Analysing Mathematical Reasoning Abilities of Neural Models (1904.01557v1)

Abstract: Mathematical reasoning---a core ability within human intelligence---presents some unique challenges as a domain: we do not come to understand and solve mathematical problems primarily on the back of experience and evidence, but on the basis of inferring, learning, and exploiting laws, axioms, and symbol manipulation rules. In this paper, we present a new challenge for the evaluation (and eventually the design) of neural architectures and similar system, developing a task suite of mathematics problems involving sequential questions and answers in a free-form textual input/output format. The structured nature of the mathematics domain, covering arithmetic, algebra, probability and calculus, enables the construction of training and test splits designed to clearly illuminate the capabilities and failure-modes of different architectures, as well as evaluate their ability to compose and relate knowledge and learned processes. Having described the data generation process and its potential future expansions, we conduct a comprehensive analysis of models from two broad classes of the most powerful sequence-to-sequence architectures and find notable differences in their ability to resolve mathematical problems and generalize their knowledge.

• The paper introduces a novel dataset split into interpolation and extrapolation tests to assess neural models' mathematical reasoning.
• It shows that Transformer models outperform recurrent networks in arithmetic tasks while struggling with complex algebraic operations.
• Empirical results expose significant generalization challenges, underscoring the need for improved compositional reasoning in neural architectures.

Mathematical Reasoning in Neural Models

The paper "Analysing Mathematical Reasoning Abilities of Neural Models" presents an empirical investigation into the capabilities of neural architectures in handling mathematical reasoning tasks. The authors develop a task suite of mathematical problems to test the sequential question and answer reasoning of neural networks, primarily focusing on sequence-to-sequence architectures.

The paper explores a structured domain of mathematics encompassing areas such as arithmetic, algebra, probability, and calculus. The construction and evaluation framework introduced allows for a rigorous assessment of the neural models' abilities to generalize beyond learned experiences, a limitation noted in current artificial intelligence systems. Through procedural generation, the authors create a diverse and potentially scalable dataset tackling a broad range of mathematical concepts, which is made available as an open-access resource.

Core Contributions

1. Dataset and Generalization Tests: The authors release a substantial dataset designed to measure mathematical reasoning, accompanied by generation code and pre-generated problems. Significantly, the dataset includes two test sets: interpolation tests that align with the training set problem types, and extrapolation tests that challenge models with problem variations exceeding those encountered during training. This segmentation allows evaluation of both in-domain proficiency and the ability to extend learned knowledge to novel contexts.
2. Experiments and Model Analysis: Experiments assess the algebraic reasoning skills of leading sequence-to-sequence architectures. The investigation includes recurrent neural models, such as LSTMs (Long Short-Term Memory networks), RMCs (Relational Memory Cores), and Transformer models. The results illustrate that while these architectures manage to solve straightforward arithmetic tasks, their performance diminishes on algebraic problems involving intermediate calculations or substantial generalization.

Insights and Observations

• Transformer Architecture: Among the models tested, the Transformer architecture typically outperforms recurrent networks. Its self-attention mechanism appears to provide enhanced capability for handling diverse mathematical structures within the composed dataset.
• Algebraic Reasoning Limitations: The paper underscores the challenges faced by current models in executing algebraic reasoning. Tasks demanding deep compositionality, like solving polynomial equations or performing operations that require intermediate computation results, displayed notable performance dips, thus highlighting areas for potential architectural innovations.
• Generalization Challenges: The extrapolation tasks revealed limited generalization, with more pronounced deficiencies in reasoning requiring the extension of learned concepts to unfamiliar problem variations. These empirical findings call attention to the need for architectures that embody intrinsic mechanisms for generalizing beyond training data confines.

Practical and Theoretical Implications

Practically, the dataset serves as a research catalyst promoting the development of more robust neural architectures capable of solving complex mathematical reasoning tasks. Theoretically, it provides a foundation for exploring computation models that more closely replicate human-like reasoning patterns. The goal is to surpass mere pattern recognition, thereby enhancing general AI's adaptive reasoning abilities through improved cognitive architectures.

Future Directions

The authors propose several avenues for extending their work. Enhancing linguistic comprehension within mathematical contexts could improve translation of word problems into mathematical formulations. Additionally, incorporating more complex mathematical constructs and visual reasoning tasks could yield insights into multi-modal neural processing abilities. Ultimately, creating architectures adept at real-world mathematical reasoning remains a frontier requiring novel computational paradigms, which more closely mimic human cognitive processes.