Algorithmic Capture, Computational Complexity, and Inductive Bias of Infinite Transformers
arXiv:2603.11161v1 Announce Type: new Abstract: We formally define Algorithmic Capture (i.e., ``grokking'' an algorithm) as the ability of a neural network to generalize to arbitrary problem sizes ($T$) with controllable error and minimal sample adaptation, distinguishing true algorithmic learning from statistical interpolation. By analyzing infinite-width transformers in both the lazy and rich regimes, we derive upper bounds on the inference-time computational complexity of the functions these networks can learn. We show that despite their universal expressivity, transformers possess an inductive bias towards low-complexity algorithms within the Efficient Polynomial Time Heuristic Scheme (EPTHS) class. This bias effectively prevents them from capturing higher-complexity algorithms, while allowing success on simpler tasks like search, copy, and sort.
arXiv:2603.11161v1 Announce Type: new Abstract: We formally define Algorithmic Capture (i.e., ``grokking'' an algorithm) as the ability of a neural network to generalize to arbitrary problem sizes ($T$) with controllable error and minimal sample adaptation, distinguishing true algorithmic learning from statistical interpolation. By analyzing infinite-width transformers in both the lazy and rich regimes, we derive upper bounds on the inference-time computational complexity of the functions these networks can learn. We show that despite their universal expressivity, transformers possess an inductive bias towards low-complexity algorithms within the Efficient Polynomial Time Heuristic Scheme (EPTHS) class. This bias effectively prevents them from capturing higher-complexity algorithms, while allowing success on simpler tasks like search, copy, and sort.
Executive Summary
This article contributes to the understanding of algorithmic capture in neural networks by formally defining it as the ability to generalize to arbitrary problem sizes with controllable error and minimal sample adaptation. The authors analyze infinite-width transformers in both the lazy and rich regimes, deriving upper bounds on their inference-time computational complexity and demonstrating an inductive bias towards low-complexity algorithms within the EPTHS class. This work has significant implications for the design of efficient and effective neural networks, particularly in applications where high-complexity algorithms are required. The findings suggest that transformers may not be suitable for tasks that require capturing complex algorithms, while being successful in simpler tasks like search, copy, and sort.
Key Points
- ▸ Algorithmic capture is formally defined as the ability to generalize to arbitrary problem sizes with controllable error and minimal sample adaptation
- ▸ Infinite-width transformers exhibit an inductive bias towards low-complexity algorithms within the EPTHS class
- ▸ Transformers may not be suitable for tasks that require capturing complex algorithms
Merits
Strength in theoretical foundations
The article provides a rigorous theoretical framework for understanding algorithmic capture in neural networks, which is essential for designing efficient and effective models.
Insights into transformer limitations
The study reveals the inductive bias of transformers, which can inform the design of neural networks for specific tasks and applications.
Demerits
Limited scope of analysis
The article focuses on infinite-width transformers and EPTHS class, which may not be representative of all neural networks or tasks.
Lack of empirical evaluation
The study does not provide empirical evidence or experimental results to support the theoretical findings, which may limit the article's impact.
Expert Commentary
This article presents a significant contribution to the understanding of algorithmic capture in neural networks, particularly in the context of infinite-width transformers. The authors' rigorous theoretical analysis provides valuable insights into the limitations of these models, which can inform the design of more efficient and effective neural networks. However, the study's scope is limited to a specific class of networks and tasks, and the lack of empirical evaluation may limit the article's impact. Nevertheless, the findings have significant implications for the development of AI systems and the design of neural architectures.
Recommendations
- ✓ Future studies should investigate the applicability of the study's findings to other neural network architectures and tasks.
- ✓ Empirical evaluations should be conducted to validate the theoretical results and provide a more comprehensive understanding of the limitations of transformers.
