Exploring the Frontiers of Softmax: Provable Optimization, Applications in Diffusion Model, and Beyond

TL;DR

This paper offers a theoretical analysis of softmax neural networks, revealing their optimization advantages and demonstrating their effectiveness in diffusion models, thereby enhancing understanding of their success in large language models and generative tasks.

Contribution

It provides a novel theoretical framework using NTK to explain softmax's advantages and applies these insights to score estimation in diffusion models, showing provable learning guarantees.

Findings

Softmax induces a good perturbation property in NTK matrices.

Softmax neural networks can learn target functions in over-parameterized regimes.

Gradient algorithms can accurately learn score functions in diffusion models.

Abstract

The softmax activation function plays a crucial role in the success of large language models (LLMs), particularly in the self-attention mechanism of the widely adopted Transformer architecture. However, the underlying learning dynamics that contribute to the effectiveness of softmax remain largely unexplored. As a step towards better understanding, this paper provides a theoretical study of the optimization and generalization properties of two-layer softmax neural networks, providing theoretical insights into their superior performance as other activation functions, such as ReLU and exponential. Leveraging the Neural Tangent Kernel (NTK) framework, our analysis reveals that the normalization effect of the softmax function leads to a good perturbation property of the induced NTK matrix, resulting in a good convex region of the loss landscape. Consequently, softmax neural networks can learn the target function in the over-parametrization regime. To demonstrate the broad applicability of our theoretical findings, we apply them to the task of learning score estimation functions in diffusion models, a promising approach for generative modeling. Our analysis shows that gradient-based algorithms can learn the score function with a provable accuracy. Our work provides a deeper understanding of the effectiveness of softmax neural networks and their potential in various domains, paving the way for further advancements in natural language processing and beyond.