High-Order Oracle Complexity of Smooth and Strongly Convex Optimization

In this note, we consider the complexity of optimizing a highly smooth (Lipschitz k-th order derivative) and strongly convex function, via calls to a k-th order oracle which returns the value and first k derivatives of the function at a given point, and where the dimension is unrestricted. Extending the techniques introduced in Arjevani et al. [2019], we prove that the worst-case oracle complexity for any fixed k to optimize the function up to accuracy ε is on the order of left(fracμk Dk-1λright)^frac23k+1+loglogleft(frac1εright) (in sufficiently high dimension, and up to log factors independent of ε), where μk is the Lipschitz constant of the k-th derivative, D is the initial distance to the optimum, and λ is the strong convexity parameter.