AI Insight
Researchers have developed a new accelerated Newton method for minimizing convex functions that achieves a global convergence rate of O(1/k³) using only one linear system solve per iteration. This is the first second-order optimization method to reach this convergence rate without requiring auxiliary procedures like cubic regularization, nonlinear parameter searches, or dual corrections. The method works with predetermined parameters, can be implemented in a Hessian-free manner with inexact solvers, and extends to more general settings including arbitrary geometries and composite optimization problems.
Why it matters
This advancement could significantly improve computational efficiency for large-scale optimization problems in machine learning, scientific computing, and engineering applications. By eliminating the need for expensive auxiliary computations while maintaining fast convergence, the method offers a more practical approach to solving complex optimization problems that arise in training neural networks, inverse problems, and other domains requiring iterative minimization.
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⚠️ Preprint – Noch nicht peer-reviewed
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Abstract: We develop a new direct accelerated Newton method for minimizing convex functions with Lipschitz continuous Hessian. The algorithm uses only primal variables and performs just one linear solve per iteration. With a simple predetermined choice of parameters, it achieves the global convergence rate of $O(1/k^3)$ in terms of the functional residual. To the best of our knowledge, this is the first second-order method for this problem class attaining this rate while relying solely on one linear system solve per iteration (without solving auxiliary nonlinear regularized subproblems, such as cubic regularization, performing nonlinear parameter searches, or using dual extragradient corrections). Our method can be implemented in a Hessian-free way, using an inexact linear system solver, while preserving the fast global rate. We further extend our construction to arbitrary geometry through Bregman divergence, and to composite optimization problems.