Complexity Lower Bounds of Adaptive Gradient Algorithms for Non-convex Stochastic Optimization under Relaxed Smoothness

Abstract

Recent results in non-convex stochastic optimization demonstrate the convergence of popular adaptive algorithms (e.g., AdaGrad) under the $(L_0, L_1)$-smoothness condition, but the rate of convergence is a higher-order polynomial in terms of problem parameters like the smoothness constants. The complexity guaranteed by such algorithms to find an $\epsilon$-stationary point may be significantly larger than the optimal complexity of $\Theta \left( \Delta L \sigma^2 \epsilon^{-4} \right)$ achieved by SGD in the $L$-smooth setting, where $\Delta$ is the initial optimality gap, $\sigma^2$ is the variance of stochastic gradient. However, it is currently not known whether these higher-order dependencies can be tightened. To answer this question, we investigate complexity lower bounds for several adaptive optimization algorithms in the $(L_0, L_1)$-smooth setting, with a focus on the dependence in terms of problem parameters $\Delta, L_0, L_1$. We provide complexity bounds for three variations of AdaGrad, which show at least a quadratic dependence on problem parameters $\Delta, L_0, L_1$. Notably, we show that the decorrelated variant of AdaGrad-Norm requires at least $\Omega \left( \Delta^2 L_1^2 \sigma^2 \epsilon^{-4} \right)$ stochastic gradient queries to find an $\epsilon$-stationary point. We also provide a lower bound for SGD with a broad class of adaptive stepsizes. Our results show that, for certain adaptive algorithms, the $(L_0, L_1)$-smooth setting is fundamentally more difficult than the standard smooth setting, in terms of the initial optimality gap and the smoothness constants.