Calibrating Neural Networks' parameters through Optimal Contraction in a Prediction Problem
Valdes Gonzalo
TL;DR
This work introduces a contraction-based framework for calibrating neural network parameters by translating recurrent dynamics into an activation-domain contraction. A regularized loss coupled with the activation-domain reformulation yields a tractable analytic form for the first-order conditions, which reduce to two Sylvester-type equations that can be solved iteratively. Under explicit bounds on the regularization parameter and condition numbers, the authors prove existence and uniqueness of optimal parameters and provide an iterative scheme that converges to the fixed point; these results extend to constrained RNNs and, with caveats, to feedforward networks. A toy polynomial regression example demonstrates practical effectiveness, and the approach is positioned as compatible with classical statistical tools and potential extensions beyond regression. The work suggests that including loops or skip connections may ease training by creating regions where contraction guarantees hold.
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.