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Paper · 2603.06248 · 2026

Gradient Flow Polarizes Softmax Outputs towards Low-Entropy Solutions

Nicolas Flammarion, Aditya Varre, Mark Rofin

arXiv · PDF · Open in the Atlas

Code that ran

We lifted 10 functions out of this paper's own repositories and ran 0 of them in a sandbox. "Ran" means the function executed on a synthesized input and returned a value. It is not a reproduction of the paper's results.

RepositoryRoleRan
tml-epfl/softmax canonical 0 of 10
FunctionStatusWhere it lives
evaluate Not yet run tml-epfl/softmax/exp_classification.py
pointer only (licence: NONE) · get_code("de1a6b54bb6d7eb2")
evaluate Not yet run tml-epfl/softmax/exp_induction.py
pointer only (licence: NONE) · get_code("195ef9d2094e5fa7")
get_metrics_for_batch Not yet run tml-epfl/softmax/exp_classification.py
pointer only (licence: NONE) · get_code("dbc815742c417081")
get_metrics_for_batch Not yet run tml-epfl/softmax/exp_induction.py
pointer only (licence: NONE) · get_code("179263c917ddf20b")
get_metrics_for_dataset Not yet run tml-epfl/softmax/exp_induction.py
pointer only (licence: NONE) · get_code("c50658a8f12d5f40")
linear_attention Not yet run tml-epfl/softmax/src/attention.py
pointer only (licence: NONE) · get_code("a91bbe5c96744514")
print_induction_sequence Not yet run tml-epfl/softmax/src/datasets.py
pointer only (licence: NONE) · get_code("dea3bf88d257b9b2")
sigmoid_attention Not yet run tml-epfl/softmax/src/attention.py
pointer only (licence: NONE) · get_code("05229e47182eaaf7")
softmax_attention Not yet run tml-epfl/softmax/src/attention.py
pointer only (licence: NONE) · get_code("cac10aee0417a8c9")
train Not yet run tml-epfl/softmax/exp_classification.py
pointer only (licence: NONE) · get_code("0dbeea554881e4bd")

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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(Vσ(a)), where V and 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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