Gradient flow in parameter space is equivalent to linear interpolation in output space

• URL: http://arxiv.org/abs/2408.01517v2
• Date: Wed, 04 Jun 2025 22:45:16 GMT
• Title: Gradient flow in parameter space is equivalent to linear interpolation in output space
• Authors: Thomas Chen, Patrícia Muñoz Ewald,
• Abstract summary: We prove that the standard flow in parameter space that underlies many training algorithms in deep learning can be continuously deformed into an adapted gradient flow.<n>For the $L2$ loss, if the Jacobian of the outputs with respect to the parameters is full rank, then the time variable can be reparametrized so that the resulting flow is simply linear.<n>For the cross-entropy loss, under the same rank condition and assuming the labels have positive components, we derive an explicit formula for the unique global minimum.
• Score: 1.189367612437469
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We prove that the standard gradient flow in parameter space that underlies many training algorithms in deep learning can be continuously deformed into an adapted gradient flow which yields (constrained) Euclidean gradient flow in output space. Moreover, for the $L^{2}$ loss, if the Jacobian of the outputs with respect to the parameters is full rank (for fixed training data), then the time variable can be reparametrized so that the resulting flow is simply linear interpolation, and a global minimum can be achieved. For the cross-entropy loss, under the same rank condition and assuming the labels have positive components, we derive an explicit formula for the unique global minimum.

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