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Oracle Complexity of Second-Order Methods for Finite-Sum Problems
Yossi Arjevani · Ohad Shamir

Mon Aug 07 01:30 AM -- 05:00 AM (PDT) @ Gallery #11

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.

Author Information

Yossi Arjevani (Weizmann Institute of Science)
Ohad Shamir (Weizmann Institute of Science)

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