How Deep Neural Networks Learn Compositional Data: The Random Hierarchy Model (2307.02129v5)

Abstract: Deep learning algorithms demonstrate a surprising ability to learn high-dimensional tasks from limited examples. This is commonly attributed to the depth of neural networks, enabling them to build a hierarchy of abstract, low-dimensional data representations. However, how many training examples are required to learn such representations remains unknown. To quantitatively study this question, we introduce the Random Hierarchy Model: a family of synthetic tasks inspired by the hierarchical structure of language and images. The model is a classification task where each class corresponds to a group of high-level features, chosen among several equivalent groups associated with the same class. In turn, each feature corresponds to a group of sub-features chosen among several equivalent ones and so on, following a hierarchy of composition rules. We find that deep networks learn the task by developing internal representations invariant to exchanging equivalent groups. Moreover, the number of data required corresponds to the point where correlations between low-level features and classes become detectable. Overall, our results indicate how deep networks overcome the curse of dimensionality by building invariant representations, and provide an estimate of the number of data required to learn a hierarchical task.

• The paper presents the Random Hierarchy Model to quantify sample complexity in deep networks, demonstrating a scaling law of n_c m^L.
• It shows that deep architectures overcome the curse of dimensionality by developing invariant representations through hierarchical compositional structures.
• Empirical analyses reveal sigmoid-like error reductions and identify a critical training threshold (P*) for effective learning.

Assessing Deep Neural Networks with the Random Hierarchy Model

This paper introduces the Random Hierarchy Model (RHM) to explore the sample complexity and learning dynamics of deep neural networks (DNNs) in capturing hierarchical compositional structures often found in natural data, such as language and imagery. The core objective is to quantify how many training examples are required for DNNs to effectively learn such structures.

Fundamental Contributions and Model Design

The RHM is an innovative synthetic task designed to mimic the inherently hierarchical nature of language and images. Each dataset instance is derived through a composition of nested features, ranging from high-level categories down to low-level details. This model captures the essence of hierarchical learning by encapsulating data in layers similar to the composition seen in natural tasks.

Key contributions include:

• Sample Complexity Estimation: The paper establishes that the sample complexity for deep networks scales as $P^* \sim n_c m^L$, where $n_c$ is the number of classes, $m$ is the degree of synonymic groups, and $L$ is the hierarchy depth. This showcases how deep architectures can overcome the curse of dimensionality by leveraging hierarchical structures.
• Graceful Handling of Curse of Dimensionality: Contrary to shallow networks, which require data exponential in dimensionality $d$, deep networks utilizing the RHM only require polynomially many samples proportional to $n_c m^L$.
• Internal Representations and Synonymic Invariance: Deep networks cultivate representations invariant to permuting synonymous groups of low-level features. This invariant representation is crucial in distinguishing between hierarchical compositions without being overwhelmed by dimensional constraints.

Empirical Insights and Analytical Approximations

The insights from empirical experiments using DNNs on the RHM reveal nuanced behaviors and processes:

• Convolutional and Dense Architectures: The investigations showcase that DNNs exhibit sigmoid-like error reductions versus training set size, emphasizing an inherent transition point $P^*$.
• Emergence of Synonymic Invariance: Via measures like synonymic sensitivity, the papers detail how layers, starting from the second layer onward, develop invariance to feature permutations that do not affect overall class identity.
• Comparison with Traditional Paradigms: The paper juxtaposes hierarchical learning in DNNs against low-dimensional feature representations, asserting that capturing high-level invariances reduces computational burdens significantly.

Implications on Deep Learning Theory

The paper proposes theoretical implications for the field of deep learning:

• Theoretical Grounds for Depth Advantage: Providing evidence that deeper networks, through hierarchical data paradigms, lower complexity bounds demonstrates a clear theoretical understanding, which can influence how new architectures are conceptualized.
• Guideline for Dataset Sample Complexity: By demonstrating the power and efficiency of hierarchical composition, there emerges an understanding of necessary data quantity benchmarks that could guide future dataset constructions and benchmarks.

Future Directions

Looking forward, the Random Hierarchy Model posits paths for the development of robust unsupervised learning frameworks, as its inherent structure does not explicitly require label feedback beyond a certain layering. Furthermore, this model could extend to other domains like reinforcement learning and generative modelling, where structured problem spaces are prevalent.

In conclusion, the RHM offers a comprehensive approach for understanding hierarchical learning and its ramifications on sample complexity and network depth, providing a quantitative framework to analyze and design deep neural networks efficaciously.