Language Models are Symbolic Learners in Arithmetic (2410.15580v1)

Abstract: LLMs are thought to struggle with arithmetic learning due to the inherent differences between LLMing and numerical computation, but concrete evidence has been lacking. This work responds to this claim through a two-side experiment. We first investigate whether LLMs leverage partial products during arithmetic learning. We find that although LLMs can identify some partial products after learning, they fail to leverage them for arithmetic tasks, conversely. We then explore how LLMs approach arithmetic symbolically by breaking tasks into subgroups, hypothesizing that difficulties arise from subgroup complexity and selection. Our results show that when subgroup complexity is fixed, LLMs treat a collection of different arithmetic operations similarly. By analyzing position-level accuracy across different training sizes, we further observe that it follows a U-shaped pattern: LLMs quickly learn the easiest patterns at the first and last positions, while progressively learning the more difficult patterns in the middle positions. This suggests that LLMs select subgroup following an easy-to-hard paradigm during learning. Our work confirms that LLMs are pure symbolic learners in arithmetic tasks and underscores the importance of understanding them deeply through subgroup-level quantification.

LLMs as Symbolic Learners in Arithmetic: A Detailed Examination

The paper "LLMs are Symbolic Learners in Arithmetic" by Chunyuan Deng et al. explores the intriguing question of whether LLMs like GPT-4o and Claude can effectively perform arithmetic tasks and if so, how these models learn such tasks. The paper provides a nuanced understanding that LLMs, traditionally viewed as tools for language understanding and generation, approach arithmetic not as bona fide calculators, but rather as symbolic pattern matchers.

Key Findings and Methodologies

The authors present a dual-faceted experimental framework to examine whether LLMs utilize partial products or symbolic learning for arithmetic tasks. Initially, they explore if LLMs leverage partial product computation, a standard approach in multiplication. The models were tested on various methods of partial product generation like the standard multiplication, lattice method, and Egyptian multiplication, to name a few. The paper finds that although LLMs can identify some partial products post learning, this does not enhance their ability to solve arithmetic problems, indicating that partial products per se are not directly used in arithmetic learning.

The research further dissects how LLMs attempt arithmetic tasks symbolically. By breaking down tasks into discrete subgroups, the authors propose hypotheses focused on subgroup complexity, defined by domain space cardinality, label space entropy, and subgroup quality. These dimensions help evaluate difficulty levels and task learnability. A notable observation is the U-shaped curve for position-level accuracy in token predictions, suggesting an "easy-to-hard" learning pattern as models progressively learn more challenging symbolic patterns.

By implementing methodical rule perturbations and examining the effects of ensemble group characteristics, the research underlines that LLMs primarily operate as symbolic learners, focusing on the patterns rather than the actual computation of values. Traditional approaches assume that errors in arithmetic stem from failure to perform calculations whereas this paper points to an alternative cognitive model rooted in symbolic understanding and pattern abstraction.

Implications and Future Directions

From a practical standpoint, these insights underscore the limitations of LLMs in applications requiring high-fidelity arithmetic computations without symbolic aids or auxiliary systems. The realization pertains to the inherently symbolic backbone of LLMs which affects their adaptation to purely mathematical tasks. A significant implication is for the development of future LLM versions more suited for operations needing strict numerical precision, possibly integrating explicit mathematical reasoning components or developing hybrid models that can seamlessly transition between semantic and quantitative tasks.

The paper invites further research to validate these findings across different arithmetic complexities, expanded digit ranges, and task varieties beyond the confines tested, such as those represented in natural language word problems. Future investigations could also explore how structured interventions in training or task partitioning could optimize symbolic learning mechanisms or allow these models to transcend their inherent limitations.

In conclusion, the paper by Deng et al. is a compelling exploration into the depths of how LLMs perceive and manage arithmetic tasks, offering both a confirmation of their symbolic prowess and a better-defined boundary of their current capabilities. While symbolic learning provides some utility in arithmetic, pushing the boundaries requires bridging current gaps with innovative approaches that combine symbolic interpretation with computational exactitude.