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It is proved that bounded variance alone is too weak: even with bounded iterates, it cannot yield high-probability average-iterate rates, and without bounded iterates it may not even guarantee convergence in expectation.
We establish the stochastic-gradient query complexity of sampling smooth strongly log-concave distributions in every fixed dimension $d\geq1$. For $\mu$-strongly convex, $L$-smooth potentials with unknown minimizers $x_f^\star$ in the ball $\mathbb{B}(0,\mu^{-1/2})$, and unbiased gradient oracles with variance at most...
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