AI 中文总结
研究非凸优化的无参数三次正则化牛顿法，在广义光滑性条件下推导找到$(\varepsilon, \delta)$ - 二阶驻点的神谕复杂度界，该条件更弱且复杂度界优于现有无参数二阶方法，特定情况与最优依赖匹配。
AI 中文摘要
我们分析了用于非凸优化的三次正则化牛顿法的一种变体。该变体无参数，即无需问题相关参数的先验知识。在广义光滑性条件$\|\nabla^3 f(x)\| \leq L_0 + L_1 \|\nabla f(x)\|$下，我们推导了找到$(\varepsilon, \delta)$ - 二阶驻点的神谕复杂度界。此假设比现有二阶方法分析中使用的广义光滑性条件更弱，且复杂度界改进了无参数二阶方法的现有保证。特别地，当$L_1 = zero$时，该界与对$L_0$以及$\varepsilon$、$\delta$和初始函数值差距的最优依赖相匹配，至多相差加性对数项。为建立此界，我们推导了泰勒型不等式并证明其与广义光滑性条件等价。
英文摘要
We analyze a variant of the cubic-regularized Newton method for nonconvex optimization. This variant is parameter-free in that it requires no prior knowledge of problem-dependent parameters. Under the generalized smoothness condition $\|\nabla^3 f(x)\| \leq L_0 + L_1 \|\nabla f(x)\|$, we derive an oracle complexity bound for finding an $(\varepsilon, δ)$-second-order stationary point. This assumption is weaker than the generalized smoothness conditions used in existing analyses of second-order methods, while the complexity bound improves upon existing guarantees for parameter-free second-order methods. In particular, when $L_1 = 0$, the bound matches the optimal dependence on $L_0$ as well as on $\varepsilon$, $δ$, and the initial function value gap, up to additive logarithmic terms. To establish this bound, we derive Taylor-type inequalities and prove their equivalence to the generalized smoothness condition.
Comments24 pages, 1 figure
