We study the query complexity of obtaining a relative Fisher information guarantee for sampling from a log-smooth non-log-concave distribution; this is a sampling analog of finding an approximate stationary point in optimization. Our algorithm is based on the proximal sampler, which is an implicit discretization of the Langevin diffusion, and requires an implementation of the backward step known as the restricted Gaussian oracle (RGO). We show that by leveraging the recent results for log-concave sampling with high-accuracy guarantees in Rényi divergence, we can obtain an approximate RGO implementation that -- when used with the proximal sampler -- yields a complexity guarantee in relative Fisher information that inherits the same dimension dependence as log-concave sampling, and improves upon prior work for non-log-concave sampling. We also show a converse reduction that any improvement in the dimension dependence in relative Fisher information for non-log-concave sampling will yield an improved dimension dependence for high-accuracy log-concave sampling.

翻译：我们研究了从对数光滑的非对数凹分布中采样以达到相对Fisher信息保证的查询复杂度；这是优化问题中寻找近似稳定点的采样对应。我们的算法基于近端采样器，即Langevin扩散的隐式离散化，并需要实现被称为受限高斯预言机（RGO）的后向步骤。我们证明，通过利用对数凹采样在Rényi散度下具有高精度保证的最新结果，可以获得近似RGO实现——该实现与近端采样器结合使用时，能在相对Fisher信息上提供继承了对数凹采样相同维度依赖的复杂度保证，并改进了非对数凹采样的先前工作。我们还展示了一个逆向归约：任何改善非对数凹采样在相对Fisher信息中维度依赖性的进展，都将导致高精度对数凹采样的维度依赖性得到改进。