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Concentration of the Langevin Algorithm's Stationary Distribution

Dec 2022

A canonical algorithm for log-concave sampling is the Langevin Algorithm, akathe Langevin Diffusion run with some discretization stepsize $\eta > 0$. Thisdiscretization leads the Langevin Algorithm to have a stationary distribution$\pi_{\eta}$ which differs from the stationary distribution $\pi$ of theLangevin Diffusion, and it is an important challenge to understand whether thewell-known properties of $\pi$ extend to $\pi_{\eta}$. In particular, whileconcentration properties such as isoperimetry and rapidly decaying tails areclassically known for $\pi$, the analogous properties for $\pi_{\eta}$ are openquestions with direct algorithmic implications. This note provides a first stepin this direction by establishing concentration results for $\pi_{\eta}$ thatmirror classical results for $\pi$. Specifically, we show that for anynontrivial stepsize $\eta > 0$, $\pi_{\eta}$ is sub-exponential (respectively,sub-Gaussian) when the potential is convex (respectively, strongly convex).Moreover, the concentration bounds we show are essentially tight. Key to our analysis is the use of a rotation-invariant moment generatingfunction (aka Bessel function) to study the stationary dynamics of the LangevinAlgorithm. This technique may be of independent interest because it enablesdirectly analyzing the discrete-time stationary distribution $\pi_{\eta}$without going through the continuous-time stationary distribution $\pi$ as anintermediary.

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