Max Hird


PhD Student, University College London 

Headshot of Max Hird

Talk Title

Preconditioning for MCMC


Linear transformation of the state variable (linear preconditioning) is a common technique that often drastically improves the practical performance of a Markov chain Monte Carlo algorithm. Despite this, however, quantifying the benefits of linear preconditioning is not well-studied theoretically, and rigorous guidelines for choosing pre-conditioners are not always readily available.  Mixing time bounds for various samplers (HMC, MALA, Unadjusted HMC, Unadjusted Langevin) have been produced in recent works for the class of strongly log-concave and Lipschitz target distributions and depend strongly on a quantity known as the condition number. We study linear preconditioning for this class of distributions, and under appropriate assumptions we provide bounds on the condition number after using a given linear preconditioner. Nonlinear transformations are less common but potentially far more powerful, and are often known by other names, such as Riemannian Manifold methods and the Mirror Langevin. We propose nonlinear preconditioning as a unifying framework in which to conceive of, and compare these techniques. (Joint work with Sam Livingstone)