Reparameterizing Mirror Descent as Gradient Descent

Most of the recent successful applications of neural networks have been based on training with gradient descent updates. However, for some small networks, other mirror descent updates learn provably more efficiently when the target is sparse. We present a general framework for casting a mirror descent update as a gradient descent update on a different set of parameters. In some cases, the mirror descent reparameterization can be described as training a modified network with standard backpropagation. The reparameterization framework is versatile and covers a wide range of mirror descent updates, even cases where the domain is constrained. Our construction for the reparameterization argument is done for the continuous versions of the updates. Finding general criteria for the discrete versions to closely track their continuous counterparts remains an interesting open problem. ⇤ An earlier version of this manuscript (with additional results on the matrix case) appeared as "Interpolating Between Gradient Descent and Exponentiated Gradient Using Reparameterized Gradient Descent" as a preprint. 2 The normalized version is called EG and the two-sided version EGU ± . More about this later. 34th Conference on Neural Information Processing Systems (NeurIPS 2020), Vancouver, Canada.