Calibrating Neural Networks' parameters through Optimal Contraction in a Prediction Problem
This addresses the challenge of parameter optimization in neural networks for researchers, though it appears incremental as it builds on existing regularization and contraction methods.
The study tackles the problem of ensuring existence and uniqueness of optimal parameters in neural networks by transforming RNNs into contractions and using a specific regularization term, resulting in conditions for unique optimal parameters that can be found with a simple algorithm to any precision, with easier fulfillment as neuron count increases.
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.