A central challenge in gradient-free MCMC is designing algorithms that simultaneously bypass manual tuning, scale efficiently with dimension, and adapt to local target geometry. While adaptive strategies can auto-tune generic frameworks like random walk Metropolis, they offer slow, linear-order scaling of mixing times with dimension. Elliptical slice sampling (ESS) offers a promising alternative: it is tuning-free, adjusts to local geometry, and can achieve nearly dimension-free scaling under favorable conditions. However, its efficiency degrades rapidly if there is a mismatch between the target distribution and the distribution used to generate the ellipse-defining auxiliary variables, precluding its use in high-dimensional settings. We demonstrate that a careful synthesis of ESS and diminishing adaptation directly resolves these bottlenecks. The resulting adaptive generalized elliptical slice sampler (AGESS) self-corrects from a slow-mixing to a fast-mixing regime, while preserving ergodicity across a wide variety of target densities satisfying mild regularity conditions. The algorithm's utility is demonstrated across a broad collection of challenging applications, including generalized regression, deep Gaussian process surrogate modeling, and high-dimensional sparse regression. Together, our theoretical results and the case studies give evidence of the efficiency and robustness of AGESS across target distributions that are non-elliptical, non-differentiable, multi-modal, or high-dimensional.

翻译：无梯度MCMC中的核心挑战在于设计能够同时避免手动调参、随维度高效扩展并适应当地目标几何结构的算法。虽然自适应策略可以自动调整随机游走Metropolis等通用框架，但其混合时间随维度呈缓慢的线性阶扩展。椭圆切片采样（ESS）提供了一种有前景的替代方案：它无需调参、能适应局部几何结构，并在有利条件下可实现近乎与维度无关的扩展。然而，若目标分布与用于生成椭圆定义辅助变量的分布不匹配，其效率将迅速下降，从而限制其在高维场景中的应用。我们证明，通过将ESS与递减自适应进行巧妙融合可直接解决这些瓶颈。由此产生的自适应广义椭圆切片采样器（AGESS）能从慢速混合状态自我修正为快速混合状态，同时在满足温和正则化条件的各类目标密度上保持遍历性。该算法的实用性在涵盖广义回归、深度高斯过程代理模型及高维稀疏回归等众多具有挑战性的应用中得到了验证。综合理论结果与案例研究，我们获得了AGESS在非椭圆、非可微、多模态或高维目标分布中高效性与鲁棒性的实证依据。