We report the development of GPT-4, a large-scale, multimodal model which can accept image and text inputs and produce text outputs. While less capable than humans in many real-world scenarios, GPT-4 exhibits human-level performance on various professional and academic benchmarks, including passing a simulated bar exam with a score around the top 10% of test takers. GPT-4 is a Transformer-based model pre-trained to predict the next token in a document. The post-training alignment process results in improved performance on measures of factuality and adherence to desired behavior. A core component of this project was developing infrastructure and optimization methods that behave predictably across a wide range of scales. This allowed us to accurately predict some aspects of GPT-4's performance based on models trained with no more than 1/1,000th the compute of GPT-4.
We study the performance of a commercially available large language model (LLM) known as ChatGPT on math word problems (MWPs) from the dataset DRAW-1K. To our knowledge, this is the first independent evaluation of ChatGPT. We found that ChatGPT's performance changes dramatically based on the requirement to show its work, failing 20% of the time when it provides work compared with 84% when it does not. Further several factors about MWPs relating to the number of unknowns and number of operations that lead to a higher probability of failure when compared with the prior, specifically noting (across all experiments) that the probability of failure increases linearly with the number of addition and subtraction operations. We also have released the dataset of ChatGPT's responses to the MWPs to support further work on the characterization of LLM performance and present baseline machine learning models to predict if ChatGPT can correctly answer an MWP. We have released a dataset comprised of ChatGPT's responses to support further research in this area.
Using the option delta systematically, we derive tighter lower and upper bounds of the Black-Scholes implied volatility than those in Tehranchi [SIAM J. Financ. Math. 7 (2016), 893-916]. As an application, we propose a Newton-Raphson algorithm on the log price that converges rapidly for all price ranges when using a new lower bound as an initial guess. Our new algorithm is a better alternative to the widely used naive Newton-Raphson algorithm, whose convergence is slow for extreme option prices.
The advent of large language models trained on code (code LLMs) has led to significant progress in language-to-code generation. State-of-the-art approaches in this area combine LLM decoding with sample pruning and reranking using test cases or heuristics based on the execution results. However, it is challenging to obtain test cases for many real-world language-to-code applications, and heuristics cannot well capture the semantic features of the execution results, such as data type and value range, which often indicates the correctness of the program. In this work, we propose LEVER, a simple approach to improve language-to-code generation by learning to verify the generated programs with their execution results. Specifically, we train verifiers to determine whether a program sampled from the LLMs is correct or not based on the natural language input, the program itself and its execution results. The sampled programs are reranked by combining the verification score with the LLM generation probability, and marginalizing over programs with the same execution results. On four datasets across the domains of table QA, math QA and basic Python programming, LEVER consistently improves over the base code LLMs(4.6% to 10.9% with code-davinci-002) and achieves new state-of-the-art results on all of them.
Mathematical language in scientific communications and educational scenarios is important yet relatively understudied compared to natural languages. Recent works on mathematical language focus either on representing stand-alone mathematical expressions, especially in their natural tree format, or mathematical reasoning in pre-trained natural language models. Existing works on jointly modeling and generating natural and mathematical languages simply treat mathematical expressions as text, without accounting for the rigid structural properties of mathematical expressions. In this paper, we propose a series of modifications to existing language models to jointly represent and generate text and math: representing mathematical expressions as sequences of node tokens in their operator tree format, using math symbol and tree position embeddings to preserve the semantic and structural properties of mathematical expressions, and using a constrained decoding method to generate mathematically valid expressions. We ground our modifications in GPT-2, resulting in a model MathGPT, and demonstrate that it outperforms baselines on mathematical expression generation tasks.
Developing automatic Math Word Problem (MWP) solvers is a challenging task that demands the ability of understanding and mathematical reasoning over the natural language. Recent neural-based approaches mainly encode the problem text using a language model and decode a mathematical expression over quantities and operators iteratively. Note the problem text of a MWP consists of a context part and a question part, a recent work finds these neural solvers may only perform shallow pattern matching between the context text and the golden expression, where question text is not well used. Meanwhile, existing decoding processes fail to enforce the mathematical laws into the design, where the representations for mathematical equivalent expressions are different. To address these two issues, we propose a new encoder-decoder architecture that fully leverages the question text and preserves step-wise commutative law. Besides generating quantity embeddings, our encoder further encodes the question text and uses it to guide the decoding process. At each step, our decoder uses Deep Sets to compute expression representations so that these embeddings are invariant under any permutation of quantities. Experiments on four established benchmarks demonstrate that our framework outperforms state-of-the-art neural MWP solvers, showing the effectiveness of our techniques. We also conduct a detailed analysis of the results to show the limitations of our approach and further discuss the potential future work. Code is available at https://github.com/sophistz/Question-Aware-Deductive-MWP.
Large language models have been demonstrated to be valuable in different fields. ChatGPT, developed by OpenAI, has been trained using massive amounts of data and simulates human conversation by comprehending context and generating appropriate responses. It has garnered significant attention due to its ability to effectively answer a broad range of human inquiries, with fluent and comprehensive answers surpassing prior public chatbots in both security and usefulness. However, a comprehensive analysis of ChatGPT's failures is lacking, which is the focus of this study. Eleven categories of failures, including reasoning, factual errors, math, coding, and bias, are presented and discussed. The risks, limitations, and societal implications of ChatGPT are also highlighted. The goal of this study is to assist researchers and developers in enhancing future language models and chatbots.
Large language models have achieved impressive performance on various natural language processing tasks. However, so far they have been evaluated primarily on benchmarks where all information in the input context is relevant for solving the task. In this work, we investigate the distractibility of large language models, i.e., how the model problem-solving accuracy can be influenced by irrelevant context. In particular, we introduce Grade-School Math with Irrelevant Context (GSM-IC), an arithmetic reasoning dataset with irrelevant information in the problem description. We use this benchmark to measure the distractibility of cutting-edge prompting techniques for large language models, and find that the model performance is dramatically decreased when irrelevant information is included. We also identify several approaches for mitigating this deficiency, such as decoding with self-consistency and adding to the prompt an instruction that tells the language model to ignore the irrelevant information.
While Chain-of-Thought (CoT) prompting boosts Language Models' (LM) performance on a gamut of complex reasoning tasks, the generated reasoning chain does not necessarily reflect how the model arrives at the answer (aka. faithfulness). We propose Faithful CoT, a reasoning framework involving two stages: Translation (Natural Language query $\rightarrow$ symbolic reasoning chain) and Problem Solving (reasoning chain $\rightarrow$ answer), using an LM and a deterministic solver respectively. This guarantees that the reasoning chain provides a faithful explanation of the final answer. Aside from interpretability, Faithful CoT also improves empirical performance: it outperforms standard CoT on 9 of 10 benchmarks from 4 diverse domains, with a relative accuracy gain of 6.3% on Math Word Problems (MWP), 3.4% on Planning, 5.5% on Multi-hop Question Answering (QA), and 21.4% on Relational Inference. Furthermore, with GPT-4 and Codex, it sets the new state-of-the-art few-shot performance on 7 datasets (with 95.0+ accuracy on 6 of them), showing a strong synergy between faithfulness and accuracy.
The surprising ability of Large Language Models (LLMs) to perform well on complex reasoning with only few-shot chain-of-thought prompts is believed to emerge only in very large-scale models (100+ billion parameters). We show that such abilities can, in fact, be distilled down from GPT-3.5 ($\ge$ 175B) to T5 variants ($\le$ 11B). We propose model specialization, to specialize the model's ability towards a target task. The hypothesis is that large models (commonly viewed as larger than 100B) have strong modeling power, but are spread on a large spectrum of tasks. Small models (commonly viewed as smaller than 10B) have limited model capacity, but if we concentrate their capacity on a specific target task, the model can achieve a decent improved performance. We use multi-step math reasoning as our testbed because it is a very typical emergent ability. We show two important aspects of model abilities: (1). there exists a very complex balance/ tradeoff between language models' multi-dimensional abilities; (2). by paying the price of decreased generic ability, we can clearly lift up the scaling curve of models smaller than 10B towards a specialized multi-step math reasoning ability. We further give comprehensive discussions about important design choices for better generalization, including the tuning data format, the start model checkpoint, and a new model selection method. We hope our practice and discoveries can serve as an important attempt towards specialized smaller models in the new research paradigm set by LLMs.
Large language models (LLMs) such as GPT-4 have recently demonstrated impressive results across a wide range of tasks. LLMs are still limited, however, in that they frequently fail at complex reasoning, their reasoning processes are opaque, they are prone to 'hallucinate' facts, and there are concerns about their underlying biases. Letting models verbalize reasoning steps as natural language, a technique known as chain-of-thought prompting, has recently been proposed as a way to address some of these issues. Here we present ThoughtSource, a meta-dataset and software library for chain-of-thought (CoT) reasoning. The goal of ThoughtSource is to improve future artificial intelligence systems by facilitating qualitative understanding of CoTs, enabling empirical evaluations, and providing training data. This first release of ThoughtSource integrates seven scientific/medical, three general-domain and five math word question answering datasets.
How language models process complex input that requires multiple steps of inference is not well understood. Previous research has shown that information about intermediate values of these inputs can be extracted from the activations of the models, but it is unclear where that information is encoded and whether that information is indeed used during inference. We introduce a method for analyzing how a Transformer model processes these inputs by focusing on simple arithmetic problems and their intermediate values. To trace where information about intermediate values is encoded, we measure the correlation between intermediate values and the activations of the model using principal component analysis (PCA). Then, we perform a causal intervention by manipulating model weights. This intervention shows that the weights identified via tracing are not merely correlated with intermediate values, but causally related to model predictions. Our findings show that the model has a locality to certain intermediate values, and this is useful for enhancing the interpretability of the models.
Studying systems where many individual bodies in motion interact with one another is a complex and interesting area. Simple mechanisms that may be determined for biological, chemical, or physical reasons can lead to astonishingly complex results that require a further understanding of the moving bodies. With the increasing interaction between computation and various scientific areas, it has become more useful, feasible, and important to create models for these systems. Here, we present two families of models, deterministic and nondeterministic, along with three distinct and realistic interaction mechanisms. These are combined in a unique way to provide the groundwork for particle system models across multiple disciplines. This work has applications that range from biology to chemistry and physics. In addition to describing the motivations and math behind all the models, software is provided that allows researchers to quickly adjust and implement what is described here.
Mathematical reasoning is a fundamental aspect of human intelligence and is applicable in various fields, including science, engineering, finance, and everyday life. The development of artificial intelligence (AI) systems capable of solving math problems and proving theorems has garnered significant interest in the fields of machine learning and natural language processing. For example, mathematics serves as a testbed for aspects of reasoning that are challenging for powerful deep learning models, driving new algorithmic and modeling advances. On the other hand, recent advances in large-scale neural language models have opened up new benchmarks and opportunities to use deep learning for mathematical reasoning. In this survey paper, we review the key tasks, datasets, and methods at the intersection of mathematical reasoning and deep learning over the past decade. We also evaluate existing benchmarks and methods, and discuss future research directions in this domain.
Deep neural networks have strong capabilities of memorizing the underlying training data, which can be a serious privacy concern. An effective solution to this problem is to train models with differential privacy, which provides rigorous privacy guarantees by injecting random noise to the gradients. This paper focuses on the scenario where sensitive data are distributed among multiple participants, who jointly train a model through federated learning (FL), using both secure multiparty computation (MPC) to ensure the confidentiality of each gradient update, and differential privacy to avoid data leakage in the resulting model. A major challenge in this setting is that common mechanisms for enforcing DP in deep learning, which inject real-valued noise, are fundamentally incompatible with MPC, which exchanges finite-field integers among the participants. Consequently, most existing DP mechanisms require rather high noise levels, leading to poor model utility. Motivated by this, we propose Skellam mixture mechanism (SMM), an approach to enforce DP on models built via FL. Compared to existing methods, SMM eliminates the assumption that the input gradients must be integer-valued, and, thus, reduces the amount of noise injected to preserve DP. Further, SMM allows tight privacy accounting due to the nice composition and sub-sampling properties of the Skellam distribution, which are key to accurate deep learning with DP. The theoretical analysis of SMM is highly non-trivial, especially considering (i) the complicated math of differentially private deep learning in general and (ii) the fact that the mixture of two Skellam distributions is rather complex, and to our knowledge, has not been studied in the DP literature. Extensive experiments on various practical settings demonstrate that SMM consistently and significantly outperforms existing solutions in terms of the utility of the resulting model.
Geometry problem solving is a well-recognized testbed for evaluating the high-level multi-modal reasoning capability of deep models. In most existing works, two main geometry problems: calculation and proving, are usually treated as two specific tasks, hindering a deep model to unify its reasoning capability on multiple math tasks. However, in essence, these two tasks have similar problem representations and overlapped math knowledge which can improve the understanding and reasoning ability of a deep model on both two tasks. Therefore, we construct a large-scale Unified Geometry problem benchmark, UniGeo, which contains 4,998 calculation problems and 9,543 proving problems. Each proving problem is annotated with a multi-step proof with reasons and mathematical expressions. The proof can be easily reformulated as a proving sequence that shares the same formats with the annotated program sequence for calculation problems. Naturally, we also present a unified multi-task Geometric Transformer framework, Geoformer, to tackle calculation and proving problems simultaneously in the form of sequence generation, which finally shows the reasoning ability can be improved on both two tasks by unifying formulation. Furthermore, we propose a Mathematical Expression Pretraining (MEP) method that aims to predict the mathematical expressions in the problem solution, thus improving the Geoformer model. Experiments on the UniGeo demonstrate that our proposed Geoformer obtains state-of-the-art performance by outperforming task-specific model NGS with over 5.6% and 3.2% accuracies on calculation and proving problems, respectively.
Math word problem (MWP) solving is an important task in question answering which requires human-like reasoning ability. Analogical reasoning has long been used in mathematical education, as it enables students to apply common relational structures of mathematical situations to solve new problems. In this paper, we propose to build a novel MWP solver by leveraging analogical MWPs, which advance the solver's generalization ability across different kinds of MWPs. The key idea, named analogy identification, is to associate the analogical MWP pairs in a latent space, i.e., encoding an MWP close to another analogical MWP, while moving away from the non-analogical ones. Moreover, a solution discriminator is integrated into the MWP solver to enhance the association between the representations of MWPs and their true solutions. The evaluation results verify that our proposed analogical learning strategy promotes the performance of MWP-BERT on Math23k over the state-of-the-art model Generate2Rank, with 5 times fewer parameters in the encoder. We also find that our model has a stronger generalization ability in solving difficult MWPs due to the analogical learning from easy MWPs.
Current math word problem (MWP) solvers are usually Seq2Seq models trained by the (one-problem; one-solution) pairs, each of which is made of a problem description and a solution showing reasoning flow to get the correct answer. However, one MWP problem naturally has multiple solution equations. The training of an MWP solver with (one-problem; one-solution) pairs excludes other correct solutions, and thus limits the generalizability of the MWP solver. One feasible solution to this limitation is to augment multiple solutions to a given problem. However, it is difficult to collect diverse and accurate augment solutions through human efforts. In this paper, we design a new training framework for an MWP solver by introducing a solution buffer and a solution discriminator. The buffer includes solutions generated by an MWP solver to encourage the training data diversity. The discriminator controls the quality of buffered solutions to participate in training. Our framework is flexibly applicable to a wide setting of fully, semi-weakly and weakly supervised training for all Seq2Seq MWP solvers. We conduct extensive experiments on a benchmark dataset Math23k and a new dataset named Weak12k, and show that our framework improves the performance of various MWP solvers under different settings by generating correct and diverse solutions.
Solving math word problems is the task that analyses the relation of quantities and requires an accurate understanding of contextual natural language information. Recent studies show that current models rely on shallow heuristics to predict solutions and could be easily misled by small textual perturbations. To address this problem, we propose a Textual Enhanced Contrastive Learning framework, which enforces the models to distinguish semantically similar examples while holding different mathematical logic. We adopt a self-supervised manner strategy to enrich examples with subtle textual variance by textual reordering or problem re-construction. We then retrieve the hardest to differentiate samples from both equation and textual perspectives and guide the model to learn their representations. Experimental results show that our method achieves state-of-the-art on both widely used benchmark datasets and also exquisitely designed challenge datasets in English and Chinese. \footnote{Our code and data is available at \url{https://github.com/yiyunya/Textual_CL_MWP}
Socratic questioning is an educational method that allows students to discover answers to complex problems by asking them a series of thoughtful questions. Generation of didactically sound questions is challenging, requiring understanding of the reasoning process involved in the problem. We hypothesize that such questioning strategy can not only enhance the human performance, but also assist the math word problem (MWP) solvers. In this work, we explore the ability of large language models (LMs) in generating sequential questions for guiding math word problem-solving. We propose various guided question generation schemes based on input conditioning and reinforcement learning. On both automatic and human quality evaluations, we find that LMs constrained with desirable question properties generate superior questions and improve the overall performance of a math word problem solver. We conduct a preliminary user study to examine the potential value of such question generation models in the education domain. Results suggest that the difficulty level of problems plays an important role in determining whether questioning improves or hinders human performance. We discuss the future of using such questioning strategies in education.