High-accuracy, randomized, and unrestricted-query exact-value complexity

Determine the deterministic exact-value minimax complexity of smooth convex optimization in the high-accuracy regime beyond the cubic-saturation threshold, and establish corresponding complexity results for randomized adaptive algorithms and for algorithms permitted to make queries at unrestricted locations, extending beyond the bounded-query deterministic setting analyzed in the paper.

Background

The paper proves matching upper and lower bounds of order d√(βR²/ε) for deterministic adaptive algorithms that receive exact scalar function values and whose queries and outputs are confined to B₂ᵈ(R), but its lower bound saturates at order d^4/3/log^1/3(ed) when the target accuracy becomes very high. The saturation results from the construction’s geometric condition m³ log(ed) ≲ d, so the paper does not characterize the minimax complexity at smaller ε beyond this threshold.

The unresolved scope also includes two changes to the algorithmic model: randomized adaptive policies and unrestricted query locations. The lower-bound construction is tailored to a realized transcript of one deterministic policy and relies on bounded query norms for spherical-cap avoidance; consequently, it does not provide a randomized lower bound or an unrestricted-query lower bound.