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

Abstract

This paper presents a theoretically grounded optimization framework for neural network training that integrates an Exponentially Decaying Learning Rate with Lyapunov-based stability analysis. We develop a dynamic learning rate algorithm and prove that it induces connected and stable descent paths through the loss landscape by maintaining the connectivity of super-level sets Sλ = θ ∈  ℝn : ℒ(θ) ≥ λ.  Under the condition that the Lyapunov function V(θ) = ℒ(θ) satisfies  Δ V(θ) ⋅ Δ ℒ(θ) ≥ 0, we establish that these super-level sets are not only connected but also equiconnected across epochs, providing uniform topological stability.  We further derive convergence guarantees using a second-order Taylor expansion and demonstrate that our exponentially scheduled learning rate with gradient-based modulation leads to a monotonic decrease in loss. The proposed algorithm incorporates this schedule into a stability-aware update mechanism that adapts step sizes based on both curvature and energy-level geometry.  This work formalizes the role of topological structure in convergence dynamics and introduces a provably stable optimization algorithm for high-dimensional, non-convex neural networks.