[Paper Review] Analysing Mathematical Reasoning Abilities of Neural Models

David Saxton, Edward Grefenstette+2 more|arXiv (Cornell University)|Apr 2, 2019

Topic ModelingComputer Science28 references87 citations

TL;DR

The paper introduces a large, procedurally generated dataset of free-form mathematical questions to evaluate neural sequence-to-sequence models on algebraic and symbolic reasoning, comparing recurrent and Transformer architectures and analyzing generalization. It reports that Transformer models generally outperform recurrent ones and that extrapolation, intermediate computations, and true algorithmic reasoning remain challenging for current models.

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.

Motivation & Objective

• Create a scalable, free-form, text-based dataset of mathematics problems to probe neural reasoning and symbol manipulation.
• Assess how well state-of-the-art sequence models generalize across problem types and to harder extrapolated scenarios.
• Identify model strengths, weaknesses, and failure modes in algebraic generalization and composition of subroutines.

Proposed method

• Procedurally generate a diverse set of math questions across modules (algebra, arithmetic, calculus, probability, etc.).
• Represent questions and answers as free-form character sequences to allow broad expressivity.
• Evaluate two broad model classes (recurrent architectures and Transformer) on input–output generation of answers.
• Implement an encoder–decoder setup (attention-based for LSTM and full Transformer) with autoregressive character-level decoding.
• Use a fixed computation budget (thinking steps) and hyperparameter sweeps to compare performance across architectures.
• Score answers by exact string match (0 or 1) per question across interpolation and extrapolation test sets.

Experimental results

Research questions

• RQ1 Can neural sequence models learn and generalize mathematical reasoning across multiple topics under free-form input/output?
• RQ2 What are the relative strengths and failure modes of recurrent versus Transformer models for symbolic mathematics?
• RQ3 To what extent do models generalize to harder or larger-scale problems not seen during training (extrapolation)?
• RQ4 Do models rely on shallow heuristics or demonstrate capabilities akin to algebraic generalization when solving composed problems?

Key findings

• Transformers achieve higher average accuracy than recurrent models on many modules, especially after sufficient thinking steps.
• Relational Memory Cores do not outperform LSTMs and can be less data-efficient.
• Attentional LSTMs improve over simple LSTMs, but gains vary across tasks; increasing thinking steps helps some models.
• Polynomial manipulation and mixed arithmetic are notably harder, with Transformers showing advantages in some polynomial tasks.
• Extrapolation performance is limited, indicating models struggle with genuine algebraic generalization beyond training distributions.
• On real exam questions, the Transformer model achieved 14/40, roughly an E grade, highlighting gaps between benchmark tasks and real-world mathematics tests.