-
The Heavy-Tail Phenomenon in SGD
In recent years, various notions of capacity and complexity have been pr...
read it
-
On the Heavy-Tailed Theory of Stochastic Gradient Descent for Deep Neural Networks
The gradient noise (GN) in the stochastic gradient descent (SGD) algorit...
read it
-
Recent advances in deep learning theory
Deep learning is usually described as an experiment-driven field under c...
read it
-
A Tail-Index Analysis of Stochastic Gradient Noise in Deep Neural Networks
The gradient noise (GN) in the stochastic gradient descent (SGD) algorit...
read it
-
Stochastic Optimization with Heavy-Tailed Noise via Accelerated Gradient Clipping
In this paper, we propose a new accelerated stochastic first-order metho...
read it
-
Why ADAM Beats SGD for Attention Models
While stochastic gradient descent (SGD) is still the de facto algorithm ...
read it
-
Fractional Underdamped Langevin Dynamics: Retargeting SGD with Momentum under Heavy-Tailed Gradient Noise
Stochastic gradient descent with momentum (SGDm) is one of the most popu...
read it
Hausdorff Dimension, Stochastic Differential Equations, and Generalization in Neural Networks
Despite its success in a wide range of applications, characterizing the generalization properties of stochastic gradient descent (SGD) in non-convex deep learning problems is still an important challenge. While modeling the trajectories of SGD via stochastic differential equations (SDE) under heavy-tailed gradient noise has recently shed light over several peculiar characteristics of SGD, a rigorous treatment of the generalization properties of such SDEs in a learning theoretical framework is still missing. Aiming to bridge this gap, in this paper, we prove generalization bounds for SGD under the assumption that its trajectories can be well-approximated by a Feller process, which defines a rich class of Markov processes that include several recent SDE representations (both Brownian or heavy-tailed) as its special case. We show that the generalization error can be controlled by the Hausdorff dimension of the trajectories, which is intimately linked to the tail behavior of the driving process. Our results imply that heavier-tailed processes should achieve better generalization; hence, the tail-index of the process can be used as a notion of “capacity metric”. We support our theory with experiments on deep neural networks illustrating that the proposed capacity metric accurately estimates the generalization error, and it does not necessarily grow with the number of parameters unlike the existing capacity metrics in the literature.
READ FULL TEXT
Comments
There are no comments yet.