Adaptive Momentum and Nonlinear Damping for Neural Network Training

We propose a continuous-time scheme for large-scale optimization that introduces individual, adaptive momentum coefficients regulated by the kinetic energy of each model parameter. This approach automatically adjusts to local landscape curvature to maintain stability without sacrificing convergence speed. We demonstrate that our adaptive friction can be related to cubic damping, a suppression mechanism from structural dynamics. Furthermore, we introduce two specific optimization schemes by augmenting the continuous dynamics of mSGD and Adam with a cubic damping term. Empirically, our methods demonstrate robustness and match or outperform Adam on training ViT, BERT, and GPT2 tasks where mSGD typically struggles. We further provide theoretical results establishing the exponential convergence of the proposed schemes.

Adaptive Stepsizing for Stochastic Gradient Langevin Dynamics in Bayesian Neural Networks

Bayesian neural networks (BNNs) require scalable sampling algorithms to approximate posterior distributions over parameters. Existing stochastic gradient Markov Chain Monte Carlo (SGMCMC) methods are highly sensitive to the choice of stepsize and adaptive variants such as pSGLD typically fail to sample the correct invariant measure without addition of a costly divergence correction term. In this work, we build on the recently proposed `SamAdams' framework for timestep adaptation (Leimkuhler, Lohmann, and Whalley 2025), introducing an adaptive scheme: SA-SGLD, which employs time rescaling to modulate the stepsize according to a monitored quantity (typically the local gradient norm). SA-SGLD can automatically shrink stepsizes in regions of high curvature and expand them in flatter regions, improving both stability and mixing without introducing bias. We show that our method can achieve more accurate posterior sampling than SGLD on high-curvature 2D toy examples and in image classification with BNNs using sharp priors.