We propose an adaptive MCMC method that learns a linear preconditioner which is dense in its off-diagonal elements but sparse in its parametrisation. Due to this sparsity, we achieve a per-iteration computational complexity of $O(m^2d)$ for a user-determined parameter $m$, compared with the $O(d^2)$ complexity of existing adaptive strategies that can capture correlation information from the target. Diagonal preconditioning has an $O(d)$ per-iteration complexity, but is known to fail in the case that the target distribution is highly correlated, see \citet[Section 3.5]{hird2025a}. Our preconditioner is constructed using eigeninformation from the target covariance which we infer using online principal components analysis on the MCMC chain. It is composed of a diagonal matrix and a product of carefully chosen reflection matrices. On various numerical tests we show that it outperforms diagonal preconditioning in terms of absolute performance, and that it outperforms traditional dense preconditioning and multiple diagonal plus low-rank alternatives in terms of time-normalised performance.

Thus, diagonal preconditioning methods remain popular.

Our4 results certify that there exists an optimal linear pre-conditioner for quadratically convex constraint sets. As such,5 adaptivegradient methods can be minimax (rate) optimal. Inonline algorithms, the common practice [4,5,6,7,2]6 is to measure regret with respect to the "best" post-hoc regularizer (i.e. In this setting, the constraint set corresponds to the set of classifiers of interest, and the geometry of the gradients34 corresponds tothegeometry ofthefeatures (orcovariates). A generalized online mirror descent with applications to classification and52 regression.

Preconditioning is a common method applied to modify Markov chain Monte Carlo algorithms with the goal of making them more efficient. In practice it is often extremely effective, even when the preconditioner is learned from the chain. We analyse and compare the finite-time computational costs of schemes which learn a preconditioner based on the target covariance or the expected Hessian of the target potential with that of a corresponding scheme that does not use preconditioning. We apply our results to the Unadjusted Langevin Algorithm (ULA) for an appropriately regular target, establishing non-asymptotic guarantees for preconditioned ULA which learns its preconditioner. Our results are also applied to the unadjusted underdamped Langevin algorithm in the supplementary material. To do so, we establish non-asymptotic guarantees on the time taken to collect $N$ approximately independent samples from the target for schemes that learn their preconditioners under the assumption that the underlying Markov chain satisfies a contraction condition in the Wasserstein-2 distance. This approximate independence condition, that we formalize, allows us to bridge the non-asymptotic bounds of modern MCMC theory and classical heuristics of effective sample size and mixing time, and is needed to amortise the costs of learning a preconditioner across the many samples it will be used to produce.