# A homotopy training algorithm for fully connected neural networks

## TL;DR

This paper introduces a Homotopy Training Algorithm (HTA) that incrementally constructs fully connected neural networks from simple models, improving optimization and performance, demonstrated on VGG models with significant error reduction.

## Contribution

The paper proposes a novel HTA method that adaptively builds neural networks along a continuous path, enhancing optimization and enabling automatic structure discovery.

## Key findings

- HTA reduces test error rate by 11.86% on VGG13 with batch normalization.
- HTA effectively finds optimal network structures.
- Numerical results confirm improved training outcomes.

## Abstract

In this paper, we present a Homotopy Training Algorithm (HTA) to solve optimization problems arising from fully connected neural networks with complicated structures. The HTA dynamically builds the neural network starting from a simplified version to the fully connected network via adding layers and nodes adaptively. Therefore, the corresponding optimization problem is easy to solve at the beginning and connects to the original model via a continuous path guided by the HTA, which provides a high probability to get a global minimum. By gradually increasing the complexity of model along the continuous path, the HTA gets a rather good solution to the original loss function. This is confirmed by various numerical results including VGG models on CIFAR-10. For example, on the VGG13 model with batch normalization, HTA reduces the error rate by 11.86% on test dataset comparing with the traditional method. Moreover, the HTA also allows us to find the optimal structure for a fully connected neural network by building the neutral network adaptively.