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.

翻译：能量守恒下降（ECD）算法（De Luca & Silverstein, 2022）近期被提出作为一种全局非凸优化方法。与梯度下降不同，适当配置的ECD动力学能够逃离严格局部极小点并收敛至全局最小值，使其对机器学习优化具有吸引力。我们首次对ECD进行理论研究，在本篇首章中聚焦于一维情形。我们形式化定义了具有能量守恒噪声的随机ECD动力学（sECD），以及ECD哈密顿量的量子类比（qECD），为通过哈密顿量模拟实现量子算法奠定基础。针对正双阱目标函数，我们计算了从局部到全局最小值的期望穿越时间。我们证明，sECD和qECD均相较于各自的梯度下降基线（随机梯度下降及其量子化版本）实现指数级加速。对于具有高势垒的目标函数，qECD相较于sECD获得进一步加速。