On stochastic gradient Langevin dynamics with dependent data streams in the logconcave case
This work addresses sampling challenges in machine learning and statistics, particularly for log-concave distributions, but is incremental as it extends existing methods to dependent data settings.
The paper tackles the problem of sampling from a probability distribution using stochastic gradient Langevin dynamics with dependent data streams, obtaining an upper bound on the Wasserstein-2 distance between the algorithm's iterates and the target distribution with explicit constants based on Lipschitz and convexity properties.
We study the problem of sampling from a probability distribution $π$ on $\rset^d$ which has a density \wrt\ the Lebesgue measure known up to a normalization factor $x \mapsto \rme^{-U(x)} / \int_{\rset^d} \rme^{-U(y)} \rmd y$. We analyze a sampling method based on the Euler discretization of the Langevin stochastic differential equations under the assumptions that the potential $U$ is continuously differentiable, $\nabla U$ is Lipschitz, and $U$ is strongly concave. We focus on the case where the gradient of the log-density cannot be directly computed but unbiased estimates of the gradient from possibly dependent observations are available. This setting can be seen as a combination of a stochastic approximation (here stochastic gradient) type algorithms with discretized Langevin dynamics. We obtain an upper bound of the Wasserstein-2 distance between the law of the iterates of this algorithm and the target distribution $π$ with constants depending explicitly on the Lipschitz and strong convexity constants of the potential and the dimension of the space. Finally, under weaker assumptions on $U$ and its gradient but in the presence of independent observations, we obtain analogous results in Wasserstein-2 distance.