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。针对一组需要自适应调整的问题参数，我们定义了“适应性代价”（PoA），该指标大致衡量因参数不确定性导致的次优性倍数增长。当初始距离最优点的距离未知但梯度范数界已知时，我们证明期望次优性的PoA至少为对数阶，中位数次优性的PoA至少为双对数阶。当距离和梯度范数均存在不确定性时，我们证明PoA必须与不确定性水平呈多项式关系。我们的下界几乎匹配现有上界，并证明不存在无参数免费午餐。