Geometry of Lightning Self-Attention: Identifiability and Dimension (2408.17221v2)

Abstract: We consider function spaces defined by self-attention networks without normalization, and theoretically analyze their geometry. Since these networks are polynomial, we rely on tools from algebraic geometry. In particular, we study the identifiability of deep attention by providing a description of the generic fibers of the parametrization for an arbitrary number of layers and, as a consequence, compute the dimension of the function space. Additionally, for a single-layer model, we characterize the singular and boundary points. Finally, we formulate a conjectural extension of our results to normalized self-attention networks, prove it for a single layer, and numerically verify it in the deep case.

• The paper establishes identifiability and computes the neuromanifold dimension for lightning self-attention models.
• It leverages algebraic geometry to analyze singularities, boundary points, and fiber structures in both single-layer and deep networks.
• The study extends to traditional self-attention by examining normalization effects, with numerical verifications supporting the conjectured dimensions.

Geometry of Lightning Self-Attention: Identifiability and Dimension

Abstract

The paper "Geometry of Lightning Self-Attention: Identifiability and Dimension" by Nathan W. Henry, Giovanni Luca Marchetti, and Kathlen Kohn explores the function spaces defined by self-attention networks devoid of normalization. Leveraging algebraic geometry, the authors investigate the identifiability and dimensionality of deep attention models by describing the generic fibers of their parametrization and deriving the function space dimension. A conjectural extension to normalized self-attention networks is also formulated, partially proved for a single layer, and numerically verified for deeper networks.

Introduction and Related Work

Self-attention mechanisms are fundamental to transformer architectures in modern machine learning, excelling in multiple domains due to their ability to model long-range dependencies. The paper centers on the geometry of neuromanifolds, influenced significantly by the gradient flow during training. Previous studies have primarily focused on fully-connected and convolutional networks. This research uniquely considers lightning self-attention models, characterized by un-normalized attention weights, thus reducing computational complexity from quadratic to linear in sequence length.

The paper's central focus is on uncovering the geometric properties of neuromanifolds associated with lightning self-attention mechanisms. These networks, tri-linear in weights and cubical in input, offer a fertile ground for algebraic geometry methods, enabling the computation of geometric quantities such as the neuromanifold's dimension. This dimension plays a crucial role in statistical learning theory, influencing the sample complexity of learnability.

Summary of Contributions

The authors achieve the following major results:

1. Single-layer Lightning Self-Attention (Section \ref{sec:singlay}): For a single-layer model, they analyze the fibers of the parametrization and establish the Euclidean closedness of neuromanifolds, identifying singular and boundary points.
2. Deep Lightning Self-Attention (Section \ref{sec:deep}): Through a novel reparametrization involving virtual weights, the paper computes the generic fibers for deep networks. The dimension of these neuromanifolds is derived under a bottleneck architecture assumption.
3. Traditional Self-Attention (Section \ref{sec:tradattn}): The paper extends to traditional self-attention by introducing softmax normalization back into the model, proving a conjecture for the dimension of the neuromanifold in the single-layer case and verifying it numerically in the deep network case.

Results and Detailed Analysis

Single-layer Identifiability and Geometry

For a single-layer lightning self-attention mechanism, the fiber of the parametrization map is determined, proving that under generic conditions, the fibers are one-dimensional. The dimension of the neuromanifold is computed as follows:

$$\text{dim}(\mathcal{M}_{d, d', a}) = \begin{cases} 2ad + dd' - a^2 - 1 & \text{if} \ a \leq d \ d^2 + dd' - 1 & \text{otherwise}. \end{cases}$$

Further geometric analysis reveals that the neuromanifold is Euclidean closed, with boundary points and singularities explicitly characterized. The closedness guarantees that limit points of the training dynamics lie within the neuromanifold, ensuring robust convergence.

Deep Networks

Employing a recursive formulation (Equation \ref{eq:indformula}), the paper confirms that for deep networks, the fibers are principally affected by the reparametrization symmetries. The deep networks’ neuromanifold dimension is:

$$2 \alpha_1 d_0 - \alpha_1^2 + \delta(d_0 + d_l) - \delta^2 - l + \sum_{1 < i \leq l}(2\alpha_i \delta - \alpha_i^2).$$

This dimension result is contingent on the bottleneck architecture, with practical implications for model complexity and learnability.

Traditional Self-Attention

For single-layer traditional self-attention mechanisms, the paper proves that the fibers of the parametrization are singletons, indicating a one-to-one mapping. It conjectures that a similar result holds for deep networks, supported by numerical verifications. Empirical results align with the conjectured dimensions, substantiating the theoretical findings.

Implications and Future Directions

The insights presented have profound implications for understanding the training dynamics and expressivity of self-attention networks. The dimensionality results assist in estimating appropriate dataset sizes, fostering informed model selection and optimization strategies.

Future work may explore extensions to more complex architectural features such as multi-head attention, residual connections, and positional encodings. Additionally, further advancements in applying algebraic geometry tools to neural networks promise to unlock deeper theoretical understandings, potentially improving the robustness and efficiency of modern AI systems.