The dimension exponent of first-order query complexity for well-conditioned log-concave sampling

Theorem 1.1: in the exact first-order oracle model with unrestricted computation between queries, for every $\varepsilon>0$ the worst-case query budget $Q(d)$ for TV error $1/10$ satisfies $Q(d)\le C_\varepsilon d^{\varepsilon}$ for all $d\ge2$, on every execution; and $Q(d)\ge c\log d$ for arbitrary randomized adaptive algorithms. Hence the optimal dimension exponent is $\gamma_*=0$. Not shown: the growth rate of $Q(d)$ beyond the exponent (no polylogarithmic bound is claimed), dependence on the accuracy or condition number, or any running-time bound; the constants and approximation orders depend on $\varepsilon$.

Theorem 1.1: in the exact first-order oracle model with unrestricted computation between queries, for every $\varepsilon>0$ the worst-case query budget $Q(d)$ for TV error $1/10$ satisfies $Q(d)\le C_\varepsilon d^{\varepsilon}$ for all $d\ge2$, on every execution; and $Q(d)\ge c\log d$ for arbitrary randomized adaptive algorithms. Hence the optimal dimension exponent is $\gamma_*=0$. Not shown: the growth rate of $Q(d)$ beyond the exponent (no polylogarithmic bound is claimed), dependence on the accuracy or condition number, or any running-time bound; the constants and approximation orders depend on $\varepsilon$.