AI 中文总结
该研究证明了带一致有界梯度噪声的光滑非凸随机优化的紧下界，匹配现有上界并解决了相关公开问题，且证明由AI生成、人类仅负责校验与文稿润色。
AI 中文摘要
我们证明了带一致有界梯度噪声的光滑非凸随机优化问题的一个精确下界。在K=1的新鲜样本模型中，所有随机自适应算法都需要Ω(ΔL/ε² + ΔLσ²/ε⁴)次查询，才能找到期望梯度范数至多为ε的点。该结果与标准上界相匹配，且据我们所知，解决了Arjevani等人2023年提出的问题：几乎确定有界的预言机误差是否能实现比有界方差更优的收敛速率。该证明是在两小时的会话中，通过Codex的Ultra模式下的GPT-5.6 Sol独立生成的。人类作者提供了提示词，仅负责检查证明以及修改润色手稿。
英文摘要
We prove a sharp lower bound for smooth nonconvex stochastic optimization with uniformly bounded gradient noise. In the \(K=1\) fresh-sample model, every randomized adaptive algorithm requires $$Ω\left( \frac{ΔL}{ε^2} + \frac{ΔLσ^2}{ε^4} \right)$$ queries to find a point with expected gradient norm at most \(ε\). This matches the standard upper bound and, to the best of our knowledge, resolves the question raised by [Arjevani et al. 2023] of whether almost-surely bounded oracle error permits a better rate than bounded variance. The proof was independently generated with GPT-5.6 Sol in Codex's Ultra mode during a two-hour session. The human author supplied the prompt and was responsible only forchecking the proof and revising and polishing the manuscript.
