Iterative Numerical Optimization for Stable Training of Nonlinear Deep Learning Models

Abstract

The training of deep neural networks can be seen as a high-dimensional nonlinear and nonconvex optimization problem, where issues such as numerical instability, ill-conditioned problem formulation, oscillatory behavior of parameter evolution, and sensitivity to learning rate choice can negatively impact convergence properties. While gradient-based methods of optimization play a vital role in deep learning techniques, traditional approaches to their implementation focus more on the algorithm than on stability considerations. This research presents a state-dependent dynamical systems formulation for achieving robust iterative optimization of non-linear deep learning networks. The method involves extending the classical notion of gradient flow through the introduction of a state-dependent positive definite stabilizing operator that influences the optimization path depending on the geometry of the objective function locally. A Lyapunov function derived from the training objective is used to prove stability in a continuous-time setting. Under common conditions of smoothness, boundedness, and positive definiteness, the dynamics will be proven to produce non-increasing trajectories towards the stationary set. A discrete iterative optimization algorithm is then formulated based on numerical time discretization, along with a sufficient condition for the monotone decrease of the objective function. In this way, the formulation forms a link between nonlinear dynamic systems, Lyapunov stability theory, and iterative optimization of deep learning algorithms. Numerical verification using nonlinear test functions and deep learning training tasks is suggested to examine convergence properties, gradient stability, and computational efficiency. The research suggests a mathematically justified approach to introducing state-dependent stabilization to nonconvex optimization problems.