Semimartingale Bracket Descent: Generator-Normalized Neural Learning Beyond Differentiable Parameter Spaces

Abstract

Semimartingale Bracket Descent (SBD) is a derivative-free learning framework that replaces conventional gradients with stochastic sensitivity inferred from parameter–loss covariation. Using a Markov generator and its carré-du-champ operator, SBD defines a generator-bracket derivative that recovers projected gradients for diffusions, nonlocal secant fields for jump processes, and graph-based regression on discrete parameter spaces. Training uses paired forward-loss evaluations and stochastic parameter displacements, requiring no backpropagation through the objective. A discrete version adapts transition intensities using the estimated learning field. The main novelty is using a normalized state–loss energy measure as a unified optimization primitive across diffusion, Lévy, and discrete Markov dynamics, enabling learning on smooth, nonsmooth, discontinuous, and fully discrete parameter spaces.

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