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 $\epsilon$ is on the order of $\left(\frac{\mu_k D^{k-1}}{\lambda}\right)^{\frac{2}{3k+1}}+\log\log\left(\frac{1}{\epsilon}\right)$ (in sufficiently high dimension, and up to log factors independent of $\epsilon$), where $\mu_k$ is the Lipschitz constant of the $k$-th derivative, $D$ is the initial distance to the optimum, and $\lambda$ is the strong convexity parameter.