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Second-order optimization methods, such as cubic regularized Newton methods,are known for their rapid convergence rates; nevertheless, they becomeimpractical in high-dimensional problems due to their substantial memoryrequirements and computational costs. One promising approach is to executesecond-order updates within a lower-dimensional subspace, giving rise tosubspace second-order methods. However, the majority of existing subspacesecond-order methods randomly select subspaces, consequently resulting inslower convergence rates depending on the problem's dimension $d$. In thispaper, we introduce a novel subspace cubic regularized Newton method thatachieves a dimension-independent global convergence rate of${O}\left(\frac{1}{mk}+\frac{1}{k^2}\right)$ for solving convex optimizationproblems. Here, $m$ represents the subspace dimension, which can besignificantly smaller than $d$. Instead of adopting a random subspace, ourprimary innovation involves performing the cubic regularized Newton updatewithin the Krylov subspace associated with the Hessian and the gradient of theobjective function. This result marks the first instance of adimension-independent convergence rate for a subspace second-order method.Furthermore, when specific spectral conditions of the Hessian are met, ourmethod recovers the convergence rate of a full-dimensional cubic regularizedNewton method. Numerical experiments show our method converges faster thanexisting random subspace methods, especially for high-dimensional problems.

Krylov Cubic Regularized Newton: A Subspace Second-Order Method with Dimension-Free Convergence Rate