  Machine learning problems have an intrinsic geometric structure as central
objects including a neural network's weight space and the loss function
associated with a particular task can be viewed as encoding the intrinsic
geometry of a given machine learning problem. Therefore, geometric concepts can
be applied to analyze and understand theoretical properties of machine learning
strategies as well as to develop new algorithms. In this paper, we address
three seemingly unrelated open questions in machine learning by viewing them
through a unified framework grounded in differential geometry. Specifically, we
view the weight space of a neural network as a manifold endowed with a
Riemannian metric that encodes performance on specific tasks. By defining a
metric, we can construct geodesic, minimum length, paths in weight space that
represent sets of networks of equivalent or near equivalent functional
performance on a specific task. We, then, traverse geodesic paths while
identifying networks that satisfy a second objective. Inspired by the geometric
insight, we apply our geodesic framework to 3 major applications: (i) Network
sparsification (ii) Mitigating catastrophic forgetting by constructing networks
with high performance on a series of objectives and (iii) Finding high-accuracy
paths connecting distinct local optima of deep networks in the non-convex loss
landscape. Our results are obtained on a wide range of network architectures
(MLP, VGG11/16) trained on MNIST, CIFAR-10/100. Broadly, we introduce a
geometric framework that unifies a range of machine learning objectives and
that can be applied to multiple classes of neural network architectures.
