We design step-size schemes that make stochastic gradient descent (SGD) adaptive to (i) the noise $\sigma^2$ in the stochastic gradients and (ii) problem-dependent constants. When minimizing smooth, strongly-convex functions with condition number $\kappa$, we first prove that $T$ iterations of SGD with Nesterov acceleration and exponentially decreasing step-sizes can achieve a near-optimal $\tilde{O}(\exp(-T/\sqrt{\kappa}) + \sigma^2/T)$ convergence rate. Under a relaxed assumption on the noise, with the same step-size scheme and knowledge of the smoothness, we prove that SGD can achieve an $\tilde{O}(\exp(-T/\kappa) + \sigma^2/T)$ rate. In order to be adaptive to the smoothness, we use a stochastic line-search (SLS) and show (via upper and lower-bounds) that SGD converges at the desired rate, but only to a neighbourhood of the solution. Next, we use SGD with an offline estimate of the smoothness and prove convergence to the minimizer. However, its convergence is slowed down proportional to the estimation error and we prove a lower-bound justifying this slowdown. Compared to other step-size schemes, we empirically demonstrate the effectiveness of exponential step-sizes coupled with a novel variant of SLS.

通过利用指数步长和随机线性搜索等技术，使得随机梯度下降算法适应不同噪声水平和问题相关的常数，可以在强凸函数的条件下，取得与理论最优相近的收敛速度，同时能够有效地处理噪声和数据不凸的情况。

面向噪声自适应、问题自适应（加速）随机梯度下降