Use the trajectory martingale decomposition to separate predictable training updates from genuinely unpredictable residual updates, then scale the residual according to its estimated response to future loss. The method targets stochastic or event-driven optimization with history-dependent samples and predicts that response-weighted residual energy, rather than total gradient variance, controls update noise and instability.
Paper: The Memory Hidden in Response Fluctuations: Trajectory-Level Fluctuation-Response Theory and Inequalities for Non-Markovian Jump DynamicsarXiv:2608.20328
Construct optimizer variables as interconnected Hamiltonian subsystems: parameters store potential energy, momentum stores kinetic energy, and a skew coupling transfers energy between them without net creation. Positive-semidefinite resistance removes energy and provides an explicit damping knob, separating conservative exploration from dissipative convergence.
Put a gradient-Gibbs prior on differences between connected neural parameters rather than on individual parameters, and evolve the parameters with Langevin steps generated from randomly selected strictly convex component energies. The aggregate regularizer may be non-convex, but every sampled component has controlled curvature and outward drift, providing a practical stability mechanism for noisy training.
Maintain a small population of neural-network parameter replicas and interleave ordinary gradient steps with Boltzmann/Kac-style binary collisions. Each collision preserves the pair's mean parameter vector and relative-distance norm while randomly rotating the relative direction, with collision frequency proportional to a regularized negative power of replica distance.