MiniMMD: Minimization of MMD for Sampling, Inference and Integration

Representing, comparing, and measuring the distance between probability distributions is a key task in computational statistics and machine learning. The choice of representation and the associated distance determine properties of the methods in which they are used: for example, certain distances can allow one to encode robustness or smoothness of the problem. Kernel methods offer flexible and rich Hilbert space representations of distributions that allow the modeller to enforce properties through the choice of kernel, and estimate associated distances at efficient nonparametric rates. In particular, the maximum mean discrepancy (MMD), a kernel-based distance constructed by comparing Hilbert space mean functions, has received significant attention due to its computational tractability and is favoured by practitioners. In this thesis, we conduct a thorough study of kernel-based distances with a focus on efficient computation, with core contributions in Chapters 3 to 6. Part I of the thesis is focused on the MMD, specifically on improved MMD estimation. In Chapter 3 we propose a theoretically sound, improved estimator for MMD in simulation-based inference. Then, in Chapter 4, we propose an MMD-based estimator for conditional expectations, a ubiquitous task in statistical computation. Closing Part I, in Chapter 5 we study the problem of calibration when MMD is applied to the task of integration. In Part II, motivated by the recent developments in kernel embeddings beyond the mean, we introduce a family of novel kernel-based discrepancies: kernel quantile discrepancies. These address some of the pitfalls of MMD, and are shown through both theoretical results and an empirical study to offer a competitive alternative to MMD and its fast approximations. We conclude with a discussion on broader lessons and future work emerging from the thesis.

Minimum maximum mean discrepancy (MMD) estimation has emerged as a robust and likelihood-free alternative to maximum likelihood estimation for parameter estimation. Yet, despite its practical success, the associated optimization problem remains poorly understood, with theoretical guarantees for existing algorithms hinging on convexity assumptions that rarely hold in practice. We address this gap by proposing a preconditioned gradient descent (PGD) scheme, establishing its asymptotic global convergence under explicit gradient-dominance and projection-residual conditions. Our approach is inspired by recent progress on MMD gradient flows, a nonparametric descent scheme on the space of probability measures. We provide extensive empirical evidence that our PGD scheme outperforms standard gradient descent across a range of challenging parameter estimation and composite hypothesis testing problems.

Over the past 20 years, kernel discrepancies have been leveraged as a highly powerful tool for quantifying the disagreement of distributions, with numerous successful applications in two-sample, goodness-of-fit, and independence testing, among others. Their fastest estimators are known to converge at a parametric rate---$n^{-1/2}$---under mild conditions. While this rate is known to be minimax optimal on $\mathbb R^d$ under strict assumptions with bounded kernels, little is known about its optimality beyond the finite-dimensional Euclidean setting with unbounded kernels. In this work, we prove that the minimax lower bound of estimation of the most popular kernel discrepancies (maximum mean discrepancy, Hilbert-Schmidt independence criterion and kernel Stein discrepancy; MMD, HSIC, KSD) is $n^{-1/2}$ on general topological spaces, and under mild assumptions on the kernel; the same rates are shown (as corollaries) to hold for the estimation of the mean embedding and the centered cross-covariance operator. Our results settle the question of optimal estimation of these kernel discrepancies.