On the Oracle Complexity of Higher-Order Smooth Non-Convex Finite-Sum Optimization

• URL: http://arxiv.org/abs/2103.05138v1
• Date: Mon, 8 Mar 2021 23:33:58 GMT
• Title: On the Oracle Complexity of Higher-Order Smooth Non-Convex Finite-Sum Optimization
• Authors: Nicolas Emmenegger, Rasmus Kyng and Ahad N. Zehmakan
• Abstract summary: We prove lower bounds for higher-order methods in smooth non finite-sum optimization. We show that a pth-order regularized method cannot profit from the finitesum objective. We show that a new second-order smoothness assumption can be seen as an analogue to the first-order mean-squared smoothness.
• Score: 1.6973426830397942
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We prove lower bounds for higher-order methods in smooth non-convex finite-sum optimization. Our contribution is threefold: We first show that a deterministic algorithm cannot profit from the finite-sum structure of the objective, and that simulating a pth-order regularized method on the whole function by constructing exact gradient information is optimal up to constant factors. We further show lower bounds for randomized algorithms and compare them with the best known upper bounds. To address some gaps between the bounds, we propose a new second-order smoothness assumption that can be seen as an analogue of the first-order mean-squared smoothness assumption. We prove that it is sufficient to ensure state-of-the-art convergence guarantees, while allowing for a sharper lower bound.

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