Calibrating Neural Networks' parameters through Optimal Contraction in a Prediction Problem

TL;DR

This paper proposes a mathematical framework that guarantees the existence and uniqueness of optimal neural network parameters by transforming the training problem into a contraction mapping, facilitating more reliable and efficient training.

Contribution

It introduces a novel contraction-based approach for neural network parameter calibration, providing analytical solutions and convergence conditions for both RNNs and FNNs.

Findings

Optimal parameters exist and are unique under certain conditions.

The approach simplifies training by ensuring convergence regions.

As the number of neurons increases, convergence conditions become easier to satisfy.

Abstract

This study introduces a novel approach to ensure the existence and uniqueness of optimal parameters in neural networks. The paper details how a recurrent neural networks (RNN) can be transformed into a contraction in a domain where its parameters are linear. It then demonstrates that a prediction problem modeled through an RNN, with a specific regularization term in the loss function, can have its first-order conditions expressed analytically. This system of equations is reduced to two matrix equations involving Sylvester equations, which can be partially solved. We establish that, if certain conditions are met, optimal parameters exist, are unique, and can be found through a straightforward algorithm to any desired precision. Also, as the number of neurons grows the conditions of convergence become easier to fulfill. Feedforward neural networks (FNNs) are also explored by including linear constraints on parameters. According to our model, incorporating loops (with fixed or variable weights) will produce loss functions that train easier, because it assures the existence of a region where an iterative method converges.