We present an information-theoretic approach to lower bound the oracle complexity of nonsmooth black box convex optimization, unifying previous lower bounding techniques by identifying a combinatorial problem, namely string guessing, as a single source of hardness. As a measure of complexity we use distributional oracle complexity, which subsumes randomized oracle complexity as well as worst-case oracle complexity. We obtain strong lower bounds on distributional oracle complexity for the box $[-1,1]^n$, as well as for the $L^p$-ball for $p \geq 1$ (for both low-scale and large-scale regimes), matching worst-case upper bounds, and hence we close the gap between distributional complexity, and in particular, randomized complexity, and worst-case complexity. Furthermore, the bounds remain essentially the same for high-probability and bounded-error oracle complexity, and even for combination of the two, i.e., bounded-error high-probability oracle complexity. This considerably extends the applicability of known bounds.

翻译：本文提出了一种基于信息论的方法，用于为非光滑黑盒凸优化的预言复杂度建立下界。通过识别一个组合问题（即字符串猜测）作为单一困难源，统一了先前的下界推导技术。我们采用分布预言复杂度作为度量标准，该标准涵盖了随机预言复杂度以及最坏情况预言复杂度。针对盒形区域 $[-1,1]^n$ 以及 $L^p$ 球（$p \geq 1$，包括低尺度和大尺度情形），我们获得了分布预言复杂度的强下界，这些下界与最坏情况上界相匹配，从而填补了分布复杂度（特别是随机复杂度）与最坏情况复杂度之间的空白。此外，这些下界对于高概率和有限误差预言复杂度，甚至两者结合（即有限误差高概率预言复杂度）仍基本保持不变。这显著扩展了已知下界的适用范围。