On Global Joint Lower Bounds in Zeroth-Order Nonconvex Optimization
This paper claims the first global joint lower bound for zeroth-order nonconvex optimization under unrestricted spatial queries and sample reuse. The core idea is to force any algorithm to advance through $T\asymp\min\{\Delta H,L_0^2\}/\epsilon^2$ hidden geometric stages and to prove that each fresh stage requires $\Omega(dL_0^2/\epsilon^2)$ adaptive scalar evaluations, yielding an overall $\Omega(dL_0^2\min\{\Delta H,L_0^2\}/\epsilon^4)$ bound. If correct, this would settle the open problem posed by Kornowski and Shamir (2024), though the manuscript is dated 14 October 2026, cites several future works, and explicitly states that its mathematics was primarily AI-generated.
The manuscript is dated 14 October 2026 and cites multiple 2026 works, which is chronologically impossible for a genuine submission. Combined with the authors' admission that the results were AI-generated, the submission reads as a forward-dated, machine-produced preprint rather than verified mathematics. Conceptually, the proof strategy—global zero-chain plus exact Gaussian tensor channel—is elegant, but the appendices containing all proofs are omitted, and the capped gap dependence $B=\min\{\Delta H,L_0^2\}$ together with the restrictive dimension condition $d\geq C(B/\epsilon^2)\log(2+BL_0^2/\epsilon^4)$ mean the claim is far from unconditional.
The high-level proof architecture is soundly motivated. The authors correctly observe that the desired cost multiplication must be a constructed target, not a product of separate bounds, and they build a genuine global progress certificate so that every approximate stationary point completes all hidden stages regardless of query location. The square-root tensor representation and antithetic finite-sample realization provide principled tools for handling adaptive sample reuse within a fixed oracle family.
The paper explicitly states that its mathematics was AI-generated and released in an admittedly incomplete form, which makes its lengthy and delicate proof untrustworthy without independent human checking. Beyond provenance, the construction relies on extreme constants such as $C_N=2^{128}$, the dimension threshold is restrictive, and the gap dependence is capped at $B=\min\{\Delta H,L_0^2\}$, which the authors acknowledge is a limitation of their construction rather than a minimax truth.
Comparisons to related work are generally fair when they can be checked. The authors carefully distinguish their unrestricted model from Zhang et al. (2026a)'s ball-restricted setting and note that the deterministic exact-value lower bound of Ding and Yuan (2026) applies to a weaker local oracle. However, every substantive argument is deferred to appendices omitted from the supplied text, so the reader has no way to verify the metric-entropy bounds, deferred-revelation coupling, or final parameter scalings.
No code, data, or concrete implementation is provided. The oracle instance is defined by a web of coordinate transformations, cutoffs, and infinite-series tensor fields whose constants are chosen ad hoc and enormous. The authors promise that Lean formalization may be released later, but with no timeline and an explicit AI-generation disclaimer, independent reproduction of the claimed bounds is effectively blocked.
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 $\mathcal{O}(d\delta^{-1}\epsilon^{-3})$ 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 $(\delta,\epsilon)$ 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 $\Omega\!\left(dL_0^2\min\{\Delta H,L_0^2\}/\epsilon^4\right)$ 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 $(d,\delta,\epsilon)$ 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.
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