o-极小结构中可定义函数梯度法的全局收敛性
Global convergence of the gradient method for functions definable in o-minimal structures
AI总结：
针对实域o-极小结构中可定义且具局部Lipschitz梯度的函数，研究变步长梯度法的全局收敛性，证明轨迹有界时的收敛性、步长大于正常数时的梯度范数衰减速率，及梯度连续可微等附加条件下步长恒定时的几乎必然局部极小收敛性。
摘要：
我们考虑变步长梯度法，用于最小化实域上o-极小结构中可定义且具有局部Lipschitz梯度的可微函数。我们证明：若连续梯度轨迹有界，则全局收敛成立；若步长大于某正常数，最小梯度范数以o(1/k)的速率消失。若梯度额外满足连续可微、所有鞍点为严格鞍点且步长为常数，则对任意有界初始点集，几乎必然收敛到局部极小点。
英文摘要：
We consider the gradient method with variable step size for minimizing functions that are definable in o-minimal structures on the real field and differentiable with locally Lipschitz gradients. We prove that global convergence holds if continuous gradient trajectories are bounded, with the minimum gradient norm vanishing at the rate $o(1/k)$ if the step sizes are greater than a positive constant. If additionally the gradient is continuously differentiable, all saddle points are strict, and the step sizes are constant, then convergence to a local minimum holds almost surely over any bounded set of initial points.
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