提出能量守恒下降法，实现非凸优化的量子与经典加速。
Classical and Quantum Speedups for Non-Convex Optimization via Energy Conserving Descent
- 设计能量守恒噪声动态与量子哈密顿量模拟
- 在双阱目标函数下，期望击中时间呈指数级缩短
- 量子版本比经典版本更快，尤其在高屏障场景
能量守恒下降（ECD）算法被提出作为全局非凸优化方法。与梯度下降不同，适当配置的ECD能逃离严格局部极小值并收敛到全局最小值，适用于机器学习优化。本文首次对ECD进行理论分析，聚焦一维情形。我们形式化了带能量守恒噪声的随机ECD（sECD），以及ECD哈密顿量的量子版本（qECD），为量子算法提供基础。针对正双阱目标函数，计算从局部极小到全局极小的期望击中时间。证明sECD和qECD均对各自基线（随机梯度下降及其量子化）实现指数加速；在高势垒情况下，qECD进一步优于sECD。
原文摘要 · Abstract (English)
The Energy Conserving Descent (ECD) algorithm was recently proposed (De Luca & Silverstein, 2022) as a global non-convex optimization method. Unlike gradient descent, appropriately configured ECD dynamics escape strict local minima and converge to a global minimum, making it appealing for machine learning optimization. We present the first analytical study of ECD, focusing on the one-dimensional setting for this first installment. We formalize a stochastic ECD dynamics (sECD) with energy-preserving noise, as well as a quantum analog of the ECD Hamiltonian (qECD), providing the foundation for a quantum algorithm through Hamiltonian simulation. For positive double-well objectives, we compute the expected hitting time from a local to the global minimum. We prove that both sECD and qECD yield exponential speedup over respective gradient descent baselines--stochastic gradient descent and its quantization. For objectives with tall barriers, qECD achieves a further speedup over sECD.
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