AI 中文总结
本文结合 CaCuTe 与 SESOP 提出 CaCuSESOP 方法，在 Lipschitz Hessian 下实现全局最优速率 O(k^{-2}) 和加速局部线性收敛，无需强凸参数等先验信息。
AI 中文摘要
我们将 CaCuTe 与顺序子空间优化（SESOP）相结合，用于在 Hessian 的 Lipschitz 连续性下进行光滑凸最小化。所得到的方法 CaCuSESOP 在每次迭代中通过固定数量的 Hessian-向量乘积实现了全局速率 $O(k^{-2})$，同时消除了经典 SESOP 的关键限制。特别地，原始目标的精确最小化被替换为低维三次上界模型的最小化，该模型只需以可计算的精度求解。三次正则化参数通过回溯选择。如果最小点处的 Hessian 是正定的，CaCuSESOP 自动达到加速的局部线性速率，其复杂度为 $O(\sqrt{\kappa_\star}\log(1/\varepsilon))$。局部强凸参数、条件数 $\kappa_\star$ 和最优值均未提供给算法。我们给出了进入该区域和调整重启调度的有限界限。最后，我们证明了对于所考虑的固定预算梯度/Hessian-向量乘积线性跨度预言机类，全局 $O(k^{-2})$ 速率是最优的，即使具有动量和无限制内存也是如此。
英文摘要
We combine CaCuTe with sequential subspace optimization (SESOP) for smooth convex minimization under Lipschitz continuity of the Hessian. The resulting method, CaCuSESOP, achieves the global rate $O(k^{-2})$ with a fixed number of Hessian--vector products per iteration, while removing key limitations of classical SESOP. In particular, exact minimization of the original objective is replaced by minimization of a low-dimensional cubic upper model, which needs only be solved to a computable accuracy. The cubic regularization parameter is selected by backtracking. If the Hessian at the minimizer is positive definite, CaCuSESOP automatically attains an accelerated local linear rate with complexity $O(\sqrt{κ_\star}\log(1/\varepsilon))$. Neither the local strong-convexity parameter, the condition number $κ_\star$, nor the optimal value is supplied to the algorithm. We give finite bounds for entering this regime and adapting the restart schedule. Finally, we prove that the global $O(k^{-2})$ rate is optimal for the considered fixed-budget gradient/Hessian--vector-product linear-span oracle class, even with momentum and unrestricted memory.
