  Deep neural networks (DNNs) have achieved state-of-the-art performance across
a variety of traditional machine learning tasks, e.g., speech recognition,
image classification, and segmentation. The ability of DNNs to efficiently
approximate high-dimensional functions has also motivated their use in
scientific applications, e.g., to solve partial differential equations (PDE)
and to generate surrogate models. In this paper, we consider the supervised
training of DNNs, which arises in many of the above applications. We focus on
the central problem of optimizing the weights of the given DNN such that it
accurately approximates the relation between observed input and target data.
Devising effective solvers for this optimization problem is notoriously
challenging due to the large number of weights, non-convexity, data-sparsity,
and non-trivial choice of hyperparameters. To solve the optimization problem
more efficiently, we propose the use of variable projection (VarPro), a method
originally designed for separable nonlinear least-squares problems. Our main
contribution is the Gauss-Newton VarPro method (GNvpro) that extends the reach
of the VarPro idea to non-quadratic objective functions, most notably,
cross-entropy loss functions arising in classification. These extensions make
GNvpro applicable to all training problems that involve a DNN whose last layer
is an affine mapping, which is common in many state-of-the-art architectures.
In our four numerical experiments from surrogate modeling, segmentation, and
classification GNvpro solves the optimization problem more efficiently than
commonly-used stochastic gradient descent (SGD) schemes. Also, GNvpro finds
solutions that generalize well, and in all but one example better than
well-tuned SGD methods, to unseen data points.
