Fractional deep neural network via constrained optimization
Abstract
This paper introduces a novel algorithmic framework for a deep neural network (DNN), which in a mathematically rigorous manner, allows us to incorporate history (or memory) into the network—it ensures all layers are connected to one another. This DNN, called Fractional-DNN, can be viewed as a time-discretization of a fractional in time non-linear ordinary differential equation (ODE). The learning problem then is a minimization problem subject to that fractional ODE as constraints. We emphasize that an analogy between the existing DNN and ODEs, with standard time derivative, is well-known by now. The focus of our work is the Fractional-DNN. Using the Lagrangian approach, we provide a derivation of the backward propagation and the design equations. We test our network on several datasets for classification problems. Fractional-DNN offers various advantages over the existing DNN. The key benefits are a significant improvement to the vanishing gradient issue due to the memory effect, and better handling of nonsmooth data due to the network’s ability to approximate non-smooth functions.
Document Details
- Document Type
- Pub Defense Publication
- Publication Date
- Dec 01, 2020
- Source ID
- 10.1088/2632-2153/aba8e7
Entities
People
- Deepanshu Verma
- Harbir Antil
- Rainald Löhner
- Ratna Khatri
Organizations
- Air Force Office of Scientific Research
- National Science Foundation Division of Mathematical Sciences