Theoretical Analysis of Compositional Structure in Sequence Processing Models

Core Concepts

Compositional functions can be precisely defined using a modular framework that separates the neural and symbolic components, allowing for a quantitative analysis of the compositional complexity of different sequence processing models.

Abstract

The paper proposes a novel definition of "compositional functions" that separates the neural and symbolic components. This allows for a concrete understanding of the expressiveness and generalization of such functions, and the introduction of "compositional complexity" to quantify the complexity of the compositions.
The authors demonstrate the flexibility of this definition by showing how various existing sequence processing models, such as recurrent, convolutional, and attention-based models, fit this definition and can be analyzed in terms of their compositional complexity.
Specifically, the authors:

Propose a general neuro-symbolic definition of compositional functions and their compositional complexity.
Analyze the compositional complexity of different model classes, including recurrent, convolutional, and transformer-based models.
Provide theoretical guarantees for the expressivity and systematic generalization of compositional models, connecting the proposed notion of compositional complexity to their ability to generalize.
Establish that input-agnostic compositional models have limitations in approximating functions with input-dependent compositional structure.
The analysis provides a theoretical framework for understanding the role of compositional structure in sequence processing models and their ability to generalize in a systematic manner.

Stats

The paper does not contain any specific numerical data or metrics. It focuses on a theoretical analysis of compositional structure in sequence processing models.

Quotes

"Compositionality is assumed to be integral to language processing."
"We propose a general modular definition of "compositional functions" to facilitate concrete understanding of the expressiveness and generalization of such functions, and propose the notion of "compositional complexity" to quantify the complexity of such functions."
"Given these definitions of compositional functions and compositional complexity, we precisely characterize the expressiveness and systematic generalization of such functions."

Key Insights Distilled From

What makes Models Compositional? A Theoretical View: With Supplement

by Parikshit Ra... at arxiv.org 05-07-2024

https://arxiv.org/pdf/2405.02350.pdf

Deeper Inquiries

How can the proposed theoretical framework be extended to incorporate other types of sequence processing models, such as graph neural networks or memory-augmented networks

The proposed theoretical framework for compositional functions can be extended to incorporate other types of sequence processing models, such as graph neural networks (GNNs) or memory-augmented networks.
For graph neural networks, the extension would involve adapting the definition of compositional functions to account for the graph structure. Instead of sequential data, the input would be a graph with nodes and edges. The token encoder would encode node features, and the computation directed acyclic graph (cDAG) would operate on the graph structure, considering the connections between nodes. The span processor and read-out function would need to be modified to handle graph data, aggregating information from neighboring nodes and predicting output based on the graph structure.
Memory-augmented networks, on the other hand, could be integrated by incorporating memory access mechanisms into the compositional framework. The cDAG would include operations for reading from and writing to the memory, allowing the model to store and retrieve information for processing sequences. The span processor and read-out function would interact with the memory component, enabling the model to utilize stored information for compositional processing.
By extending the framework to include these types of models, researchers can analyze and compare the compositional complexity of a broader range of sequence processing architectures, providing insights into how different models capture and utilize compositional structure in data.

What are the practical implications of the limitations of input-agnostic compositional models in approximating functions with input-dependent compositional structure

The limitations of input-agnostic compositional models in approximating functions with input-dependent compositional structure have significant practical implications.

Model Performance: Input-agnostic models may struggle to capture the nuanced relationships and dependencies present in data with input-dependent structures. This limitation can lead to suboptimal performance in tasks that require understanding complex compositional patterns, such as natural language processing or image understanding.

Generalization: Input-agnostic models may fail to generalize well to unseen examples that exhibit input-dependent compositional structures. This lack of generalization can hinder the model's ability to adapt to new data distributions or tasks, limiting its practical utility in real-world applications.

Interpretability: Models that cannot capture input-dependent compositional structures may provide less interpretable results. Understanding how the model processes and represents data becomes challenging when the compositional relationships are not accurately captured.

Efficiency: Inefficient approximation of input-dependent functions by input-agnostic models can lead to increased computational complexity and resource requirements. This inefficiency can impact the scalability and deployment of the models in production environments.

Addressing these limitations by designing models that can effectively capture input-dependent compositional structures is crucial for enhancing the performance, generalization, interpretability, and efficiency of sequence processing architectures in various applications.

Can the insights from this work be leveraged to design new sequence processing architectures that better capture compositional structure and enable improved systematic generalization

The insights from this work can be leveraged to design new sequence processing architectures that better capture compositional structure and enable improved systematic generalization.

Hybrid Models: By combining the strengths of different architectures, such as incorporating input-dependent mechanisms from graph neural networks and memory-augmented networks into existing models, designers can create hybrid models that leverage compositional structure more effectively.

Attention Mechanisms: Enhancing attention mechanisms in models like transformers to incorporate input-dependent sparsity patterns can improve their ability to capture compositional relationships. This can lead to more efficient and accurate processing of sequential data.

Structured Representations: Designing models that explicitly represent and manipulate compositional structures within the data can improve the model's ability to generalize systematically. This could involve incorporating hierarchical processing mechanisms or specialized modules for handling specific compositional patterns.

Interpretable Architectures: Developing architectures that not only capture compositional structure effectively but also provide insights into how the model processes data can enhance interpretability and trust in the model's decisions.

By integrating these insights into the design of new sequence processing architectures, researchers can advance the field towards more robust, efficient, and interpretable models that excel in capturing and utilizing compositional structure for improved systematic generalization.