A Theory for Length Generalization in Learning to Reason (2404.00560v1)

Abstract: Length generalization (LG) is a challenging problem in learning to reason. It refers to the phenomenon that when trained on reasoning problems of smaller lengths or sizes, the resulting model struggles with problems of larger sizes or lengths. Although LG has been studied by many researchers, the challenge remains. This paper proposes a theoretical study of LG for problems whose reasoning processes can be modeled as DAGs (directed acyclic graphs). The paper first identifies and proves the conditions under which LG can be achieved in learning to reason. It then designs problem representations based on the theory to learn to solve challenging reasoning problems like parity, addition, and multiplication, using a Transformer to achieve perfect LG.

• The paper establishes theoretical conditions for models to generalize from short to long reasoning tasks using causal functions over finite input spaces.
• It introduces key concepts like maximal input element distance (R) and (n, r)-consistency to manage recursive reasoning steps in DAG-structured problems.
• Empirical validation with Transformer models on tasks such as parity and arithmetic demonstrates perfect length generalization, reinforcing the theory's practical impact.

A Theory for Length Generalization in Learning to Reason

The phenomenon of length generalization (LG) presents a notable hurdle in the domain of machine learning's ability to reason. In this context, LG refers to a model's difficulty in extrapolating reasoning abilities from training on smaller problem sizes to accurately handling larger, more complex ones. The paper by Changnan Xiao and Bing Liu addresses this conundrum by presenting a theoretical analysis focused on reasoning tasks that can be represented using Directed Acyclic Graphs (DAGs). It establishes a set of conditions under which LG can be effectively managed, providing a foundation for developing models that maintain performance as the complexity of tasks increases.

Key Contributions and Theoretical Insights

The core contribution of the paper is the establishment of theoretical conditions essential for achieving LG in reasoning tasks structured as DAGs. It introduces the concept of reasoning problems, decomposable into individual steps that can be captured as causal processes on DAGs. A central theme is identifying the finite character of such problems and analyzing how these features contribute to successful LG.

1. Causal Functions and Finite Input Spaces: The paper begins by exploring the causal functions embedded in reasoning tasks represented by DAGs. It establishes that a critical condition for LG is that these causal functions must operate over a finite input space. Intuitively, this finiteness ensures that a model trained on finite samples can predict unseen data reliably within the same structured problem domain, especially when working recursively through complex problems.
2. Maximal Input Element Distance $R$: A novel and significant parameter introduced is the maximal input element distance $R$, defined for reasoning steps within a sequence. The authors argue that generalization is attainable when $R$ is finite. This parameter pertains to the maximal separation in the sequence's order between any two elements necessary to perform a calculation or infer a subsequent step, and its finite nature simplifies learning of the recursive calculation over different problem lengths.
3. (n, r)-Consistency: The research extends to problems where $R = \infty$, usually posing greater challenges for LG due to seemingly unbounded elements necessary for reasoning. The introduction of $(n, r)$-consistency offers a structured way to manage and decompose such problems effectively, ensuring that a group of sub-sequences adequately covers the reasoning step and maintains its reasoning capacity even across varying problem sizes.
4. Empirical Validation with Transformer Models: The paper complements its theoretical claims with empirical evidence using Transformer architectures. It demonstrates that by respecting the identified conditions, models can learn reasoning tasks, such as parity, addition, and multiplication, and exhibit perfect LG. This practical aspect reinforces the theoretical findings, showing that adopting specified problem representations and constraints facilitates scalability in performance.

Implications and Future Directions

This work outlines both practical and theoretical approaches to mitigating the LG issue, offering a pathway to improving AI's reasoning capabilities. Practically, adhering to conditions like ensuring finite causal function inputs and employing the $(n, r)$-consistent formulation in reasoning problems can guide better model design. Theoretically, the concepts explored—particularly maximal input element distance $R$ and recursive problem-solving paradigms—set the stage for refined learning algorithms tailored for scalable reasoning.

As the discussion of reasoning models and LG broadens, several avenues remain for future research. One key direction lies in exploring reasoning tasks that fall outside the DAG framework, such as those involving temporal and spatial dependencies more intricate than present DAG representations allow. Further investigation into the necessity, beyond sufficiency, of the conditions provided could prove invaluable, potentially leading to a deeper understanding of reasoning capabilities in machine learning models and laying the groundwork for developing architectures beyond current paradigms.

The advancement of these theoretical underpinnings not only leverages existing models for reasoning capabilities but holds the promise of opening new paradigms where AI can engage in more complex, nuanced problem-solving scenarios, replicating or exceeding human-like reasoning under varied and extended conditions.