Gibbs Gradient Descent: A Langevin Approach to Optimization

Many modern generative modeling and inference methods require optimizing objectives defined on parametric probability distributions. This leads to costly nested procedures, where each gradient step in first-order optimization techniques requires approximate sampling from the iterate parametric distribution. Gibbs gradient descent (GGD) was recently proposed as an efficient ``single-loop'' alternative by coupling sampling and optimization dynamics (Marion et al., 2025). However, its theoretical understanding remains limited, especially in the presence of particle approximations. This work provides a comprehensive analysis of GGD for the ideal infinite-particle system and its finite-particle approximation, explicitly quantifying the effect of particle noise via a sharp persistent $\mathcal{O}(1/n)$ variance term. We then identify a finite-observable structural class for the Gibbs gradient, which contains our motivating examples, and introduce a control-variate version of GGD that reduces the particle-induced variance at the observable level. Our results reveal a fundamental trade-off between optimization and sampling, driven by the stochastic error induced by finite particle approximations.