Stochastic Gradient Langevin Dynamics (SGLD) is a combination of a Robbins-Monro type algorithm with Langevin dynamics. In this paper, the SGLD method with fixed step size @math is considered in order to sample from a logconcave target distribution @math , known up to a normalisation factor. We assume that unbiased estimates of the gradient from possibly dependent observations are available. It is shown that, for all @math , the Wasserstein- @math distance of the @math th iterate of the SGLD algorithm from @math is dominated by @math with appropriate constants @math and @math , whereas for the case of i.i.d. data the upper bound is improved to @math .
It is pointed out in @cite_20 that the ergodicity property of ) is sensitive to the step size @math . Lemma 6.3 of @cite_4 gives an example in which the Euler-Maruyama discretization is transient. As pointed out in @cite_4 , under discretization, the minorization condition is insensitive with appropriate sampling rate while the Lyapunov condition may be lost. An invariant measure exists if the two conditions hold simultaneously, see Theorem 7.3 of @cite_4 and also Theorem 3.2 of @cite_20 for similar discussions. In our paper we follow the footsteps of @cite_24 in imposing strong convexity of @math together with Lipschitzness of its gradient and thus we do obtain ergodicity of ).
@cite_24 @cite_4 @cite_20
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The ergodic properties of SDEs, and various time discretizations for SDEs, are studied. The ergodicity of SDEs is established by using techniques from the theory of Markov chains on general state spaces, such as that expounded by Meyn-Tweedie. Application of these Markov chain results leads to straightforward proofs of geometric ergodicity for a variety of SDEs, including problems with degenerate noise and for problems with locally Lipschitz vector fields. Applications where this theory can be usefully applied include damped-driven Hamiltonian problems (the Langevin equation), the Lorenz equation with degenerate noise and gradient systems. The same Markov chain theory is then used to study time-discrete approximations of these SDEs. The two primary ingredients for ergodicity are a minorization condition and a Lyapunov condition. It is shown that the minorization condition is robust under approximation. For globally Lipschitz vector fields this is also true of the Lyapunov condition. However in the locally Lipschitz case the Lyapunov condition fails for explicit methods such as Euler-Maruyama; for pathwise approximations it is, in general, only inherited by specially constructed implicit discretizations. Examples of such discretization based on backward Euler methods are given, and approximation of the Langevin equation studied in some detail. In this paper we consider a continuous-time method of approximating a given distribution using the Langevin diusion dLtdWt 1 2 r log (Lt)dt. We ®nd conditions under this diusion converges exponentially quickly to or does not: in one dimension, these are essentially that for distributions with exponential tails of the form (x) exp (y |x| , 0< <1, exponential convergence occurs if and only if 1. We then consider conditions under which the discrete approximations to the diusion converge. We ®rst show that even when the diusion itself converges, naive discretizations need not do so. We then consider a Metropolis-adjusted' version of the algorithm, and ®nd conditions under which this also converges at an exponential rate: perhaps surprisingly, even the Metropolized version need not converge exponentially fast even if the diusion does. We brie y discuss a truncated form of the algorithm which, in practice, should avoid the diculties of the other forms.