How regularization affects the geometry of loss functions

What neural networks learn depends fundamentally on the geometry of the underlying loss function. We study how different regularizers affect the geometry of this function. One of the most basic geometric properties of a smooth function is whether it is Morse or not. For nonlinear deep neural networks, the unregularized loss function $L$ is typically not Morse. We consider several different regularizers, including weight decay, and study for which regularizers the regularized function $L_\epsilon$ becomes Morse.

The critical locus of overparameterized neural networks

Many aspects of the geometry of loss functions in deep learning remain mysterious. In this paper, we work toward a better understanding of the geometry of the loss function $L$ of overparameterized feedforward neural networks. In this setting, we identify several components of the critical locus of $L$ and study their geometric properties. For networks of depth $\ell \geq 4$, we identify a locus of critical points we call the star locus $S$. Within $S$ we identify a positive-dimensional sublocus $C$ with the property that for $p \in C$, $p$ is a degenerate critical point, and no existing theoretical result guarantees that gradient descent will not converge to $p$. For very wide networks, we build on earlier work and show that all critical points of $L$ are degenerate, and give lower bounds on the number of zero eigenvalues of the Hessian at each critical point. For networks that are both deep and very wide, we compare the growth rates of the zero eigenspaces of the Hessian at all the different families of critical points that we identify. The results in this paper provide a starting point to a more quantitative understanding of the properties of various components of the critical locus of $L$.

Secondary gradient descent in higher codimension

In this paper, we analyze discrete gradient descent and $\epsilon$-noisy gradient descent on a special but important class of functions. We find that when used to minimize a function $L: \mathbb{R}^n \rightarrow \mathbb{R}$ in this class we consider, discrete gradient descent can exhibit strikingly different behavior from continuous gradient descent. On long time scales, discrete gradient descent and continuous gradient descent tend toward different global minima of $L$. Discrete gradient descent preferentially finds global minima at which the graph of the function $L$ is shallowest, while gradient flow shows no such preference.

Discrete gradient descent differs qualitatively from gradient flow

We consider gradient descent on functions of the form $L_1 = |f|$ and $L_2 = f^2$, where $f: \mathbb{R}^n \rightarrow \mathbb{R}$ is any smooth function with 0 as a regular value. We show that gradient descent implemented with a discrete step size $\tau$ behaves qualitatively differently from continuous gradient descent. We show that over long time scales, continuous and discrete gradient descent on $L_1$ find different minima of $L_1$, and we can characterize the difference - the minima that tend to be found by discrete gradient descent lie in a secondary critical submanifold $M' \subset M$, the locus within $M$ where the function $K=|\nabla f|^2 \big|_M$ is minimized. In this paper, we explain this behavior. We also study the more subtle behavior of discrete gradient descent on $L_2$.