Multilevel-in-Layer Training for Deep Neural Network Regression

TL;DR

This paper introduces a multilevel-in-width training method for deep neural networks inspired by Algebraic Multigrid techniques, which acts as an effective regularizer to improve generalization in PDE regression tasks.

Contribution

It proposes a novel multilevel-in-width training framework for neural networks, adapting AMG's FAS scheme to enhance regularization and hierarchical learning.

Findings

Improves generalization performance on PDE regression problems.

Applicable to various layer types including fully-connected and convolutional.

Demonstrates effectiveness as a regularizer in deep learning models.

Abstract

A common challenge in regression is that for many problems, the degrees of freedom required for a high-quality solution also allows for overfitting. Regularization is a class of strategies that seek to restrict the range of possible solutions so as to discourage overfitting while still enabling good solutions, and different regularization strategies impose different types of restrictions. In this paper, we present a multilevel regularization strategy that constructs and trains a hierarchy of neural networks, each of which has layers that are wider versions of the previous network's layers. We draw intuition and techniques from the field of Algebraic Multigrid (AMG), traditionally used for solving linear and nonlinear systems of equations, and specifically adapt the Full Approximation Scheme (FAS) for nonlinear systems of equations to the problem of deep learning. Training through V-cycles then encourage the neural networks to build a hierarchical understanding of the problem. We refer to this approach as \emph{multilevel-in-width} to distinguish from prior multilevel works which hierarchically alter the depth of neural networks. The resulting approach is a highly flexible framework that can be applied to a variety of layer types, which we demonstrate with both fully-connected and convolutional layers. We experimentally show with PDE regression problems that our multilevel training approach is an effective regularizer, improving the generalize performance of the neural networks studied.