Patricia Muñoz Ewald

Thomas Chen, Patricia Muñoz Ewald

In this paper, we explicitly determine local and global minimizers of the $\mathcal{L}^2$ cost function in underparametrized Deep Learning (DL) networks; our main goal is to shed light on their geometric structure and properties. We accomplish this by a direct construction, without invoking the gradient descent flow at any point of this work. We specifically consider $L$ hidden layers, a ReLU ramp activation function, an $\mathcal{L}^2$ Schatten class (or Hilbert-Schmidt) cost function, input and output spaces $\mathbb{R}^Q$ with equal dimension $Q\geq1$, and hidden layers also defined on $\mathbb{R}^{Q}$; the training inputs are assumed to be sufficiently clustered. The training input size $N$ can be arbitrarily large - thus, we are considering the underparametrized regime. More general settings are left to future work. We construct an explicit family of minimizers for the global minimum of the cost function in the case $L\geq Q$, which we show to be degenerate. Moreover, we determine a set of $2^Q-1$ distinct degenerate local minima of the cost function. In the context presented here, the concatenation of hidden layers of the DL network is reinterpreted as a recursive application of a {\em truncation map} which "curates" the training inputs by minimizing their noise to signal ratio.

Thomas Chen, Patrícia Muñoz Ewald

We explicitly construct zero loss neural network classifiers. We write the weight matrices and bias vectors in terms of cumulative parameters, which determine truncation maps acting recursively on input space. The configurations for the training data considered are (i) sufficiently small, well separated clusters corresponding to each class, and (ii) equivalence classes which are sequentially linearly separable. In the best case, for $Q$ classes of data in $\mathbb{R}^M$, global minimizers can be described with $Q(M+2)$ parameters.

Thomas Chen, Chun-Kai Kevin Chien, Patricia Muñoz Ewald et al.

We prove that overparametrized neural networks are able to generalize with a test error that is independent of the level of overparametrization, and independent of the Vapnik-Chervonenkis (VC) dimension. We prove explicit bounds that only depend on the metric geometry of the test and training sets, on the regularity properties of the activation function, and on the operator norms of the weights and norms of biases. For overparametrized deep ReLU networks with a training sample size bounded by the input space dimension, we explicitly construct zero loss minimizers without use of gradient descent, and prove that the generalization error is independent of the network architecture.