Neural Networks and the Chomsky Hierarchy (2207.02098v3)

Abstract: Reliable generalization lies at the heart of safe ML and AI. However, understanding when and how neural networks generalize remains one of the most important unsolved problems in the field. In this work, we conduct an extensive empirical study (20'910 models, 15 tasks) to investigate whether insights from the theory of computation can predict the limits of neural network generalization in practice. We demonstrate that grouping tasks according to the Chomsky hierarchy allows us to forecast whether certain architectures will be able to generalize to out-of-distribution inputs. This includes negative results where even extensive amounts of data and training time never lead to any non-trivial generalization, despite models having sufficient capacity to fit the training data perfectly. Our results show that, for our subset of tasks, RNNs and Transformers fail to generalize on non-regular tasks, LSTMs can solve regular and counter-language tasks, and only networks augmented with structured memory (such as a stack or memory tape) can successfully generalize on context-free and context-sensitive tasks.

• The paper presents an extensive empirical study testing 20,910 models on 15 tasks categorized by the Chomsky hierarchy.
• It reveals that memory-augmented networks like Stack-RNNs and Tape-RNNs outperform standard RNNs and LSTMs on complex language tasks.
• The study highlights the critical role of structured memory and training regimes in extending neural networks' generalization capabilities.

Essay on "Neural Networks and the Chomsky Hierarchy"

This paper by Gregoire Delétang et al. investigates the generalization ability of neural networks through the lens of formal language theory. The research bridges the gap between machine learning and the theory of computation by employing the Chomsky hierarchy as a framework to systematically evaluate the capabilities of various neural network architectures in sequence prediction tasks. The paper provides an extensive empirical analysis with 20,910 models across 15 tasks, categorizing these tasks into levels defined by the Chomsky hierarchy: regular, deterministic context-free (DCF), context-sensitive (CS), and others.

Core Contributions

The authors present several key contributions to the understanding of neural network generalization:

1. Empirical Study of Generalization:
   • The paper evaluates different neural network architectures, including RNNs, LSTMs, Transformers, and memory-augmented networks like Stack-RNNs and Tape-RNNs, against tasks representative of various levels of the Chomsky hierarchy.
   • It provides a comprehensive benchmark suite to evaluate length generalization, testing models on sequences significantly longer than those seen during training.
2. Chomsky Hierarchy and Neural Networks:
   • The paper reveals a hierarchy for neural networks analogous to the Chomsky hierarchy. For instance, RNNs can generalize on regular tasks, while LSTMs, despite their theoretical Turing completeness, can successfully solve more complex counting tasks but not higher-level context-free tasks.
   • Memory-augmented networks, which incorporate external memory structures like stacks or tapes, are shown to be more capable of handling tasks higher up the hierarchy.
3. Training and Architectural Insights:
   • The paper highlights the importance of the training regime and the choice of architecture in determining what tasks a model can generalize across. For example, while RNNs and LSTMs struggle with non-regular tasks, augmenting them with structured memory can enhance their computational capabilities.
   • The research introduces practical insights into the limitations and potential of neural architectures, shedding light on why certain models fail to generalize despite having sufficient capacity to fit the training data perfectly.

Detailed Analysis of Architectures

RNNs and LSTMs:

• RNNs: Demonstrated competency in solving regular tasks but fell short on more complex tasks. Their finite state control mechanisms are insufficient for tasks requiring stack or tape-like memory structures.
• LSTMs: Success in tasks involving counting (e.g., Bucket Sort) was noted. However, their generalized performance did not improve with deterministic context-free tasks due to the inability to implement higher-level grammar rules despite possessing Turing completeness as shown theoretically.

Transformers:

• The results indicate significant limitations when applying Transformers to non-regular tasks unless they are permutation-invariant. Neither increasing training data nor model parameters significantly improved their performance on such tasks, highlighting their misalignment with the Chomsky hierarchy for tasks requiring extended sequences or hierarchical dependencies.

Memory-Augmented Networks:

• Stack-RNNs: Exhibited powerful generalization on deterministic context-free tasks by leveraging stacks to handle nested structures and hierarchies.
• Tape-RNNs: Successfully managed context-sensitive tasks by employing tapes for extended memory operations, simulating computational mechanisms akin to Turing machines.

Implications of Findings

1. Architectural Evolution:
   • The findings suggest a clear path for advancing neural architectures by incorporating more structured and flexible memory models. Differentiable memory structures like tapes and stacks can be pivotal for scaling neural networks' computational abilities.
2. Theoretical and Practical Impact:
   • The empirical alignment of neural network performance with the Chomsky hierarchy provides a concrete basis for developing models tailored to the complexity of the task at hand.
   • The research counters the prevailing belief that "bigger is always better" in neural network design, pointing towards the nuanced requirement for specific inductive biases and structured memory to achieve true generalization.
3. Future Directions in AI:
   • This work sets a foundation for next-generation AI systems capable of reliable generalization in out-of-distribution scenarios, an essential attribute for robust, real-world applications of machine learning.
   • Scaling laws and the effectiveness of various training protocols, signposted here, offer a strategic roadmap for future explorations aiming to close the gap between theoretical capabilities and practical performance of AI models.

In conclusion, Delétang et al.'s investigation delineates a clear hierarchy in the practical capabilities of neural networks and paves the way for more sophisticated architectures that can generalize effectively across various levels of computational complexity. By methodically evaluating these models through the Chomsky hierarchy, the paper enriches the theoretical understanding and provides actionable insights for the improvement of neural network designs.