We study the composite convex optimization problems with a Quasi-Self-Concordant smooth component. This problem class naturally interpolates between classic Self-Concordant functions and functions with Lipschitz continuous Hessian. Previously, the best complexity bounds for this problem class were associated with trust-region schemes and implementations of a ball-minimization oracle. In this paper, we show that for minimizing Quasi-Self-Concordant functions we can use instead the basic Newton Method with Gradient Regularization. For unconstrained minimization, it only involves a simple matrix inversion operation (solving a linear system) at each step. We prove a fast global linear rate for this algorithm, matching the complexity bound of the trust-region scheme, while our method remains especially simple to implement. Then, we introduce the Dual Newton Method, and based on it, develop the corresponding Accelerated Newton Scheme for this problem class, which further improves the complexity factor of the basic method. As a direct consequence of our results, we establish fast global linear rates of simple variants of the Newton Method applied to several practical problems, including Logistic Regression, Soft Maximum, and Matrix Scaling, without requiring additional assumptions on strong or uniform convexity for the target objective.

翻译：我们研究具有拟自协和光滑分量的复合凸优化问题。该问题类自然地插值了经典自协和函数与具有Lipschitz连续Hessian阵的函数。此前，此类问题的最佳复杂度界与信赖域方案和球最小化预言机的实现相关联。本文表明，对于最小化拟自协和函数，我们可以改用带梯度正则化的基本牛顿法。对于无约束最小化，该方法每一步仅涉及简单的矩阵求逆运算（求解线性系统）。我们证明了该算法的快速全局线性速率，匹配信赖域方案的复杂度界，同时我们的方法尤其易于实现。接着，我们引入对偶牛顿法，并基于此为此问题类开发相应的加速牛顿方案，进一步改善了基本方法的复杂度因子。作为我们结果的直接推论，我们建立了应用于若干实际问题（包括逻辑回归、Soft最大化和矩阵缩放）的牛顿法简单变体的快速全局线性速率，而无需对目标函数施加强凸性或一致凸性的额外假设。