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arXiv · 2603.06248

Gradient Flow Polarizes Softmax Outputs towards Low-Entropy Solutions

Aditya Varre·Mark Rofin·Nicolas Flammarion
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Abstract

Understanding the intricate non-convex training dynamics of softmax-based models is crucial for explaining the empirical success of transformers. In this article, we analyze the gradient flow dynamics of the value-softmax model, defined as ${L}(\mathbf{V} \sigma(\mathbf{a}))$, where $\mathbf{V}$ and $\mathbf{a}$ are a learnable value matrix and attention vector, respectively. As the matrix times softmax vector parameterization constitutes the core building block of self-attention, our analysis provides direct insight into transformer's training dynamics. We reveal that gradient flow on this structure inherently drives the optimization toward solutions characterized by low-entropy outputs. We demonstrate the universality of this polarizing effect across various objectives, including logistic and square loss. Furthermore, we discuss the practical implications of these theoretical results, offering a formal mechanism for empirical phenomena such as attention sinks and massive activations.

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Aditya Varre, Mark Rofin, Nicolas Flammarion. 2026-03-06. Gradient Flow Polarizes Softmax Outputs towards Low-Entropy Solutions. https://arxiv.org/abs/2603.06248

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