Super Level Sets and Exponential Decay: A Synergistic Approach to Stable Neural Network Training

TL;DR

This paper introduces a theoretical framework demonstrating that an adaptive learning rate algorithm, combining exponential decay and anti-overfitting strategies, ensures stable and connected loss landscapes in neural network training.

Contribution

It establishes the first formal proof of superlevel set connectedness and equiconnectedness in neural network loss functions under adaptive learning rates.

Findings

Superlevel sets of the loss function are always connected.

The loss landscape exhibits uniform stability across training conditions.

The proposed framework enhances the understanding of dynamic learning rate effects.

Abstract

The objective of this paper is to enhance the optimization process for neural networks by developing a dynamic learning rate algorithm that effectively integrates exponential decay and advanced anti-overfitting strategies. Our primary contribution is the establishment of a theoretical framework where we demonstrate that the optimization landscape, under the influence of our algorithm, exhibits unique stability characteristics defined by Lyapunov stability principles. Specifically, we prove that the superlevel sets of the loss function, as influenced by our adaptive learning rate, are always connected, ensuring consistent training dynamics. Furthermore, we establish the "equiconnectedness" property of these superlevel sets, which maintains uniform stability across varying training conditions and epochs. This paper contributes to the theoretical understanding of dynamic learning rate mechanisms in neural networks and also pave the way for the development of more efficient and reliable neural optimization techniques. This study intends to formalize and validate the equiconnectedness of loss function as superlevel sets in the context of neural network training, opening newer avenues for future research in adaptive machine learning algorithms. We leverage previous theoretical discoveries to propose training mechanisms that can effectively handle complex and high-dimensional data landscapes, particularly in applications requiring high precision and reliability.