We investigate the stochastic-gradient query complexity of sampling smooth strongly log-concave distributions in any fixed Euclidean dimension. The potential is
μ-strongly convex and
L-smooth, with an unknown mode in the ball of radius
μ−1/2 about the origin. We have access to unbiased stochastic oracles with the variance at most
σ2. For every
σ2≥0 and total variation (TV) accuracy
0<ε≤1/10, we prove that the tight complexity of sampling a distribution within
ϵ-TV distance from the target distribution is
NTV⋆=Θ(log(1+κ)+μϵσ2),
where
κ:=μL is the condition number. Note that this complexity bound is simultaneously tight for the condition number
κ and accuracy
ϵ. Besides, our tight complexity bound is adaptive to noiseless setting
σ=0, which is
NTV⋆=Θ(log(1+κ)).