Adaptive Rollback Gradients for Noisy High-Dimensional Nonconvex Optimization Landscapes

Abstract

The optimization of high-dimensional nonconvex landscapes remains a central challenge in modern computational sciences and machine learning. Traditional stochastic gradient descent methods and their adaptive variants often struggle with pathological landscape features such as sharp minima, vast saddle point plateaus, and vanishing gradients, especially under conditions of high stochastic noise. This paper introduces the adaptive rollback gradient framework, a novel optimization methodology designed to proactively monitor optimization trajectories, detect divergence or entrapment, and dynamically roll back to previously stable parameter checkpoints. By utilizing a continuous memory buffer of recent topological states and evaluating the moving average of gradient norms against a dynamic noise threshold, the proposed method mitigates the catastrophic divergence frequently observed in complex deep neural network training and high-dimensional physics simulations. Extensive theoretical exposition is provided to detail the geometric properties of noisy nonconvex spaces and the mechanics of the rollback trigger. Empirical evaluations across complex synthetic benchmark topologies and deep learning architectures demonstrate that adaptive rollback gradients significantly improve convergence speed and reliability compared to standard optimizers. The results indicate that incorporating trajectory memory and automated state reversion provides a highly robust mechanism for escaping suboptimal regions, offering a scalable solution for next-generation optimization tasks.