Information-Theoretic Upper Bounds for Deterministic Noise in Zeroth-Order Convex Optimization

Abstract

We study deterministic adversarial noise in zeroth-order convex optimization on Euclidean balls. The maximum admissible level of noise is the largest uniform error in function-value queries for which polynomial-query optimization remains possible. We convert the Risteski-Li information-theoretic obstruction for approximately convex optimization into deterministic noisy-oracle upper bounds on this quantity. The conversion gives the Lipschitz convex MALN upper bound with the Risteski-Li dimension dependence. A localized conic-collar embedding gives the corresponding Lipschitz strongly convex bound. Compact randomized smoothing transfers these constructions to smooth convex objectives, producing the stated fourth-root dimension dependence, and to smooth strongly convex objectives on the associated compatibility window. At the endpoint where the smoothness and strong-convexity constants coincide, the class consists only of shifted quadratics. We prove that this endpoint class admits robust $2n$-query reconstruction at noise level of order $R\sqrt{\mu\varepsilon/n}$. Consequently, for query budgets at least $2n$, no uniform $R$-free smooth strongly convex upper bound of the usual form can extend to the endpoint. The results separate theorem validity ranges from the first-branch regimes in which the class-dependent MALN scale dominates the universal $\varepsilon/n$ branch.