We prove impossibility results for adaptivity in non-smooth stochastic convex optimization. Given a set of problem parameters we wish to adapt to, we define a "price of adaptivity" (PoA) that, roughly speaking, measures the multiplicative increase in suboptimality due to uncertainty in these parameters. When the initial distance to the optimum is unknown but a gradient norm bound is known, we show that the PoA is at least logarithmic for expected suboptimality, and double-logarithmic for median suboptimality. When there is uncertainty in both distance and gradient norm, we show that the PoA must be polynomial in the level of uncertainty. Our lower bounds nearly match existing upper bounds, and establish that there is no parameter-free lunch.

翻译：我们证明了非光滑随机凸优化中适应性的不可能性结果。给定一组我们希望适应的问题参数，我们定义了一个“适应性代价”（PoA），大致而言，它衡量了由于这些参数的不确定性而导致的次优性的乘性增长。当到最优解的初始距离未知但梯度范数边界已知时，我们证明对于期望次优性，PoA至少是对数级别的，而对于中位数次优性，则是双对数级别的。当距离和梯度范数都存在不确定性时，我们证明PoA必须是不确定性程度的多项式函数。我们的下界几乎匹配现有的上界，并表明不存在无参数免费的午餐。