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We prove lower bounds for higher-order methods in smooth non-convexfinite-sum optimization. Our contribution is threefold: We first show that adeterministic algorithm cannot profit from the finite-sum structure of theobjective, and that simulating a pth-order regularized method on the wholefunction by constructing exact gradient information is optimal up to constantfactors. We further show lower bounds for randomized algorithms and comparethem 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 ananalogue of the first-order mean-squared smoothness assumption. We prove thatit is sufficient to ensure state-of-the-art convergence guarantees, whileallowing for a sharper lower bound.

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