Year: 2026
Type: preprint
Abstract: We establish the stochastic-gradient query complexity of sampling smooth strongly log-concave distributions in every fixed dimension $d\geq1$. For $μ$-strongly convex, $L$-smooth potentials with unknown minimizers $x_f^\star$ in the ball $\mathbb{B}(0,μ^{-1/2})$, and unbiased gradient oracles with variance at most $σ^2$, the minimax worst-case expected query complexity is $Θ\left(\log(1+κ)+\frac{σ^2}{μ\varepsilon}\right)$, jointly optimal for the condition number $κ:=L/μ$, variance $σ^2\ge 0$, and TV accuracy $0<\varepsilon\leq 1/10$. In the noiseless setting $σ=0$, this tight complexity $Θ\left(\log(1+κ)\right)$ is independent of $\varepsilon$. Moreover, a gradient-only sampler generates an exact sample with ${O}(\log(1+κ))$ worst-case expected queries. Without previous ball $\mathbb{B}(0,μ^{-1/2})$, exact sampling from any initial point $x_0$ can be implemented with expected cost ${O}\left(\log(1+κ)+\log(1+\sqrtμ\|x_0-x_f^\star\|)\right)$, without knowing the initial distance. We also show that dependence on initial distance is generally unavoidable.
Source: arXiv (Cornell University)
Authors: Ou, Weiming, Wang, Xiao
Institution: Shanghai University of Finance and Economics
Subfield: Statistics and Probability
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7212933924
Cites:
Cited by:
Related to: