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Evaluating the Generalisability of Transformer Models in Mathematical Reasoning Tasks


Core Concepts
Transformer models, including GPT-4 and fine-tuned BERT variants, exhibit limited generalisability to out-of-distribution perturbations in mathematical reasoning tasks despite strong in-distribution performance.
Abstract
The paper proposes a framework for generating and perturbing detailed mathematical derivations at scale using symbolic engines. This framework is used to evaluate the generalisability of Transformer models, including GPT-4, GPT-3.5, and fine-tuned BERT variants, on two sequence classification tasks related to mathematical reasoning. The key findings are: GPT-4 can rival the in-distribution performance of fine-tuned BERT models, but still exhibits significant performance drops on out-of-distribution perturbations, particularly when the task requires decoding indirect references to mathematical entities. GPT-3.5 struggles to effectively classify mathematical reasoning, performing significantly worse than the BERT-based models on most test sets. The fine-tuned BERT models fail to generalize to perturbations and simpler examples, despite their strong in-distribution performance. This indicates their reliance on superficial patterns rather than the underlying rules of mathematical operators. Pairwise analysis reveals that BERT models struggle the most with operators that are not associated with fixed text spans or that rely on explicitly structured dependency relations, such as the substitution operator. The paper highlights the potential of using symbolic engines to generate high-quality mathematical datasets for exploring model weaknesses and improving mathematical reasoning in Transformer-based models.
Stats
The average in-distribution performance of fine-tuned BERT models surpasses GPT-3.5 and rivals GPT-4. Perturbations to the input reasoning can reduce the performance of fine-tuned BERT models by up to 80 F1 points. GPT-4 obtains 80% F1 on 2-step derivations, while the fine-tuned BERT model SciBERT-cased scores only 11% F1 on the same task. In the Calculus Classification task, GPT-4 and GPT-3.5 score below 60% F1, while the fine-tuned BERT models average 90% F1 on the static in-distribution set.
Quotes
"Surprisingly, our empirical evaluation reveals that the average in-distribution performance of fine-tuned models surpasses GPT-3.5, and rivals GPT-4." "However, perturbations to input reasoning can reduce their performance by up to 80 F1 points."

Deeper Inquiries

How can the proposed framework be extended to evaluate the generalisability of Transformer models on a broader range of mathematical reasoning tasks, such as symbolic integration, differential equations, or theorem proving?

The proposed framework can be extended to evaluate the generalisability of Transformer models on a broader range of mathematical reasoning tasks by incorporating specific task structures and complexities into the data generation and evaluation process. For tasks like symbolic integration, the framework can generate derivations that involve integration operators and equations with varying levels of complexity. This would require defining new symbolic rules and operators related to integration, and generating datasets that include integration steps and related annotations. Similarly, for differential equations, the framework can be extended to generate derivations involving differentiation operators and equations that require solving differential equations. By introducing new operators and rules specific to differential equations, the framework can create datasets that challenge models to reason and generalize effectively in this domain. For tasks like theorem proving, the framework can be adapted to generate derivations that involve logical reasoning steps, axioms, and proofs. This would involve defining symbolic rules for logical operations, generating datasets with theorem statements and corresponding proofs, and evaluating models based on their ability to reason and generalize in a theorem proving context. Overall, by customizing the symbolic engine, defining task-specific operators, and generating datasets that reflect the complexities of tasks like symbolic integration, differential equations, and theorem proving, the framework can be extended to evaluate the generalisability of Transformer models across a broader range of mathematical reasoning tasks.

How do the findings in this paper relate to the broader challenge of achieving robust and reliable AI systems for domains that require rigorous and controlled reasoning, such as physics, biomedicine, or software verification?

The findings in this paper shed light on the challenges and limitations of current Transformer models in handling complex mathematical reasoning tasks. These challenges are highly relevant to the broader goal of achieving robust and reliable AI systems for domains that demand rigorous and controlled reasoning, such as physics, biomedicine, and software verification. Interpretability and Explainability: The paper highlights the importance of structured reasoning and understanding indirect textual references in mathematical contexts. This is crucial for AI systems in physics and biomedicine where clear and interpretable reasoning processes are essential for decision-making. Generalisation and Adaptability: The evaluation of model generalisability to out-of-distribution perturbations underscores the need for AI systems in software verification to adapt to diverse scenarios and handle unexpected inputs effectively. Task-Specific Knowledge: The identification of challenging operators and the impact of specific perturbations on model performance emphasize the importance of domain-specific knowledge and reasoning capabilities in AI systems for physics, biomedicine, and software verification. By addressing the limitations and enhancing the generalisability of Transformer models in mathematical reasoning tasks, the findings contribute to the broader challenge of developing AI systems that can reliably and accurately reason in complex domains requiring controlled and rigorous reasoning.

What architectural modifications or training strategies could improve the generalisability of Transformer models to out-of-distribution mathematical reasoning problems?

Task-Specific Fine-Tuning: Fine-tuning Transformer models on a diverse set of mathematical reasoning tasks, including symbolic integration, differential equations, and theorem proving, can improve their ability to generalize to out-of-distribution problems in these domains. Multi-Task Learning: Training Transformer models on a combination of mathematical reasoning tasks can enhance their overall reasoning capabilities and adaptability to different types of problems. Structured Prompting: Designing structured prompts that provide context and guidance for mathematical reasoning tasks can help Transformer models better understand the underlying logic and dependencies in complex mathematical problems. Architectural Enhancements: Introducing architectural modifications such as incorporating explicit reasoning modules or attention mechanisms that focus on specific mathematical operations can improve the models' ability to handle diverse mathematical tasks. Data Augmentation: Augmenting the training data with a wide range of mathematical expressions, equations, and reasoning chains can expose the models to a more diverse set of examples, enhancing their generalisation to out-of-distribution problems. By implementing these architectural modifications and training strategies, Transformer models can be better equipped to handle out-of-distribution mathematical reasoning problems and improve their overall generalisability in complex domains.
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