AI 中文总结
针对超线性增长随机梯度的采样和优化问题，提出tSGHMC算法，在特定条件下建立非渐近误差界，推导预期超额风险上界，通过实际例子验证算法有效性，并与一阶算法比较，显示其在多任务中有更低误差和风险。
AI 中文摘要
在本文中，我们针对具有超线性增长随机梯度的采样和随机优化问题，提出了一种新颖的驯服随机梯度哈密顿蒙特卡罗（tSGHMC）算法。在一定的平均连续性条件和强凸性条件下，我们为tSGHMC建立了Wasserstein-2距离下的非渐近误差界，收敛速率为1/4。然后，我们推导了相关预期超额风险的上界估计，为tSGHMC的性能提供了理论保证。为说明该算法的有效性，我们将tSGHMC应用于实际例子，包括报童问题和条件风险价值最小化问题，使用合成和真实世界数据集。数值结果支持我们的理论发现。此外，我们将tSGHMC与其一阶对应算法，即驯服的未调整随机朗之万算法进行比较。仿真结果表明，tSGHMC在一系列任务中实现了更低的均方根误差和预期超额风险。
英文摘要
In this paper, we propose a novel tamed stochastic gradient Hamiltonian Monte Carlo (tSGHMC) algorithm for sampling and stochastic optimization problems with superlinearly growing stochastic gradients. Under a certain continuity in average condition and a strong convexity condition, we establish a non-asymptotic error bound in Wasserstein-2 distance for tSGHMC with the rate of convergence equal to $1/4$. Then, we derive an upper estimate for the associated expected excess risk, which provides a theoretical guarantee for the performance of tSGHMC. To illustrate the effectiveness of the proposed algorithm, we apply tSGHMC to practical examples, including a newsvendor problem and a Conditional Value-at-Risk minimization problem, using synthetic and real-world datasets. Numerical results support our theoretical findings. Furthermore, we compare tSGHMC with its first-order counterpart, namely, the tamed unadjusted stochastic Langevin algorithm. Simulation results demonstrate that tSGHMC achieves lower root mean square error and expected excess risk across a range of tasks.
