arXiv subjects

Max Hird

Publications and source records attributed to Max Hird.

6 recordsLinked to original sources

arXiv2026

High-dimensional Adaptive MCMC with Reduced Computational Complexity

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.

stat.CO↗

arXiv2026

A Non-asymptotic Analysis for Learning and Applying a Preconditioner in MCMC

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 various algorithms including the Unadjusted Langevin Algorithm (ULA) and the proximal sampler for an appropriately regular target, establishing non-asymptotic guarantees for versions of these algorithms that learn and use preconditioners. 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 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 samples it will help to produce.

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arXiv2024

The occlusion process: improving sampler performance with parallel computation and variational approximation

Autocorrelations in MCMC chains increase the variance of the estimators they produce. We propose the occlusion process to mitigate this problem. It is a process that sits upon an existing MCMC sampler, and occasionally replaces its samples with ones that are decorrelated from the chain. We show that this process inherits many desirable properties from the underlying MCMC sampler, such as a Law of Large Numbers, convergence in a normed function space, and geometric ergodicity, to name a few. We show how to simulate the occlusion process at no additional time-complexity to the underlying MCMC chain. This requires a threaded computer, and a variational approximation to the target distribution. We demonstrate empirically the occlusion process' decorrelation and variance reduction capabilities on two target distributions. The first is a bimodal Gaussian mixture model in 1d and 100d. The second is the Ising model on an arbitrary graph, for which we propose a novel variational distribution.

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arXiv2024

Improving the convergence of Markov chains via permutations and projections

This paper aims at improving the convergence to equilibrium of finite ergodic Markov chains via permutations and projections. First, we prove that a specific mixture of permuted Markov chains arises naturally as a projection under the KL divergence or the squared-Frobenius norm. We then compare various mixing properties of the mixture with other competing Markov chain samplers and demonstrate that it enjoys improved convergence. This geometric perspective motivates us to propose samplers based on alternating projections to combine different permutations and to analyze their rate of convergence. We give necessary, and under some additional assumptions also sufficient, conditions for the projection to achieve stationarity in the limit in terms of the trace of the transition matrix. We proceed to discuss tuning strategies of the projection samplers when these permutations are viewed as parameters. Along the way, we reveal connections between the mixture and a Markov chain Sylvester's equation as well as assignment problems, and highlight how these can be used to understand and improve Markov chain mixing. We provide two examples as illustrations. In the first example, the projection sampler (with a suitable choice of the permutation) improves upon Metropolis-Hastings in a discrete bimodal distribution with a reduced relaxation time from exponential to polynomial in the system size, while in the second example, the mixture of permuted Markov chain yields a mixing time that is logarithmic in system size (with high probability under random permutation), compared to a linear mixing time in the Diaconis-Holmes-Neal sampler. Finally, we provide numerical experiments on simple statistical physics models to illustrate the improved mixing performance of the proposed projection samplers over standard Metropolis-Hastings.

math.PR↗

arXiv2023

Quantifying the effectiveness of linear preconditioning in Markov chain Monte Carlo

We study linear preconditioning in Markov chain Monte Carlo. We consider the class of well-conditioned distributions, for which several mixing time bounds depend on the condition number $\kappa$. First we show that well-conditioned distributions exist for which $\kappa$ can be arbitrarily large and yet no linear preconditioner can reduce it. We then impose two sets of extra assumptions under which a linear preconditioner can significantly reduce $\kappa$. For the random walk Metropolis we further provide upper and lower bounds on the spectral gap with tight $1/\kappa$ dependence. This allows us to give conditions under which linear preconditioning can provably increase the gap. We then study popular preconditioners such as the covariance, its diagonal approximation, the hessian at the mode, and the QR decomposition. We show conditions under which each of these reduce $\kappa$ to near its minimum. We also show that the diagonal approach can in fact \textit{increase} the condition number. This is of interest as diagonal preconditioning is the default choice in well-known software packages. We conclude with a numerical study comparing preconditioners in different models, and showing how proper preconditioning can greatly reduce compute time in Hamiltonian Monte Carlo.

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arXiv2020

A fresh take on 'Barker dynamics' for MCMC

We study a recently introduced gradient-based Markov chain Monte Carlo method based on 'Barker dynamics'. We provide a full derivation of the method from first principles, placing it within a wider class of continuous-time Markov jump processes. We then evaluate the Barker approach numerically on a challenging ill-conditioned logistic regression example with imbalanced data, showing in particular that the algorithm is remarkably robust to irregularity (in this case a high degree of skew) in the target distribution.

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