Original title: Closing the Oracle-Complexity Gap in Derivative-Free Convex Optimization: A Near-Quadratic Lower Bound from Exact Function Values
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An optimization researcher reports that GPT-5.6 Sol Pro generated a proof, in 148 minutes, showing that deterministic zeroth-order convex optimization over the unit ball needs Ω(d²) exact function-value queries for ε-optimality, matching the 1996 Protasov upper bound of O(d²) and removing a decades-long linear gap in complexity assumptions. The result is formalized in Lean after the model returned a candidate proof, then was manually checked by the author, who had previously explored the problem for about a year and built a ten-page prompt modeled on OpenAI’s CDC proof approach. The model was asked to prove a near-quadratic lower bound at tighter accuracy, and the final argument reached order d⁻³ precision while still aligning with the original target. The author emphasizes that the core ingredients remain classical—an adversarial oracle construction and known convex-geometry-style reasoning—so the novelty lies more in rapid discovery and integration than in new theory. In discussion, commenters largely agree the theorem is real and niche but meaningful, with one noting the bound appears tight versus existing algorithms and another flagging that the claim is not yet peer reviewed. The thread also discusses that the gradient-oracle comparison is suggestive but not fully settled, and that model-level differences (Sol Pro vs Ultra) and the heavy prompting requirements matter for interpreting capability claims. Practical and philosophical reactions focus on what this means for researchers: if established techniques are AI-accessible, then human effort may shift toward choosing hard problems and framing them effectively rather than routine theorem hunting.
Commenters describe the result as specialized yet substantive, praising the Ω(d²) bound for derivative-free optimization while noting that nontrivial lower bounds are typically far harder than upper-bound runtime proofs. Some participants argue the theorem may imply practical near-tightness for gradient settings by arguing gradients can be approximated, but they treat that implication as a separate technical step. A recurring concern is rigor and reproducibility, with users pointing out that the work is not peer reviewed and that it relied on substantial prior domain setup, including the long prompt and prior research context, rather than spontaneous model discovery. The thread also interrogates AI-system labeling, asking for substantiating evidence of what Sol Pro actually differs from Ultra and whether multi-agent orchestration explains output quality. Broader reaction splits between optimism about acceleration in math research and anxiety about social impact, including fears of overhyped promises, narrowing job roles to direction-setting, and whether future breakthroughs may become easier but less human-creative. Others counter with cautious skepticism toward “no human required” claims, stressing that domain expertise, question formulation, and judgment remain central and that this does not yet settle harder, less structured problems.