonstrate the effectiveness of these approaches, yielding significant improvements over previous baselines for prompting and fine-tuning. Our results suggest that leveraging a large amount of training data can improve the generalization ability of prompts and boost the performance of fine-tuned small models in MWP solving.
The temporal aspect is a significant dimension of our reality. We notice the challenge that large language models (LLMs) face when engaging in temporal reasoning. Our preliminary experiments show that methods involving the generation of intermediate reasoning steps, such as chain-of-thought and program-aided language models, do not consistently boost the performance of complex temporal question-answering tasks. This limitation can be attributed to the LLMs' inadequate understanding of temporal information. To address this problem, we propose TempLogic, a novel framework designed specifically for temporal question-answering tasks across three levels of reasoning. TempLogic incorporates retrieval-guided context distillation, temporal data extraction, and tailor-made logic reasoning. Extensive experiments and analysis demonstrate the effectiveness of our framework in solving intricate time-bound reasoning tasks.
Chain-of-Thought (CoT) prompting enables large language models to solve complex reasoning problems by generating intermediate steps. However, confined by its inherent single-pass and sequential generation process, CoT heavily relies on the initial decisions, causing errors in early steps to accumulate and impact the final answers. In contrast, humans adopt recursive thinking when tackling complex reasoning problems, i.e., iteratively breaking the original problem into approachable sub-problems and aggregating their answers to resolve the original one. Inspired by the human cognitive process, we propose SOCRATIC QUESTIONING, a divide-and-conquer style algorithm that mimics the recursive thinking process. Specifically, SOCRATIC QUESTIONING leverages large language models to raise and answer sub-questions until collecting enough information to tackle the original question. Unlike CoT, SOCRATIC QUESTIONING explicitly navigates the thinking space, stimulates effective recursive thinking, and is more robust towards errors in the thinking process. Extensive experiments on several complex reasoning tasks, including MMLU, MATH, LogiQA, and visual question-answering demonstrate significant performance improvements over the state-of-the-art prompting methods, such as CoT, and Tree-of-Thought. The qualitative analysis clearly shows that the intermediate reasoning steps elicited by SOCRATIC QUESTIONING are similar to humans' recursively thinking process of complex reasoning problems.
In the context of multi-step reasoning, e.g., with chain-of-thought, language models (LMs) can easily assign a high likelihood to incorrect steps. As a result, decoding strategies that optimize for solution likelihood often yield incorrect solutions. To address this issue, we propose Guiding chain-of-thought ReAsoning with a CorrectnEss Discriminator (GRACE), a stepwise decoding approach that steers the decoding process towards producing correct reasoning steps. GRACE employs a discriminator trained with a contrastive loss over correct and incorrect steps, which is used during decoding to score next-step candidates based on their correctness. Importantly, GRACE only requires sampling from the LM, without the need for LM training or fine-tuning. Using models from FLAN-T5 and LLaMA families, we evaluate GRACE over four math and two symbolic reasoning tasks, where it exhibits substantial performance gains compared to greedy decoding, verifiers, and self-consistency in most settings. When further combined with self-consistency, GRACE outperforms all the baselines by sizeable margins. Human and LLM evaluations over GSM8K show that GRACE not only improves the final answer accuracy but also the correctness of the intermediate reasoning. Our implementation can be accessed at \url{https://github.com/mukhal/grace}.
Does inverse scaling only occur as a function of model parameter size, or can it also occur over the course of training? We carry out an exploratory study investigating whether, over the course of training on the language modeling task, the performance of language models at specific tasks can decrease while general performance remains high. We find that for two tasks from the Inverse Scaling Challenge - quote-repetition and redefine-math - this is indeed the case. Specifically, we find that for Pythia (Biderman et al., 2023) models with a higher number of parameters, performance decreases over the course of training at these two tasks, despite these models showing standard (positive) scaling overall. This highlights the importance of testing model performance at all relevant benchmarks any time they are trained on additional data, even if their overall performance improves.