arXiv subjects

Sam Power

Publications and source records attributed to Sam Power.

At least 19 recordsLinked to original sources

arXiv2026

Geometric Functional Inequalities for Markov Processes: An Invitation via the Langevin Diffusion

Functional inequalities provide a language for reasoning about the long-time properties of ergodic Markov processes in terms of their macroscopic geometry. This article offers an informal introduction to functional inequalities for readers who are more familiar with probabilistic tools based around couplings, Lyapunov techniques, and similar. The main ideas are developed through the overdamped Langevin diffusion, where various functionals related to this convergence admit particularly transparent expressions, and for which the key associated inequalities -- namely, the Poincaré and logarithmic Sobolev inequalities -- are especially well-studied. We explain the qualitative content of these inequalities, what they imply about convergence to equilibrium, how they transform and compose under natural operations, and how they interface with classical probabilistic approaches. Related inequalities are discussed more briefly, and an annotated guide is provided for readers wishing to enter the wider literature.

math.PR↗

arXiv2026

Non-Uniform Random Scans in Gibbs Sampling and CAVI

Gibbs sampling and coordinate ascent variational inference (CAVI) are two basic coordinate-wise methods for statistical computation. Recent analyses under strong log-concavity establish convergence rates for versions of these algorithms that update one uniformly selected block at each step. We extend both results to arbitrary fixed, strictly positive selection probabilities. The rates are governed by a selection-adapted convexity constant $λ^{\star}_θ$, defined using the block-smoothness constants and the selection probabilities $θ$. The same constant yields a contraction of relative entropy for the Gibbs sampler and a contraction of the mean-field objective gap for random-scan CAVI. The new bounds recover the uniform-scan results, and are never weaker than the naive comparison based on the smallest selection probability. They provide a principled way to adapt the scan to heterogeneous block geometry using curvature information.

stat.CO↗

arXiv2026

A Uniformly Efficient Rejection Sampler for the Multivariate Expectile-Based Distribution

The multivariate expectile-based distribution is a `spiked' perturbation of a multivariate Gaussian distribution, proposed by Arbel et al. in 2023. The rejection sampler proposed for this distribution in the initial work is valid, but its acceptance probability can degenerate badly both in high dimension and in the strongly asymmetric limit. We give a simple alternative. After affine whitening and a polar decomposition, the sampling problem reduces exactly to a univariate distribution, and an additional hyperbolic change of variables exhibits this distribution as log-concave, so that a universal and uniformly efficient construction of Devroye applies. The resulting exact sampler therefore enjoys an acceptance probability of at least $1 - \exp \left( -1 \right) = 0.632120 \ldots$, uniformly over the dimension and all admissible asymmetry parameters.

stat.CO↗

arXiv2026

Robustness of random-walk Metropolis for steep potentials

In Markov chain Monte Carlo sampling, light-tailed target distributions present something of a poisoned chalice: their light tails offer good confinement, and tend to imply good mixing properties for natural continuous-time dynamics, but the steepness of their tail decay means that they often fall outside of the scope of modern quantitative convergence theory. For usual gradient-based samplers, this reflects a genuine instability issue, whereby Metropolis acceptance rates can degrade badly. In this work, we study the gradient-free random-walk Metropolis sampler, and show that for a wide range of light-tailed targets, the acceptance probability remains stable for reasonable choices of proposal variance, from which effective and favourable mixing time estimates can be deduced. The analysis relies on a simple relationship between the first and second derivatives of the log-density of the target distribution.

math.PR↗

arXiv2026

A Complexity Bound for the Kent-Ganeiber-Mardia Sampler for the Bingham Distribution

The Bingham distribution is a family of antipodally symmetric distributions on the unit sphere, characterised by an exponential-of-quadratic change of measure with respect to the uniform distribution. Kent, Ganeiber and Mardia proposed a rejection sampler for generating samples from Bingham distributions using proposals from an angular central Gaussian (ACG) distribution. Their empirical results suggest that the least efficient regime is the high-concentration limit, where the acceptance probability is of order $d^{-1/2}$ in dimension $d$, implying a polynomial complexity guarantee. In this note, we verify this dimension-dependent prediction, establishing the uniform guarantee $\inf\{α_D:D=D^\top\in\mathbb{R}^{d\times d}\}\ge c_\star/\sqrt{d}$, where $c_\star=0.759\ldots$. A one-dimensional high-concentration limit demonstrates that the $d^{-1/2}$ rate is unimprovable and that even the constant $c_\star$ cannot be improved beyond $0.857\ldots$. The proof relies on a novel interpretation of the acceptance probability and a comparison principle for weighted sums of chi-squared random variables, which may be of independent interest.

stat.CO↗

arXiv2026

Uniform efficiency of the Ulrich-Wood sampler for the von Mises-Fisher distribution

The standard approach to stochastic simulation from the von Mises-Fisher distribution is a rejection sampler proposed by Ulrich and Wood. This note provides a theoretical justification of its efficiency, lower-bounding its acceptance probability uniformly over the dimension and concentration parameter. A novel interpretation of the proposal is given which provides some intuition for this efficiency.

stat.CO↗

arXiv2026

Wasserstein Contraction of Coordinate Ascent Variational Inference

We study the non-asymptotic contraction in Wasserstein distance of the sequential, parallel, and random-scan coordinate ascent variational inference algorithms. This is shown to hold under a functional smoothness condition of the optimality maps and a transportation-information inequality at their fixed points. Our results are sharp and general, and as opposed to those based on global strong log-concavity assumptions, they allow for local convergence on smooth, non-smooth, and discrete manifolds, including within the context of data augmentation. We consider many applications in statistical physics and Bayesian statistics. These include pairwise Markov Random field models such as Ising and Curie-Weiss, unbalanced Bayesian Gaussian Mixture Models, high-dimensional Bayesian Probit Regression, and high-dimensional Logistic Regression with Pólya--Gamma random variables (i.e. Jaakkola-Jordan's algorithm). In many of these models, these represent the first available convergence results of their kind.

stat.ML↗

arXiv2026

Towards practical PDMP sampling: Metropolis adjustments, locally adaptive step-sizes, and NUTS-based time lengths

Piecewise-Deterministic Markov Processes (PDMPs) hold significant promise for sampling from complex probability distributions. However, their practical implementation is hindered by the need to compute model-specific bounds. Conversely, while Hamiltonian Monte Carlo (HMC) offers a generally efficient approach to sampling, its inability to adaptively tune step sizes impedes its performance when sampling complex distributions like funnels. To address these limitations, we introduce three innovative concepts: (a) a Metropolis-adjusted approximation for PDMP simulation that eliminates the need for explicit bounds without compromising the invariant measure, (b) an adaptive step size mechanism compatible with the Metropolis correction, and (c) a No U-Turn Sampler (NUTS)-inspired scheme for dynamically selecting path lengths in PDMPs. These three ideas can be seamlessly integrated into a single, `doubly-adaptive' PDMP sampler with favourable robustness and efficiency properties.

stat.CO↗

arXiv2026

The sharp one-dimensional convex sub-Gaussian comparison constant

Let $X$ be an integrable real random variable with mean zero and two-sided sub-Gaussian tail $\mathbb{P}(|X|>t)\le 2e^{-t^{2}/2}$ for all $t\ge 0$. We determine the smallest constant $c_\star$ such that $X$ is dominated in convex order by $c_\star G$, where $G$ is standard normal. Equivalently, $c_\star^2$ is the sharp one-dimensional convex sub-Gaussian comparison constant appearing in the \emph{Optimization Constants in Mathematics} repository~\cite{optimization-constants-repo}. We show that $c_\star$ is given by an explicit system of one-dimensional equations and is attained by an extremal distribution that saturates the tail constraint. Numerically, $c_\star \approx 2.30952$ (so $c_\star^2 \approx 5.33386$). We also determine the analogous sharp constant under a two-sided sub-exponential tail bound, with convex domination by a scaled Laplace law. Finally, we record two higher-dimensional consequences: a sequential tensorization principle for multivariate convex domination, and a dimension-free Gaussian comparator for the cone generated by convex ridge functions (the linear convex order).

math.PR↗

arXiv2025

Some aspects of robustness in modern Markov Chain Monte Carlo

Markov Chain Monte Carlo (MCMC) is a flexible approach to approximate sampling from intractable probability distributions, with a rich theoretical foundation and comprising a wealth of exemplar algorithms. While the qualitative correctness of MCMC algorithms is often easy to ensure, their practical efficiency is contingent on the `target' distribution being reasonably well-behaved. In this work, we concern ourself with the scenario in which this good behaviour is called into question, reviewing an emerging line of work on `robust' MCMC algorithms which can perform acceptably even in the face of certain pathologies. We focus on two particular pathologies which, while simple, can already have dramatic effects on standard `local' algorithms. The first is roughness, whereby the target distribution varies so rapidly that the numerical stability of the algorithm is tenuous. The second is flatness, whereby the landscape of the target distribution is instead so barren and uninformative that one becomes lost in uninteresting parts of the state space. In each case, we formulate the pathology in concrete terms, review a range of proposed algorithmic remedies to the pathology, and outline promising directions for future research.

stat.CO↗

arXiv2025

Analysis of Multiple-try Metropolis via Poincaré inequalities

We study the Multiple-try Metropolis algorithm using the framework of Poincaré inequalities. We describe the Multiple-try Metropolis as an auxiliary variable implementation of a resampling approximation to an ideal Metropolis--Hastings algorithm. Under suitable moment conditions on the importance weights, we derive explicit Poincaré comparison results between the Multiple-try algorithm and the ideal algorithm. We characterize the spectral gap of the latter, and finally in the Gaussian case prove explicit non-asymptotic convergence bounds for Multiple-try Metropolis by comparison.

stat.CO↗

arXiv2025

Distributional Training Data Attribution: What do Influence Functions Sample?

Randomness is an unavoidable part of training deep learning models, yet something that traditional training data attribution algorithms fail to rigorously account for. They ignore the fact that, due to stochasticity in the initialisation and batching, training on the same dataset can yield different models. In this paper, we address this shortcoming through introducing distributional training data attribution (d-TDA), the goal of which is to predict how the distribution of model outputs (over training runs) depends upon the dataset. Intriguingly, we find that influence functions (IFs), a popular data attribution tool, are 'secretly distributional': they emerge from our framework as the limit to unrolled differentiation, without requiring restrictive convexity assumptions. This provides a new perspective on the effectiveness of IFs in deep learning. We demonstrate the practical utility of d-TDA in experiments, including improving data pruning for vision transformers and identifying influential examples with diffusion models.

cs.LG↗

arXiv2025

Weak Poincaré inequality comparisons for ideal and hybrid slice sampling

Using the framework of weak Poincaré inequalities, we provide a general comparison between Hybrid and Ideal Slice Sampling in terms of their corresponding Dirichlet forms. In particular, under suitable assumptions Hybrid Slice Sampling inherits fast convergence from Ideal Slice Sampling and conversely. We apply our results to analyse the convergence of the Independent Metropolis-Hastings, stepping-out and shrinkage, as well as Hit-and-Run within slice sampling algorithms.

stat.CO↗

arXiv2025

A New Proof of Sub-Gaussian Norm Concentration Inequality

We present a new method for proving the norm concentration inequality of sub-Gaussian variables. Our proof is based on an averaged version of the moment generating function, termed the averaged moment generating function. Our method applies to both vector cases to bound the vector norm and matrix cases to bound the operator norm. Compared with the widely adopted $\varepsilon$-net technique-based proof of the sub-Gaussian norm concentration inequality, our method does not rely on the union bound and promises a tighter concentration bound.

math.PR↗

arXiv2025

Prediction-Centric Uncertainty Quantification via MMD

Deterministic mathematical models, such as those specified via differential equations, are a powerful tool to communicate scientific insight. However, such models are necessarily simplified descriptions of the real world. Generalised Bayesian methodologies have been proposed for inference with misspecified models, but these are typically associated with vanishing parameter uncertainty as more data are observed. In the context of a misspecified deterministic mathematical model, this has the undesirable consequence that posterior predictions become deterministic and certain, while being incorrect. Taking this observation as a starting point, we propose Prediction-Centric Uncertainty Quantification, where a mixture distribution based on the deterministic model confers improved uncertainty quantification in the predictive context. Computation of the mixing distribution is cast as a (regularised) gradient flow of the maximum mean discrepancy (MMD), enabling consistent numerical approximations to be obtained. Results are reported on both a toy model from population ecology and a real model of protein signalling in cell biology.

stat.ME↗

arXiv2023

Weak Poincaré Inequalities for Markov chains: theory and applications

We investigate the application of Weak Poincaré Inequalities (WPI) to Markov chains to study their rates of convergence and to derive complexity bounds. At a theoretical level we investigate the necessity of the existence of WPIs to ensure \mathrm{L}^{2}-convergence, in particular by establishing equivalence with the Resolvent Uniform Positivity-Improving (RUPI) condition and providing a counterexample. From a more practical perspective, we extend the celebrated Cheeger's inequalities to the subgeometric setting, and further apply these techniques to study random-walk Metropolis algorithms for heavy-tailed target distributions and to obtain lower bounds on pseudo-marginal algorithms.

math.PR↗

arXiv2023

Explicit convergence bounds for Metropolis Markov chains: isoperimetry, spectral gaps and profiles

We derive the first explicit bounds for the spectral gap of a random walk Metropolis algorithm on $R^d$ for any value of the proposal variance, which when scaled appropriately recovers the correct $d^{-1}$ dependence on dimension for suitably regular invariant distributions. We also obtain explicit bounds on the ${\rm L}^2$-mixing time for a broad class of models. In obtaining these results, we refine the use of isoperimetric profile inequalities to obtain conductance profile bounds, which also enable the derivation of explicit bounds in a much broader class of models. We also obtain similar results for the preconditioned Crank--Nicolson Markov chain, obtaining dimension-independent bounds under suitable assumptions.

math.PR↗

arXiv2023

Adaptive Tuning for Metropolis Adjusted Langevin Trajectories

Hamiltonian Monte Carlo (HMC) is a widely used sampler for continuous probability distributions. In many cases, the underlying Hamiltonian dynamics exhibit a phenomenon of resonance which decreases the efficiency of the algorithm and makes it very sensitive to hyperparameter values. This issue can be tackled efficiently, either via the use of trajectory length randomization (RHMC) or via partial momentum refreshment. The second approach is connected to the kinetic Langevin diffusion, and has been mostly investigated through the use of Generalized HMC (GHMC). However, GHMC induces momentum flips upon rejections causing the sampler to backtrack and waste computational resources. In this work we focus on a recent algorithm bypassing this issue, named Metropolis Adjusted Langevin Trajectories (MALT). We build upon recent strategies for tuning the hyperparameters of RHMC which target a bound on the Effective Sample Size (ESS) and adapt it to MALT, thereby enabling the first user-friendly deployment of this algorithm. We construct a method to optimize a sharper bound on the ESS and reduce the estimator variance. Easily compatible with parallel implementation, the resultant Adaptive MALT algorithm is competitive in terms of ESS rate and hits useful tradeoffs in memory usage when compared to GHMC, RHMC and NUTS.

stat.CO↗