  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.
