AI 中文总结
该研究证明了在标准d-单纯形的负熵光滑凸函数类中，一阶方法收敛速率存在Ω(L/T)下界，无法加速，还将结果扩展到量子场景的负冯·诺依曼熵函数类。
AI 中文摘要
我们证明了：在标准d-单纯形上，相对于负熵为凸且L-光滑的函数类中，当d=Ω(T²)时，所有一阶方法的收敛速率都满足Ω(L/T)的下界。特别地，这表明镜像下降法在该函数类中达到对数因子内的最优性能。由于在ℓ₁-范数光滑性假设下存在现成的加速方法，这一结果可能令人惊讶。尽管Dragomir等人（《数学规划》，2022）已证明在相对光滑性下加速可能不可能，但他们的邻近函数是病态的，且与困难实例一同构造。相比之下，我们针对具有特别良好结构的特定邻近函数证明了非加速性。我们还将该结果扩展到量子场景，证明在d×d迹为1的半正定厄米矩阵的谱面上，相对于负冯·诺依曼熵为L-光滑的函数类中，存在相同的下界。
英文摘要
We prove an $Ω(L/T)$ lower bound for the convergence rate of minimization in the class of functions that are convex and $L$-smooth relative to negative entropy on the standard $d$-simplex, valid for every first-order method when $d = Ω(T^2)$. In particular, this shows that mirror descent is optimal up to a logarithmic factor in this class. This may be surprising due to the fact that accelerated methods are readily available under the assumption of smoothness in $\ell_1$-norm. While Dragomir et al. (Mathematical Programming, 2022) have already showed that acceleration might be impossible under relative smoothness, their prox-function is pathological and constructed together with the hard instance. In contrast, we show non-acceleration for a specific prox-function with particularly favorable structure. We also extend the result to the quantum setting, proving the same lower bound in the class of functions $L$-smooth relative to negative von Neumann entropy on the spectrahedron of $d \times d$ Hermitian positive-semidefinite matrices with unit trace.
Comments19 pages
