49 research outputs found
A Mean Field View of the Landscape of Two-Layers Neural Networks
Multi-layer neural networks are among the most powerful models in machine
learning, yet the fundamental reasons for this success defy mathematical
understanding. Learning a neural network requires to optimize a non-convex
high-dimensional objective (risk function), a problem which is usually attacked
using stochastic gradient descent (SGD). Does SGD converge to a global optimum
of the risk or only to a local optimum? In the first case, does this happen
because local minima are absent, or because SGD somehow avoids them? In the
second, why do local minima reached by SGD have good generalization properties?
In this paper we consider a simple case, namely two-layers neural networks,
and prove that -in a suitable scaling limit- SGD dynamics is captured by a
certain non-linear partial differential equation (PDE) that we call
distributional dynamics (DD). We then consider several specific examples, and
show how DD can be used to prove convergence of SGD to networks with nearly
ideal generalization error. This description allows to 'average-out' some of
the complexities of the landscape of neural networks, and can be used to prove
a general convergence result for noisy SGD.Comment: 103 page
Generalization Error Bounds of Gradient Descent for Learning Over-parameterized Deep ReLU Networks
Empirical studies show that gradient-based methods can learn deep neural
networks (DNNs) with very good generalization performance in the
over-parameterization regime, where DNNs can easily fit a random labeling of
the training data. Very recently, a line of work explains in theory that with
over-parameterization and proper random initialization, gradient-based methods
can find the global minima of the training loss for DNNs. However, existing
generalization error bounds are unable to explain the good generalization
performance of over-parameterized DNNs. The major limitation of most existing
generalization bounds is that they are based on uniform convergence and are
independent of the training algorithm. In this work, we derive an
algorithm-dependent generalization error bound for deep ReLU networks, and show
that under certain assumptions on the data distribution, gradient descent (GD)
with proper random initialization is able to train a sufficiently
over-parameterized DNN to achieve arbitrarily small generalization error. Our
work sheds light on explaining the good generalization performance of
over-parameterized deep neural networks.Comment: 27 pages. This version simplifies the proof and improves the
presentation in Version 3. In AAAI 202