Non-stationary phase of the MALA algorithm.

Area of Science:

• Computational Statistics
• Markov Chain Monte Carlo Methods
• Stochastic Processes

Background:

• The Metropolis-Adjusted Langevin Algorithm (MALA) is a Markov Chain Monte Carlo (MCMC) method used for sampling target distributions.
• Assessing the computational cost of MALA as a function of dimension (N) is crucial for large-scale applications.
• Existing research often focuses on product measures or stationary initial conditions.

Purpose of the Study:

• To analyze the computational cost, measured by Expected Squared Jumping Distance (ESJD), of MALA in high dimensions.
• To investigate the behavior of MALA for non-product target measures and non-stationary initial conditions.
• To extend previous analyses of MALA's efficiency to more complex and practically relevant scenarios.

Main Methods:

• Deriving a diffusion limit for the Markov chain generated by the MALA algorithm.
• Studying a class of non-product target measures derived from discretizing infinite-dimensional Hilbert space measures.
• Developing new analytical techniques to handle the non-stationary and non-product setting, involving coupled stochastic PDEs and ODEs.

Main Results:

• The diffusion limit reveals a coupled stochastic PDE and ODE system.
• The computational cost (ESJD) in the non-stationary regime scales as O(N^(1/2)).
• In contrast, the stationary regime exhibits a cost scaling of O(N^(-1/2)).

Conclusions:

• The non-stationary nature of MALA significantly impacts its computational cost in high dimensions.
• The findings provide a more realistic assessment of MALA's efficiency for complex target distributions.
• This work offers a theoretical framework for understanding MALA's performance in challenging settings.