We establish quantitative convergence to the target and uniform-in-time propagation of chaos for Langevin-regularized Stein variational gradient descent. The Stein interaction need not be small relative to the confining Langevin drift and does not generally yield a contractive particle coupling. At the mean-field level, the Stein and Langevin components dissipate the same relative entropy in the kernel-induced Stein and $2$-Wasserstein geometries, producing the squared kernel Stein discrepancy and relative Fisher information. Under a log-Sobolev inequality for the target, this yields exponential last-iterate convergence. We also derive a finite-particle entropy identity relative to the product target, giving exponential-in-time convergence of the empirical measure up to polynomial sampling errors. For propagation of chaos, we develop two complementary finite-time approaches. A synchronous coupling, combined with exponential moment estimates for the nonlinear mean-field diffusion, yields explicit single-exponential bounds in Wasserstein distance and kernel Stein discrepancy (KSD). Moving-product entropy gives joint-law relative entropy control relative to the evolving mean-field product law and, through entropy superadditivity and concentration, fixed-marginal relative entropy and total variation bounds and empirical KSD estimates. Under an additional $T_2$ inequality for the initial law, it also yields Wasserstein bounds. Combining these finite-time estimates with target convergence at a logarithmic cutoff time gives polynomial uniform-in-time propagation of chaos rates in expectation for empirical KSD and $W_2^2$, and for fixed-marginal total variation and $W_2^2$. All bounds control the last iterate in physical time. We also compare the two finite-time mechanisms and identify regimes in which each gives the sharper polynomial exponent.
Existing constant-step analysis of stochastic \Scaf{} identifies a leading $O(γ/N)$ stationary mean bias and shows that higher-order bias can persist as the client count increases, but does not identify the first client-independent contribution at coefficient level. For full-participation stochastic \Scaf{} with one-dimensional homogeneous clients, fixed local-step count $H$, and bounded additive gradient noise, we prove, uniformly over $N\ge2$, $$ \begin{aligned} \mathbb{E}_{π_{γ,N,H}}[x]-x^\star ={}& -\frac{f'''(x^\star)σ^2}{4f''(x^\star)^2}\fracγ{N}\\ &- \frac{f'''(x^\star)σ^2}{12f''(x^\star)} \frac{(H-1)(5H-1)}{H}γ^2 +O_H\!\left(\frac{γ^2}{N}+γ^3\right). \end{aligned} $$ Hence client averaging suppresses the leading $O(γ/N)$ bias but does not remove the client-independent $O(γ^2)$ component when its coefficient is nonzero. The mechanism is indirect: although the direct control contribution cancels pathwise in the linear global average, the controls still alter within-round local trajectories and their second moments. Fresh gradient noise and persistent control fluctuations therefore generate local second-moment corrections that nonquadratic curvature converts into stationary mean bias. The coefficient vanishes for quadratic objectives. Numerical experiments are consistent with the predicted coefficient, its persistence as client count increases, and the stated joint remainder. The result is restricted to the one-dimensional homogeneous fixed-$H$ setting.
Wujun Lv, Xiaoyu Wang, Yingli Wang +1stat.ML cs.LG math.AP math.PR
Hessian-free high-resolution (HFHR) dynamics augments underdamped Langevin dynamics (ULD) with reversible position diffusion for sampling problems that arise in machine learning. We establish an explicit quantitative contraction rate for HFHR dynamics under a position Poincaré inequality, weighted Hessian and Laplacian bounds, and a compact Sobolev embedding, where the potential function is not necessarily convex. An adapted time-augmented Poincaré inequality yields an explicit rate that improves upon the contraction rate of the underdamped Langevin dynamics. We also give a weak-solution construction and a self-contained spectral proof of the divergence lemma underlying the argument. For HFHR Monte Carlo (HFHRMC) algorithm, which is based on a discretization scheme of HFHR dynamics, we use a path-space Girsanov argument to obtain a non-asymptotic convergence bound and an explicit iteration complexity in total variation distance. The bounds hold for every $α\geq0$ and $γ>0$ and remain regular at the ULD endpoint. Optimizing the iteration complexity bound yields a positive, accuracy-dependent position-diffusion parameter at finite accuracy, while its leading high-accuracy order coincides with that of the optimized ULD endpoint. Our iteration complexity bound improves upon the existing work on HFHR algorithms. Numerical experiments including Bayesian learning problems on real data are provided to illustrate the effect of positive $α$ and its benefit.
Darrel K Joseph, M P Rajanmath.NA math.FA math.ST stat.ML
Inverse learning within a statistical framework has a wide range of applications. It has garnered significant attention in machine learning, artificial intelligence, and related fields, where the goal is to infer unknown parameters from indirect and noisy observations. This work investigates the stable approximation of $u^{\dagger}$ which solves the equation $Au=g$, with $A$ being a linear operator between appropriate vector spaces. We will consider the domain to be a non-reflexive Banach Space and the co-domain to be a space of real-valued functions on a metric space $X$. The function $g$ is characterized by a finite number of independently and identically distributed data points, which are assumed to follow some unknown probability measure $ρ$. We employ Tikhonov regularization with an arbitrary convex functional to obtain the regularized solution corresponding to the given data point. The convergence analysis is carried out with respect to the Bregman distance, and an upper bound for the error is derived in probability terms. The theoretical findings are then supported by numerical experiments.
Keyi Li, Yuval Kluger, Boris Landastat.ML cs.LG stat.AP stat.ME
Dataset alignment is a central step in data analysis across science and engineering, where the goal is to match observations between datasets. Entropic Optimal Transport (EOT) offers a computationally tractable framework for this task by encoding cross-dataset affinities in a transport plan. However, when two datasets are sampled from geometrically similar low-dimensional structures with substantially different sampling densities, the EOT plan may match points by relative sampling density rather than geometric proximity, yielding geometrically misleading correspondences. To address this issue, we propose a density-reweighted EOT framework in which the influence of sampling density on the transport plan can be discounted to a desired degree, ranging from standard EOT to alignment driven purely by underlying geometry. Under suitable regularity conditions, we establish convergence of the reweighted EOT plan to a family of population-level plans whose dependence on sampling density is made explicit. Through simulations, we show that our approach recovers geometrically faithful correspondences, improving over related EOT-based frameworks when datasets exhibit substantial sampling density disparity.
Statistical inverse problems have garnered significant attention in recent years due to the growing importance of statistical learning theory and functional analytic approaches in the fields of machine learning and artificial intelligence. In this paper, we investigate the stable approximation of the element $u^{\dagger}$ that satisfies the equation $Au = g$, where $A$ is a linear operator that maps a Banach space into an appropriate function space. The function $g$ is observed only through independently and identically distributed data points that are corrupted by noise and assumed to follow an unknown distribution $ρ$. We employ the Tikhonov regularization scheme, leveraging statistical learning techniques and the framework of reproducing kernel Banach spaces to estimate the solution. We establish convergence and derive the convergence rate of the estimated solution with respect to the true solution as the number of data points increases, with the rate expressed in probabilistic terms. The theoretical findings are further supported by numerical experiments that demonstrate the effectiveness of the proposed approach.
The Subspace Constrained Mean Shift (SCMS) algorithm is a popular nonparametric method for extracting density ridges, which serve as a low-dimensional representation of high-dimensional data. It is a widely held belief in the literature that SCMS trajectories converge to the classical density ridge, which we call the "static ridge", defined via the density gradient and the eigenvalues and eigenvectors of the density's Hessian. In this paper, we demonstrate that this assumption does not hold in general, as the static definition fails to account for the rotation of the trailing eigenspace along the continuous flow of the algorithm's underlying vector field. To resolve this, we propose a paradigm shift by introducing the "stable ridge", a novel geometric structure defined through the lens of dynamical systems and the Jacobian of the projected density gradient. We prove that this stable ridge is the true theoretical target of the SCMS algorithm. Building upon this foundation, we develop a generalized SCMS framework utilizing a constant step size, establishing its uniform R-linear convergence and topological surjectivity onto the stable ridge. We further derive the rates of convergence for estimating the stable ridge in terms of the Hausdorff distance. Finally, we expose that the original SCMS algorithm suffers from polynomial-time computational complexity, which is caused by implicitly coupling the step size to the smoothing bandwidth via the Mean Shift operator, and demonstrate how our generalized framework provides a statistically consistent and more efficient solution.
We study asynchronous optimization for finite-sum eigenspace computation in heterogeneous distributed systems. The theoretical foundations for asynchronous eigenspace computation remain scarce, with existing approaches offering limited coverage of dynamics directly on the Grassmannian under stale information. In this paper, we propose a Grassmannian incremental aggregation method that refreshes only arriving components and reuses cached gradients, retaining low per-update cost without global synchronization. The method employs an extrinsic polar update that preserves the intrinsic subspace geometry without requiring parallel transport of stale tangent vectors. Our analysis establishes a tight angle-dependent gradient-dominance characterization of the objective and a basin-invariance property for stale aggregated updates. These yield two-phase linear convergence, comprising an explicit broad-basin regime and a sharper local regime, with constants controlled by component spectral spreads. Experiments on serial and distributed PCA demonstrate improved sample efficiency and wall-clock convergence over representative baselines.
{AdaBoost.MH} reduces multi-class classification to a collection of binary subproblems and enjoys the classical boosting-type convergence guarantee under a weak learning condition. A more structured variant, Factorized {AdaBoost.MH}, uses base classifiers of the form $\mathbf{h}(x)=α\mathbf{v} \bm{\varphi}(x)$, where a single binary classifier $\bm{\varphi}$ is shared across all classes and the label dependence is carried by a vote vector $\mathbf{v} \in\{\pm1\}^K$. This factorization is algorithmically attractive and achieves better performance in practice, but its convergence depends on whether one can always choose a vote vector with sufficiently large induced binary weight mass. Previous work resolved this question with a lower bound $\max\{1/n,1/\sqrt{2K}\}$, which still leaves a dimension-dependent slowdown relative to the original {AdaBoost.MH} analysis. In this paper, we sharpen this combinatorial step. For the minimax quantity $\mathfrak{W}_{n,K}$ governing the factorized edge, we prove $\mathfrak{W}_{n,K} = C_{\min\{n+1,K\}}$, where $C_q=1$ for $q=1$, $C_q=q/(3q-4)$ for even $q\ge2$, and $C_q=(q+1)/(3q-1)$ for odd $q\ge2$. Since $C_q\downarrow 1/3$, our bounds show that $\mathfrak{W}_{n,K}=Θ(1)$ uniformly over $n$ and $K$. Consequently, Factorized {AdaBoost.MH} achieves the same boosting-type convergence rate as {AdaBoost.MH} up to a universal constant factor, removing the previously suggested additional dependence on $n$ or $K$ in the number of boosting rounds.
Borna Khodabandeh, Mehdi Molkaraiestat.ML cs.IT cs.LG stat.CO
We study the convergence properties of the random-sweep Gibbs sampler for Gaussian graphical models with a thin-membrane prior. We demonstrate that the convergence rate of the Gibbs sampler is significantly accelerated in the dual model, which is obtained by applying the Fourier transform to the local factors of the normal factor graph representing the original model. In both domains, we derive the exact convergence rates for homogeneous $k$-regular graphs. We prove that, for all homogeneous models whose graphical representations contain cycles, the convergence rate in the dual domain is universal and independent of the underlying graph topology. Moreover, we show that the effective convergence rate in the dual domain is governed by the algebraic connectivity of the graph, providing an additional acceleration without increasing the computational complexity per sweep. We further establish an explicit algebraic relation between the covariance structures of the primal and dual models, enabling marginal statistics of the primal model to be recovered directly from those of the dual model. Finally, numerical experiments on several graph families confirm our theoretical results and demonstrate substantial improvements in the convergence rates in various settings.
Michael Pokojovy, J. Marcus Jobe, Simon Lacoste-Julienstat.ML cs.LG stat.CO
Lloyd's $K$-means algorithm, also known as naïve $K$-means, is a widely used ad hoc optimization heuristic, designed to minimize the sum of squared errors (SSE) across all $K$-partitions of a dataset via iterative cluster refinement. In this work, we establish a novel connection between Lloyd's algorithm and the Frank-Wolfe (FW) algorithm, a prominent first-order method for projection-free optimization. We demonstrate that Lloyd's algorithm is a special case of FW. Leveraging recent advances in FW methods for concave objectives, we derive a non-asymptotic $\mathcal{O}(1/t)$ convergence rate to a local minimum of the SSE objective. To account for empty clusters, an outcome possible under Lloyd's greedy assignment, we develop an FW variant for semismooth objectives while retaining the same convergence rate that is solely controlled by the initial SSE value. We illustrate our findings with a simulation study for spherical Gaussian mixtures and a real-world image segmentation dataset.
Benjamin Capdeville, Young-Heon Kim, Soumik Palmath.PR stat.ML
Given a strongly convex function $u$, equip $R^d$ with a Riemannian metric given by the Hessian $\nabla^2 u$. This is a so-called Hessian manifold. Given a probability density $μ$ one may run a Langevin diffusion intrinsic to the manifold with stationary distribution $μ$. Such (Hessian) manifold-valued Langevin diffusions are called Mirror Langevin diffusions (MLD) which have recently become popular. One of the questions we explore is whether, given $μ$, one can choose $u$ to get an exponential convergence to equilibrium for the MLD, especially if $μ$ is not strongly log-concave. Our results are based on Lyapunov function methods and give sufficient conditions for a Poincaré or a log-Sobolev inequality to hold for the MLD. These, in turn, imply exponential convergence. We also introduce a Markov chain approximation to the MLD given by a two step Gibbs sampler with stationary distribution $μ$. This Markov chain is a variant of the Sinkhorn Markov chain introduced in arXiv:2307.16421 that is conjectured to converge to a time-inhomogeneous generalization of the MLD. Under suitable assumptions, we prove that the Markov chain has a guaranteed convergence rate in $χ^2$ that is consistent with the diffusion time scale. Our proofs are based on ideas from entropic optimal transport and strong data processing inequalities.
Zermelo's algorithm is a classical method for computing the maximum likelihood estimator in the Bradley--Terry (BT) model, but its convergence can be slow in practice. To accelerate computation, Newman introduced a family of Zermelo-type fixed-point iterations parameterized by $α$, with Zermelo's algorithm recovered at $α=1$. Empirical evidence suggests that the choice $α=0$ often converges substantially faster, making it a promising alternative, yet the mechanism underlying this acceleration remains elusive. This paper provides theoretical insight into this phenomenon through a systematic local convergence analysis. We derive closed-form expressions for local convergence factors under synchronous and asynchronous updates and analyze their dependence on $α$ via spectral analysis of the associated Jacobian matrices. For synchronous updates, we show that the algorithm may fail to converge when $α<1$, and its local convergence factor is quasi-convex in $α$ under the population BT model. In contrast, asynchronous updates are always locally convergent, and their local convergence factor is provably monotonically increasing in $α$ under the population BT model of consistently ordered bipartite comparison graphs, establishing the optimality of $α=0$ in this setting. We further establish asymptotic approximation results for the population convergence factors under the BT model, justifying their practical relevance. Numerical experiments on synthetic and real-world datasets confirm the theory. Our analysis complements existing convergence results and shows that the acceleration of $α=0$ arises not only from the parameter choice but, more importantly, from the use of asynchronous updates.
Binary Iterative Hard Thresholding (BIHT) is a simple, yet effective, greedy method for recovering a sparse vector from one-bit sign measurements. In its original form, BIHT performs a ``gradient-descent'' step, followed by hard thresholding. A convergence analysis of this algorithm was left open in the introductory work of [Jac+11] and has remained unresolved for over a decade, with subsequent sharp analyses studying a normalized variant instead, that additionally projects every iterate onto the unit sphere. This paper resolves that gap and characterizes when per-iteration normalization is algorithmically necessary. In the noiseless setting, we prove a universal, sample-optimal convergence theorem for the original BIHT algorithm. Specifically, with $\widetilde O(s/ε)$ measurements, a deterministic finite-time iterate has directional error at most $ε$, simultaneously for every $s$-sparse unit vector. This matches the optimal sample dependence achieved by normalized BIHT in prior work. Thus, in the noiseless regime, per-iterate normalization is unnecessary for optimal recovery. Under sign corruptions, we prove a sharp separation. If at most a $τ$ fraction of signs are flipped adversarially, then BIHT, without per-iterate normalization, still reaches the robust error floor at an early iterate with a matching $\widetilde O(s/ε)$ sample complexity rate as its normalized variant. This recovery, however, is not stable. We prove a scalar lower bound showing that any nontrivial corruption pattern, even one that involves only one flipped sign together with one clean sign, forces the iterates to oscillate indefinitely. Consequently, no general last-iterate convergence theorem can hold for BIHT under sign corruptions, while its normalized surrogate provably escapes this instance.
Stein variational gradient descent (SVGD) transports interacting particles toward a target distribution through deterministic kernelized dynamics. Singular Riesz kernels are attractive because they can provide quantitative population-level convergence, but at the finite-particle level the corresponding Stein energy has infinite self-interaction. We study periodic Riesz SVGD with self-interaction removed and prove a many-particle, long-time sampling theorem. Throughout the range in which the singular Stein energy is locally integrable, under a uniform bound on the initial relative entropy per particle, the time-averaged empirical-measure law converges weakly to the point mass \(δ_π\) at the target as the particle number and any diverging averaging horizon tend to infinity. We also show that the empirical-measure laws induced by invariant particle laws of finite relative entropy converge weakly to \(δ_π\), without a uniform entropy bound. Below the logarithmic singularity threshold, we obtain an explicit algebraic finite-particle error bound. These results extend the joint-entropy approach for smooth-kernel SVGD to singular interactions.
Antonin Chodron de Courcel, Matthew Rosenzweigmath.AP stat.ML
We study the long-time behavior of the Wasserstein gradient flow of the squared Maximum Mean Discrepancy (MMD) between a probability measure $ρ$ and a target measure $μ$, where the underlying kernel is given by a Coulomb potential. For $L^\infty$ target densities $μ$, we establish the existence of global weak solutions starting from arbitrary Borel probability measures and prove that the density $ρ_t$ belongs to $L^\infty$ for any $t>0$. We also show that the Hölder norm can grow exponentially in time. On the flat torus ${\mathbb{T}}^\mathsf{d}$, we prove a global metric PL inequality for every finite-Coulomb-energy source and nearly uniform target. For general bounded, uniformly positive targets, we prove exponential decay of the squared MMD without requiring a lower bound on the initial data, using a defective PL inequality. We also prove that the usual PL inequality may fail when the target vanishes only at one point and that, when $\mathsf{d}\ge2$, no PL constant can hold uniformly over all targets satisfying a prescribed lower bound. On ${\mathbb{R}}^\mathsf{d}$, for $\mathsf{d}\ge2$, under radial symmetry, source-support inclusion, and target-positivity assumptions, we establish a PL inequality and exponential convergence. On the unrestricted whole-space class, neither a multiplicative squared-MMD decay modulus uniform over the initial datum nor a global PL inequality can hold. Finally, in every dimension and in both spatial settings, we prove that every Lagrangian critical point coincides with the target when $(ρ-μ)^+$ is absolutely continuous. In dimension two, the energy supplies uniform tightness. This implies that if our constructed solutions have finite energy at some positive time, then they converge to the target narrowly and strongly in negative-order Sobolev spaces.
When analyzing a manifold learning algorithm for data lying on a smooth, compact, connected Riemannian submanifold $(\mathcal{M}, g)$ of $\mathbb{R}^d$, a key estimate for the geodesic distance $d_g$ is that there exists $K > 0$ such that $0 \leq d_g(p, q)^2 - \|p-q\|^2 \leq K d_g(p, q)^4$ for all $p, q \in \mathcal{M}$. We observe that more generally, when $\mathcal{M}$ is equipped with a smooth symmetric divergence $D$ satisfying a non-degeneracy condition and $g$ is given by $g_p := \frac{1}{2}\mathrm{Hess}_p(D(p, \cdot))$ for all $p \in \mathcal{M}$, there exists $K > 0$ such that $\left| D(p, q) - d_g(p, q)^2 \right| \leq K d_g(p, q)^4$ for all $p, q \in \mathcal{M}$. We demonstrate that this is sufficient for the pointwise convergence of graph Laplacians constructed with $D$ and discuss examples where $D$ is given by the Sinkhorn divergence on a family of probability measures parametrized by a manifold.
Sophia Seulkee Kang, Louis Sharrock, Xiaoyuan Cheng +2cs.LG stat.ML
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 \emph{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.
Collaborative learning is sustainable only when it benefits each participant. Standard federated learning optimizes a global average objective, which can under perform for clients whose data distributions differ substantially from the population. We study selfish personalization: how a designated target client can use peer gradients to minimize its own risk while avoiding negative transfer. We propose SP-CACW, a convergence-aware client-weighting framework that selects aggregation weights by minimizing an upper bound on the target client's convergence error. The resulting rule explicitly trades off peer bias against stochastic variance and can assign zero weight to harmful peers. We provide convergence guarantees under smoothness and bounded-variance assumptions and evaluate the method on MNIST, CIFAR-100, and LEAF Shakespeare, where it is competitive with or improves over strong personalized and clustering baselines.
Hanna Myleiko, Sergei Solodky, Vasyl Semenovstat.ML cs.LG math.NA
This paper investigates convergence properties of regularized Nyström subsampling applied to the unsupervised domain adaptation problem under covariate shift. We focus on the low-smoothness (misspecified) case where the target function lies outside the reproducing kernel Hilbert space. By combining Tikhonov regularization with Nyström projection onto a subsampled subspace, we obtain upper bounds on the excess risk that hold with high probability and are expressed in terms of the source condition, the effective dimension, and the sample sizes. We further extend the analysis to the setting where the Radon-Nikodym derivative between the target and source marginal distributions is unknown and must be approximated, and we identify the minimal additional sample sizes required to maintain the same convergence rate as in the oracle case.
We propose a robust gradient estimator based on per-sample gradient clipping and analyze its properties both theoretically and empirically. We show that the resulting method, per-sample clipped SGD (PS-Clip-SGD), achieves optimal in-expectation convergence rates for non-convex optimization problems under heavy-tailed gradient noise. Moreover, we establish high-probability convergence guarantees that match the in-expectation rates up to polylogarithmic factors in the failure probability. We complement our theoretical results with multiple numerical experiments. In particular, we demonstrate that PS-Clip-SGD outperforms both vanilla SGD with momentum and standard gradient clipping when training AlexNet on the CIFAR-100 dataset, even after accounting for the additional computational time caused by per-sample clipping. We also empirically show that, in the presence of gradient accumulation, applying clipping at the mini-batch level can improve training performance while incurring virtually no additional computational cost. This finding is particularly interesting, as it contradicts the common practice of applying clipping only after all accumulation steps have been completed.