AI 中文总结
该研究证明动力学朗之万动力学的标准Strang分裂（如OBABO、BAOAB）无法通过固定参数调优实现采样加速，其冷启动总变差混合的条件数依赖为线性阶，而非预期的平方根阶。
AI 中文摘要
OBABO、BAOAB方案以及动力学（欠阻尼）朗之万动力学的其他标准Strang分裂是广泛使用的马尔可夫链蒙特卡罗算法。在合适的摩擦标度下，基础扩散过程以弹道时间尺度弛豫，表明经适当调优的这些离散化方法，以条件数κ在O(√κ)次迭代内采样目标分布。我们证明，仅基于曲率界和维度的步长与摩擦的任何固定选择都无法实现这种加速：OBABO的总变差混合时间下界表明，弹道冷启动混合在光滑强凸类上一致失效，且该下界以相同阶数扩展至BAOAB及其他四种Strang分裂。该证明将非加速性从优化领域转移到采样领域。消除速度变量得到精确的噪声重球递归，根据Goujaud、Taylor和Dieuleveut的非加速定理，对于每一种调优，要么某个高斯目标的模式弛豫时间至少为κ阶，要么在光滑势上存在吸引周期；将此类势伸缩为U_R(x)=R²U(x/R)可保持其曲率界，并从初始状态（Wasserstein距离平衡态为O(R)）产生步数指数级于R²的亚稳态。利用Leimkuhler、Paulin和Whalley的收缩估计及Wasserstein到总变差的正则化估计，我们证明固定参数OBABO调优的互补上界为O(κ)步（对数因子除外）。因此，在该类的固定参数OBABO调优中，冷启动总变差混合的最优条件数依赖为线性（对数因子除外）。直接高斯计算也排除了左端点指数积分器的固定参数加速性。
英文摘要
The OBABO and BAOAB schemes and the other standard Strang splittings of kinetic (underdamped) Langevin dynamics are widely used Markov chain Monte Carlo algorithms. Under a suitable friction scaling, the underlying diffusion relaxes on a ballistic time scale, suggesting that these discretizations, suitably tuned, sample targets with condition number $κ$ in $O(\sqrtκ)$ iterations. We prove that no fixed choice of step size and friction, based only on the curvature bounds and the dimension, achieves this acceleration: total variation mixing time lower bounds for OBABO show that ballistic cold-start mixing fails uniformly over the smooth strongly convex class, and the lower bounds extend, with the same orders, to BAOAB and the other four Strang splittings. The proof transfers non-acceleration from optimization to sampling. Eliminating velocity gives an exact noisy heavy-ball recursion, and by the non-acceleration theorem of Goujaud, Taylor and Dieuleveut, for every tuning either some Gaussian target has a mode with relaxation time at least of order $κ$, or an attracting cycle exists on a smooth potential; dilating such a potential as $U_R(x)=R^2U(x/R)$ preserves its curvature bounds and produces metastability for a number of steps exponential in $R^2$, from an initial state at Wasserstein distance $O(R)$ from equilibrium. Using contraction estimates of Leimkuhler, Paulin and Whalley and a Wasserstein-to-total-variation regularization estimate, we prove a complementary upper bound of $O(κ)$ steps, up to logarithmic factors, for a fixed-parameter OBABO tuning. Hence, among fixed-parameter OBABO tunings, the optimal condition-number dependence of cold-start total variation mixing over this class is linear, up to logarithmic factors. A direct Gaussian calculation also rules out fixed-parameter acceleration for the left-endpoint exponential integrator.
Comments58 pages, 4 figures. Mixing time lower bounds ruling out ballistic (accelerated) cold-start mixing for all six standard Strang splittings of kinetic Langevin dynamics under fixed-parameter tunings, with a matching O(kappa) upper bound for OBABO
