带有内生梯度记忆锚点与恢复阻尼的加速梯度流：选择与恢复阻尼
Accelerated Gradient Flow with Endogenous Gradient-Memory Anchor: Selection and Restoring Damping
中文总结 AI 辅助
该研究提出带内生梯度记忆锚点与非线性恢复-阻尼反馈的加速梯度流，仅用一阶信息，可加速目标值衰减、收敛到极小化子，数值实验验证其机制。
中文摘要 AI 辅助
我们提出了一种带有内生梯度记忆锚点和非线性恢复-阻尼反馈的加速梯度流方法。该锚点由梯度的加权历史演化而来，而非线性反馈则由各坐标相对于锚点的位移平方进行调制。所得到的动力学仅使用目标函数的一阶信息，不涉及显式的海森（Hessian）信息。在适当假设下，我们证明了强解的全局存在性与唯一性。通过李雅普诺夫（Lyapunov）分析得到加速目标值估计：$F(x(t))-F^\bigstar=\bigO(t^{-2})$，其中非线性恢复-阻尼项提供了额外的耗散。在类似假设下，原始轨迹强收敛到一个极小化子。当解集$S$为仿射集时，极限被明确识别为初始锚点$z_0$到$S$的欧氏投影$P_S(z_0)$。特别地，对于秩亏最小二乘问题，该动力学选择最接近$z_0$的最小二乘解。我们还建立了相位变量的有限加权耗散估计。此外，在非线性反馈起作用的坐标中，若轨迹在某区间内与锚点保持分离，则对应的齐次相位动力学将获得额外的多项式衰减因子。数值实验验证了极小化子选择、加速目标值衰减及恢复-阻尼机制。
英文摘要
We introduce an accelerated gradient flow with an endogenous gradient-memory anchor and a nonlinear restoring--damping feedback. The anchor evolves from a weighted history of the gradients, while the nonlinear feedback is modulated by the squared coordinatewise displacement from the anchor. The resulting dynamics uses only first-order information from the objective function and does not involve explicit Hessian information. Under suitable assumptions, we establish global existence and uniqueness of strong solutions. A Lyapunov analysis yields the accelerated objective-value estimate $F(x(t))-F^\star=\mathcal{O}(t^{-2})$, with the nonlinear restoring--damping term contributing additional dissipation. Under similar assumptions, the primal trajectory converges strongly to a minimizer. When, in addition, the solution set $S$ is affine, the limit is identified explicitly as the Euclidean projection $P_S(z_0)$ of the initial anchor $z_0$ onto $S$. In particular, for rank-deficient least-squares problems, the dynamics selects the least-squares solution closest to $z_0$. We also establish a finite weighted dissipation estimate for the phase variable. Moreover, in coordinates where the nonlinear feedback is active, if the trajectory remains separated from the anchor over an interval, then the corresponding homogeneous phase dynamics acquires an additional polynomial decay factor. Numerical experiments illustrate the minimizer-selection, accelerated objective-value decay, and restoring--damping mechanisms.
