Furthermore, we obtain exactp-values, which are easily computed in terms of the Tulap random variable. We show that our results also apply to distribution-free hypothesis testsforcontinuous data.

In this work, we give a ${\rm poly}(d,k)$ time and sample algorithm for efficiently learning the parameters of a mixture of $k$ spherical distributions in $d$ dimensions. Unlike all previous methods, our techniques apply to heavy-tailed distributions and include examples that do not even have finite covariances. Our method succeeds whenever the cluster distributions have a characteristic function with sufficiently heavy tails. Such distributions include the Laplace distribution but crucially exclude Gaussians. All previous methods for learning mixture models relied implicitly or explicitly on the low-degree moments. Even for the case of Laplace distributions, we prove that any such algorithm must use super-polynomially many samples. Our method thus adds to the short list of techniques that bypass the limitations of the method of moments. Somewhat surprisingly, our algorithm does not require any minimum separation between the cluster means. This is in stark contrast to spherical Gaussian mixtures where a minimum $\ell_2$-separation is provably necessary even information-theoretically [Regev and Vijayaraghavan '17]. Our methods compose well with existing techniques and allow obtaining ''best of both worlds" guarantees for mixtures where every component either has a heavy-tailed characteristic function or has a sub-Gaussian tail with a light-tailed characteristic function. Our algorithm is based on a new approach to learning mixture models via efficient high-dimensional sparse Fourier transforms. We believe that this method will find more applications to statistical estimation. As an example, we give an algorithm for consistent robust mean estimation against noise-oblivious adversaries, a model practically motivated by the literature on multiple hypothesis testing. It was formally proposed in a recent Master's thesis by one of the authors, and has already inspired follow-up works.

End-to-end autonomous driving has emerged as a pivotal direction in the field of autonomous systems. Recent works have demonstrated impressive performance by incorporating high-level guidance signals to steer low-level trajectory planners. However, their potential is often constrained by inaccurate high-level guidance and the computational overhead of complex guidance modules. To address these limitations, we propose Mimir, a novel hierarchical dual-system framework capable of generating robust trajectories relying on goal points with uncertainty estimation: (1) Unlike previous approaches that deterministically model, we estimate goal point uncertainty with a Laplace distribution to enhance robustness; (2) To overcome the slow inference speed of the guidance system, we introduce a multi-rate guidance mechanism that predicts extended goal points in advance. Validated on challenging Navhard and Navtest benchmarks, Mimir surpasses previous state-of-the-art methods with a 20% improvement in the driving score EPDMS, while achieving 1.6 times improvement in high-level module inference speed without compromising accuracy. The code and models will be released soon to promote reproducibility and further development. The code is available at https://github.com/ZebinX/Mimir-Uncertainty-Driving

Outside the hypothesis testing setting, there is some work on optimal population inference under DP .

Pedestrian inertial localization is key for mobile and IoT services because it provides infrastructure-free positioning. Yet most learning-based methods depend on fixed sliding-window integration, struggle to adapt to diverse motion scales and cadences, and yield inconsistent uncertainty, limiting real-world use. We present ReNiL, a Bayesian deep-learning framework for accurate, efficient, and uncertainty-aware pedestrian localization. ReNiL introduces Inertial Positioning Demand Points (IPDPs) to estimate motion at contextually meaningful waypoints instead of dense tracking, and supports inference on IMU sequences at any scale so cadence can match application needs. It couples a motion-aware orientation filter with an Any-Scale Laplace Estimator (ASLE), a dual-task network that blends patch-based self-supervision with Bayesian regression. By modeling displacements with a Laplace distribution, ReNiL provides homogeneous Euclidean uncertainty that integrates cleanly with other sensors. A Bayesian inference chain links successive IPDPs into consistent trajectories. On RoNIN-ds and a new WUDataset covering indoor and outdoor motion from 28 participants, ReNiL achieves state-of-the-art displacement accuracy and uncertainty consistency, outperforming TLIO, CTIN, iMoT, and RoNIN variants while reducing computation. Application studies further show robustness and practicality for mobile and IoT localization, making ReNiL a scalable, uncertainty-aware foundation for next-generation positioning.

We study the problem of sampling from a distribution under local differential privacy (LDP). Given a private distribution $P \in \mathcal{P}$, the goal is to generate a single sample from a distribution that remains close to $P$ in $f$-divergence while satisfying the constraints of LDP. This task captures the fundamental challenge of producing realistic-looking data under strong privacy guarantees. While prior work by Park et al. (NeurIPS'24) focuses on global minimax-optimality across a class of distributions, we take a local perspective. Specifically, we examine the minimax risk in a neighborhood around a fixed distribution $P_0$, and characterize its exact value, which depends on both $P_0$ and the privacy level. Our main result shows that the local minimax risk is determined by the global minimax risk when the distribution class $\mathcal{P}$ is restricted to a neighborhood around $P_0$. To establish this, we (1) extend previous work from pure LDP to the more general functional LDP framework, and (2) prove that the globally optimal functional LDP sampler yields the optimal local sampler when constrained to distributions near $P_0$. Building on this, we also derive a simple closed-form expression for the locally minimax-optimal samplers which does not depend on the choice of $f$-divergence. We further argue that this local framework naturally models private sampling with public data, where the public data distribution is represented by $P_0$. In this setting, we empirically compare our locally optimal sampler to existing global methods, and demonstrate that it consistently outperforms global minimax samplers.

Standard first-order Langevin algorithms such as the unadjusted Langevin algorithm (ULA) are obtained by discretizing the Langevin diffusion and are widely used for sampling in machine learning because they scale to high dimensions and large datasets. However, they face two key limitations: (i) they require differentiable log-densities, excluding targets with non-differentiable components; and (ii) they generally fail to sample heavy-tailed targets. We propose anchored Langevin dynamics, a unified approach that accommodates non-differentiable targets and certain classes of heavy-tailed distributions. The method replaces the original potential with a smooth reference potential and modifies the Langevin diffusion via multiplicative scaling. We establish non-asymptotic guarantees in the 2-Wasserstein distance to the target distribution and provide an equivalent formulation derived via a random time change of the Langevin diffusion. We provide numerical experiments to illustrate the theory and practical performance of our proposed approach.

Uncertainty quantification in reinforcement learning can greatly improve exploration and robustness. Approximate Bayesian approaches have recently been popularized to quantify uncertainty in model-free algorithms. However, so far the focus has been on improving the accuracy of the posterior approximation, instead of studying the accuracy of the prior and likelihood assumptions underlying the posterior. In this work, we demonstrate that there is a cold posterior effect in Bayesian deep Q-learning, where contrary to theory, performance increases when reducing the temperature of the posterior. To identify and overcome likely causes, we challenge common assumptions made on the likelihood and priors in Bayesian model-free algorithms. We empirically study prior distributions and show through statistical tests that the common Gaussian likelihood assumption is frequently violated. We argue that developing more suitable likelihoods and priors should be a key focus in future Bayesian reinforcement learning research and we offer simple, implementable solutions for better priors in deep Q-learning that lead to more performant Bayesian algorithms.