Lower Bounds for Higher-Order Convex Optimization

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Naman Agarwal, Elad Hazan ;
Proceedings of the 31st Conference On Learning Theory, PMLR 75:774-792, 2018.

Abstract

State-of-the-art methods in mathematical optimization employ higher-order derivative information. We explore the limitations of higher-order optimization and prove that even for convex optimization, a polynomial dependence on the approximation guarantee and higher-order smoothness parameters is necessary. This refutes the hope that higher-order smoothness and higher-order derivatives can lead to dimension free polynomial time algorithms for convex optimization. As a special case, we show Nesterov’s accelerated cubic regularization method and higher-order methods to be nearly tight.

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