Tight Sampling Complexity with stochastic gradient oracles in Fixed Dimensions

We investigate the stochastic-gradient query complexity of sampling smooth strongly log-concave distributions in any fixed Euclidean dimension. The potential is $\mu$-strongly convex and $L$-smooth, with an unknown mode in the ball of radius ${\mu }^{-1\mathrm{/}2}$ about the origin. We have access to unbiased stochastic oracles with the variance at most ${\sigma }^2$. For every ${\sigma }^2\ge 0$ and total variation (TV) accuracy $0<\epsilon \le 1\mathrm{/}10$, we prove that the tight complexity of sampling a distribution within $\epsilon$-TV distance from the target distribution is $$N_{\text{TV}}^{\star }=\mathrm{\Theta }\text{\hspace{0.17em}⁣}(\log (1+\kappa )+\frac{{\sigma }^2}{\mu \epsilon }),$$ where $\kappa :=\frac{L}{\mu }$ is the condition number. Note that this complexity bound is simultaneously tight for the condition number $\kappa$ and accuracy $\epsilon$. Besides, our tight complexity bound is adaptive to noiseless setting $\sigma =0$, which is $N_{\text{TV}}^{\star }=\mathrm{\Theta }\text{\hspace{0.17em}⁣}(\log (1+\kappa ))$.