Near-Optimal Convex Optimization with Lazy Second-Order Oracles

We study the deterministic first-order oracle complexity of finding $ε$-stationary points in smooth nonconvex optimization when the objective satisfies higher-order smoothness assumptions. While the classical $ε^{-2}$ rate is optimal under only Lipschitz gradients, higher-order smoothness leads to accelerated first-order upper bounds, most notably the $ε^{-7/4}$ rate under Lipschitz Hessians and the $ε^{-5/3}$ rate under Lipschitz third derivatives. The matching lower bounds, however, have remained open. We resolve this gap by proving a new dimension-free first-order lower bound for higher-order smooth nonconvex functions, valid for every finite smoothness order. In particular, our construction gives a matching $Ω(ε^{-7/4})$ lower bound in the Hessian-Lipschitz case and a matching $Ω(ε^{-5/3})$ lower bound in the third-order-smooth regime. The hard instance is based on a block-chain mechanism that enforces blockwise oracle revelation while preserving the smoothness structure needed for the scalar hard instance. The lower-bound construction was discovered with the assistance of ChatGPT 5.5 Pro and subsequently verified by the authors.

We prove two lower bounds for the first order oracle complexity of minimizing a $d$-dimensional $1$-Lipschitz convex function over the unit ball with $m$ bits of memory. We first show that any such (possibly randomized) algorithm must make $\tildeΩ(\frac{d^2}{\sqrt{m}})$ oracle queries. For deterministic optimization algorithms, we show that $\tildeΩ(\min\{d^{1.6},\frac{d^{8/3}}{m^{2/3}}\})$ queries are required. For all memory regimes of interest, these improves upon the previous best known lower bounds of $\tildeΩ(\max\{\frac{d^{8/3}}{m^{4/3}},\frac{d^{4/3}}{m^{1/6}}\})$ and $\tildeΩ(\frac{d^{5/3}}{m^{1/3}})$ for randomized and deterministic algorithms respectively. Notably, due to existing upper bounds, our lower bound for deterministic algorithms is the first to show a sharp oracle complexity phase transition around $m\approx d^2$, where a polylogarithmic change in memory leads to a $\mathsf{poly}(d)$ change in the number of required oracle calls. Further, when the suboptimality is polynomially small in $d$, our lower bound randomized algorithms is the first to show that $\tildeΩ(d^2)$ memory is necessary to nearly match the optimal query complexity among algorithms without memory constraints. Previously, such a result was only known for the regime where the suboptimality is quasipolynomially small in $d$.

Recent work has sought to transfer first-order nonconvex optimization guarantees to zeroth-order methods through gradient estimators, typically incurring a factor linear in $d$. \citet{kornowski2024} obtained an $\mathcal{O}(dδ^{-1}ε^{-3})$ oracle complexity for nonsmooth nonconvex optimization and asked whether its dependence on all three parameters is jointly optimal. Although the dimension dependence and the first-order dependence on $(δ,ε)$ are separately known to be unavoidable, a matching zeroth-order lower bound has remained open. We answer this question in the unrestricted full-space setting. For smooth stochastic nonconvex optimization, we establish a scalar-oracle lower bound of $Ω\!\left(dL_0^2\min\{ΔH,L_0^2\}/ε^4\right)$ assuming a smooth, globally lower-bounded population objective and a second-moment bound on sample Lipschitz constants. A smooth reduction then yields a lower bound for Goldstein stationarity, proving the joint minimax optimality of the dependence on $(d,δ,ε)$ under explicit high-dimensional and small-accuracy conditions. Crucially, our results allow algorithms to query arbitrary points in $\mathbb{R}^d$, adaptively select sample-function labels, reuse the same sampled function at any number of locations, and output points without spatial restrictions. Thus, unlike lower bounds based on bounded query regions or fresh evaluation noise, our guarantees apply to a fully adaptive, reusable-sample scalar oracle. We also establish a separate noiseless exact-value lower bound against randomized algorithms, and hence also against deterministic methods. Our proof combines a globally defined sequential nonconvex construction with an exact Gaussian scalar channel, multistage random-field estimates, and an antithetic finite-sample realization that preserves information hardness under adaptive reuse.