Probing Large Language Models for Arithmetic Reasoning Capabilities

Core Concepts

Large Language Models struggle to perform basic arithmetic reasoning over implicitly held numerical knowledge, despite making progress in knowledge acquisition and statistical inference.

Abstract

The paper investigates the ability of Large Language Models (LLMs) to reason about implicit numerical knowledge. The authors construct a dataset of subject-element pairs with associated numerical facts, as well as quad-tuples combining these facts to probe for entailed arithmetic relationships. The key findings are: Numerical Fact Probing: LLMs, especially GPT-4, demonstrate reasonable performance in extracting numerical facts from text, but struggle with "null values" (e.g., assuming a sparrow has 4 legs). There is a clear bias towards hallucinating the most common numerical values associated with each subject. Entailed Arithmetic Relationship (EAR) Probing: When asked to compare the numerical values implied by the subject-element pairs, LLMs perform poorly, often failing to correctly infer the entailed arithmetic relationship. The performance is unstable and depends heavily on the prompt formulation, indicating a lack of genuine reasoning ability. Even the best-performing models, like GPT-4, struggle to consistently put the implicit numerical facts together to reason about their relationships. The authors argue that while LLMs can produce seemingly correct answers through statistical inference, they lack the combinatorial reasoning capabilities required for tasks involving arithmetic and logical reasoning. They emphasize that the ability to generate correct answers does not equate to genuine reasoning ability, and that simply making LLMs larger and training them on more data is an insufficient solution to this problem.

Stats

A typical bicycle has a number of wheels that is more than the number of wheels a unicycle has. A typical sparrow has a number of legs that is less than the number of wheels a tricycle has. A typical human has a number of fingers that is more than the number of wheels a bicycle has.

Quotes

"It is difficult to argue that pure statistical learning can cope with the combinatorial explosion inherent in many commonsense reasoning tasks, especially once arithmetical notions are involved." "Bigger is not always better and chasing purely statistical improvements is flawed at the core, since it only exacerbates the dangerous conflation of the production of correct answers with genuine reasoning ability."

Key Insights Distilled From

Can Large Language Models put 2 and 2 together? Probing for Entailed Arithmetical Relationships

by D. Panas,S. ... at arxiv.org 05-01-2024

https://arxiv.org/pdf/2404.19432.pdf

Deeper Inquiries

How can we design hybrid architectures that combine the strengths of statistical learning in LLMs with the systematic reasoning capabilities of symbolic systems?

To design hybrid architectures that leverage the strengths of statistical learning in Large Language Models (LLMs) while incorporating systematic reasoning capabilities of symbolic systems, we can adopt a Neuro-Symbolic (NeSy) approach. This approach aims to integrate the statistical power of LLMs with the structured reasoning abilities of symbolic systems. One way to achieve this is through hybrid enablement, where LLMs are combined with external solvers that excel in symbolic reasoning tasks. By integrating these external solvers into the architecture, LLMs can benefit from their systematic reasoning capabilities when faced with complex logical or arithmetic tasks. This hybrid model allows for a more comprehensive approach to reasoning, utilizing the statistical learning of LLMs for data processing and the symbolic reasoning of external solvers for logical inference. Another approach is to develop integrated architectural solutions that combine the statistical and symbolic components within the LLM itself. By enhancing the architecture of LLMs to incorporate symbolic reasoning modules, we can create models that are capable of both statistical learning and systematic reasoning. This integration can enable LLMs to perform more robust arithmetic and logical reasoning tasks by leveraging the strengths of both paradigms. Overall, designing hybrid architectures that blend statistical learning with systematic reasoning involves integrating external solvers or developing internal modules within LLMs to combine the strengths of both approaches, ultimately enhancing the models' reasoning capabilities across a wide range of tasks.

What are the fundamental limitations of current LLM architectures that prevent them from performing robust arithmetic and logical reasoning, and how can these limitations be addressed?

Current LLM architectures face several fundamental limitations that hinder their ability to perform robust arithmetic and logical reasoning tasks. One key limitation is the reliance on statistical inference, which may not be sufficient for handling the combinatorial explosion inherent in many arithmetic and logical reasoning problems. LLMs excel at processing large amounts of text data and making predictions based on patterns in the data, but they may struggle with complex reasoning tasks that require systematic logic and arithmetic operations. Additionally, the lack of explicit numerical and logical reasoning training data in the pre-training phase of LLMs contributes to their limitations in these areas. Without sufficient exposure to explicit arithmetic and logical reasoning tasks during training, LLMs may not develop the necessary skills to perform such tasks accurately. To address these limitations and enhance the arithmetic and logical reasoning capabilities of LLMs, several strategies can be implemented. One approach is to incorporate explicit numerical and logical reasoning tasks into the training data of LLMs, exposing them to a diverse range of arithmetic and logical problems to improve their reasoning abilities. This can help LLMs learn to perform arithmetic operations, solve logical puzzles, and make inferences based on numerical data. Furthermore, developing specialized prompts and fine-tuning techniques that focus on arithmetic and logical reasoning can help improve the performance of LLMs in these specific domains. By tailoring the training process to emphasize arithmetic and logical tasks, LLMs can become more adept at handling complex reasoning challenges and producing accurate outputs. Overall, addressing the limitations of current LLM architectures in arithmetic and logical reasoning requires a combination of explicit training data, specialized prompts, and fine-tuning strategies to enhance the models' reasoning capabilities and improve their performance on arithmetic and logical tasks.

Given the observed biases and instabilities in LLM responses, how can we develop techniques to better calibrate and control the reasoning processes of these models to ensure reliable and trustworthy outputs?

To mitigate biases and instabilities in LLM responses and improve the reliability and trustworthiness of their outputs, several techniques can be employed to calibrate and control the reasoning processes of these models effectively. One approach is to implement adversarial training methods that expose LLMs to diverse and challenging scenarios to enhance their robustness and reduce biases. By training LLMs on a wide range of data that includes counterexamples and edge cases, we can help the models learn to handle unexpected inputs and make more accurate and unbiased predictions. Additionally, developing techniques for prompt engineering and fine-tuning can help control the reasoning processes of LLMs and guide them towards more reliable outputs. By crafting specific prompts that steer the models towards logical and unbiased reasoning paths, we can influence their decision-making processes and improve the quality of their responses. Furthermore, implementing post-processing techniques such as ensemble methods or model interpretation tools can help identify and correct biases in LLM outputs. By analyzing the predictions of multiple models or interpreting the decision-making processes of LLMs, we can detect and mitigate biases to ensure more trustworthy outputs. Overall, by combining adversarial training, prompt engineering, fine-tuning, and post-processing techniques, we can develop strategies to calibrate and control the reasoning processes of LLMs effectively, leading to more reliable and trustworthy outputs across a variety of tasks and domains.