Newton-type methods enjoy fast local convergence and strong empirical performance, but achieving global guarantees comparable to first-order methods remains challenging. Even for simple strongly convex problems, no straightforward variant of Newton's method matches the global complexity of gradient descent. While more sophisticated variants can improve iteration complexity, they typically require solving difficult subproblems with high per-iteration costs, leading to worse overall complexity. These limitations stem from treating the subproblem as an afterthought, either as a black box, yielding overly complex and impractical formulations, or in isolation, without regard to its role in advancing the optimization of the main objective. By tightening the integration between the inner iterations of the subproblem solvers and the outer iterations of the optimization algorithm, we introduce simple Newton-type variants, called Faithful-Newton framework, which, in a sense, remain faithful to the overall simplicity of classical Newton's method by retaining simple linear system subproblems. The key conceptual difference, however, is that the quality of the subproblem solution is directly assessed based on its effectiveness in reducing optimality, which in turn enables desirable convergence complexities across a variety of settings. Under standard assumptions, we show that our variants, depending on parameter choices, achieve global superlinear convergence, condition-number-independent linear convergence, and/or local quadratic convergence, even when using inexact Newton steps, for strongly convex problems; and competitive iteration complexity for general convex problems. Numerical experiments further demonstrate that our proposed methods perform competitively compared with several alternative Newton-type approaches.

翻译：牛顿型方法具有快速的局部收敛性和强大的实证性能，但实现与一阶方法相媲美的全局保证仍然具有挑战性。即使对于简单的强凸问题，也没有任何牛顿方法的直接变体能匹配梯度下降的全局复杂度。虽然更复杂的变体可以改进迭代复杂度，但它们通常需要求解具有高单次迭代成本的困难子问题，从而导致更差的总体复杂度。这些局限性源于将子问题视为事后考虑：要么作为黑箱处理，产生过于复杂且不实用的公式；要么孤立处理，无视其在推进主目标优化中的作用。通过加强子问题求解器内部迭代与优化算法外部迭代之间的整合，我们引入了一类简单的牛顿型变体，称为忠实牛顿框架。该框架在某种意义上通过保留简单的线性系统子问题，保持了经典牛顿方法的整体简洁性。然而，关键的概念差异在于，子问题解的质量直接根据其降低最优性的有效性进行评估，这反过来使得在各种设置下实现理想的收敛复杂度成为可能。在标准假设下，我们证明，对于强凸问题，我们的变体根据参数选择可实现全局超线性收敛、条件数无关的线性收敛和/或局部二次收敛（即使使用不精确牛顿步）；对于一般凸问题，则具有竞争力的迭代复杂度。数值实验进一步表明，与几种替代的牛顿型方法相比，我们提出的方法具有竞争性的性能表现。