The Price of Adaptivity in Stochastic Convex Optimization
This work addresses the fundamental limits of adaptivity in optimization for researchers and practitioners, establishing that there is no parameter-free lunch, which is incremental as it builds on and tightens prior bounds.
The paper tackles the problem of adaptivity in non-smooth stochastic convex optimization by proving impossibility results, showing that the price of adaptivity is at least logarithmic for expected suboptimality and polynomial when uncertainty exists in both distance and gradient norm, nearly matching existing upper bounds.
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. En route, we also establish tight upper and lower bounds for (known-parameter) high-probability stochastic convex optimization with heavy-tailed and bounded noise, respectively.