Sampling from a high-dimensional distribution is a fundamental task in statistics, engineering, and the sciences. A canonical approach is the Langevin Algorithm, i.e., the Markov chain for the discretized Langevin Diffusion. This is the sampling analog of Gradient Descent. Despite being studied for several decades in multiple communities, tight mixing bounds for this algorithm remain unresolved even in the seemingly simple setting of log-concave distributions over a bounded domain. This paper completely characterizes the mixing time of the Langevin Algorithm to its stationary distribution in… Sampling from a high-dimensional distribution is a fundamental task in statistics, engineering, and the sciences. A canonical approach is the Langevin Algorithm, i.e., the Markov chain for the discretized Langevin Diffusion. This is the sampling analog of Gradient Descent. Despite being studied for several decades in multiple communities, tight mixing bounds for this algorithm remain unresolved even in the seemingly simple setting of log-concave distributions over a bounded domain. This paper completely characterizes the mixing time of the Langevin Algorithm to its stationary distribution in… Read More