Metropolis-Adjusted Langevin Algorithm with SPSA-Approximated Gradients

In Metropolis Hastings methods, gradient-based proposals, such as Metropolis-Adjusted Langevin Algorithm (MALA), are widely used due to the fast convergence in distribution compared with non-gradient-based sampling methods like random walk proposals. On the other hand, the application of MALA is constrained by the accessibility of gradients. To extend the application scenario of MALA, we propose to use SPSA-based approximated gradients in MALA and name the algorithm MALA-SPSA. We prove the mixing time of MALA-SPSA to prove its efficiency in theory. Numerical experiments are conducted to verify the performance of MALA-SPSA.