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In this paper, we propose a unified two-phase scheme to accelerate anyhigh-order regularized tensor approximation approach on the smooth part of acomposite convex optimization model. The proposed scheme has the advantage ofnot needing to assume any prior knowledge of the Lipschitz constants for thegradient, the Hessian and/or high-order derivatives. This is achieved by tuningthe parameters used in the algorithm \textit{adaptively} in its process ofprogression, which has been successfully incorporated in high-order nonconvexoptimization (CartisGouldToint2018,Birgin-Gardenghi-Martinez-Santos-Toint-2017). In general, we show that theadaptive high-order method has an iteration bound of $O\left( 1 /\epsilon^{1/(p+1)} \right)$ if the first $p$-th order derivative information isused in the approximation, which has the same iteration complexity as in thatof the nonadaptive version in (Baes-2009, Nesterov-2018) where the Lipschitzconstants are assumed to be known and the subproblems are assumed to be solvedexactly. Thus, our results partially address the problem of incorporatingadaptive strategies into the high-order {\it accelerated} methods raised byNesterov in (Nesterov-2018), although our strategies cannot assure theconvexity of the auxiliary problem and such adaptive strategies are alreadypopular in high-order nonconvex optimization (CartisGouldToint2018,Birgin-Gardenghi-Martinez-Santos-Toint-2017). Our numerical experiment resultsshow a clear effect of real acceleration displayed in the adaptive Newton'smethod with cubic regularization on a set of regularized logistic regressioninstances.

A Unified Adaptive Tensor Approximation Scheme to Accelerate Composite Convex Optimization