A Note on Inexact Condition for Cubic Regularized Newton's Method
This is an incremental improvement for researchers in nonconvex optimization, fixing a practical issue in an existing method.
The paper addresses the non-implementable inexactness condition in the cubic-regularized Newton's method by proving the same convergence rate under a new adaptive condition that depends only on the current iterate, using a proof technique that controls function value decrease over total iterations.
This note considers the inexact cubic-regularized Newton's method (CR), which has been shown in \cite{Cartis2011a} to achieve the same order-level convergence rate to a secondary stationary point as the exact CR \citep{Nesterov2006}. However, the inexactness condition in \cite{Cartis2011a} is not implementable due to its dependence on future iterates variable. This note fixes such an issue by proving the same convergence rate for nonconvex optimization under an inexact adaptive condition that depends on only the current iterate. Our proof controls the sufficient decrease of the function value over the total iterations rather than each iteration as used in the previous studies, which can be of independent interest in other contexts.