Oracle Complexity of Second-Order Methods for Finite-Sum Problems

Finite-sum optimization problems are ubiquitous in machine learning, and are commonly solved using first-order methods which rely on gradient computations. Recently, there has been growing interest in \emph{second-order} methods, which rely on both gradients and Hessians. In principle, second-order methods can require much fewer iterations than first-order methods, and hold the promise for more efficient algorithms. Although computing and manipulating Hessians is prohibitive for high-dimensional problems in general, the Hessians of individual functions in finite-sum problems can often be efficiently computed, e.g. because they possess a low-rank structure. Can second-order information indeed be used to solve such problems more efficiently? In this paper, we provide evidence that the answer -- perhaps surprisingly -- is negative, at least in terms of worst-case guarantees. However, we also discuss what additional assumptions and algorithmic approaches might potentially circumvent this negative result.