Implicit Bias of Gradient Descent on Reparametrized Models: On Equivalence to Mirror Descent
This provides a theoretical framework for analyzing optimization dynamics in machine learning, though it is incremental by building on prior results in implicit bias.
The paper tackles the problem of understanding the implicit bias of gradient descent in overparametrized models by characterizing when gradient flow is equivalent to mirror descent, showing that under commuting parametrizations, gradient flow corresponds to continuous mirror descent with a related Legendre function, and vice versa using Nash's embedding theorem.
As part of the effort to understand implicit bias of gradient descent in overparametrized models, several results have shown how the training trajectory on the overparametrized model can be understood as mirror descent on a different objective. The main result here is a characterization of this phenomenon under a notion termed commuting parametrization, which encompasses all the previous results in this setting. It is shown that gradient flow with any commuting parametrization is equivalent to continuous mirror descent with a related Legendre function. Conversely, continuous mirror descent with any Legendre function can be viewed as gradient flow with a related commuting parametrization. The latter result relies upon Nash's embedding theorem.