We provide a nonasymptotic analysis of the convergence of the stochastic gradient Hamiltonian Monte Carlo (SGHMC) to a target measure in Wasserstein-2 distance without assuming log-concavity. Our analysis quantifies key theoretical properties of the SGHMC as a sampler under local conditions which significantly improves the findings of previous results. In particular, we prove that the Wasserstein-2 distance between the target and the law of the SGHMC is uniformly controlled by the step-size of the algorithm, therefore demonstrate that the SGHMC can provide high-precision results uniformly in the number of iterations. The analysis also allows us to obtain nonasymptotic bounds for nonconvex optimization problems under local conditions and implies that the SGHMC, when viewed as a nonconvex optimizer, converges to a global minimum with the best known rates. We apply our results to obtain nonasymptotic bounds for scalable Bayesian inference and nonasymptotic generalization bounds.

翻译：我们在不假设对数凹性的前提下，提供了随机梯度哈密顿蒙特卡洛（SGHMC）在Wasserstein-2距离下收敛到目标测度的非渐近分析。该分析量化了SGHMC在局部条件下作为采样器的关键理论性质，显著改进了先前结果。具体而言，我们证明了目标测度与SGHMC分布之间的Wasserstein-2距离由算法步长一致控制，从而表明SGHMC能在迭代次数上一致地提供高精度结果。该分析还允许我们在局部条件下获得非凸优化问题的非渐近界，并揭示出将SGHMC视为非凸优化器时，其以当前已知最优速率收敛至全局最小值。我们将所得结果应用于可扩展贝叶斯推断的非渐近界以及非渐近泛化界的推导。