Making Gradient Descent Optimal for Strongly Convex Stochastic Optimization
Date of this Version
Proceedings of the 29th International Conference on Machine Learning
Stochastic gradient descent (SGD) is a simple and popular method to solve stochastic optimization problems which arise in machine learning. For strongly convex problems, its convergence rate was known to be O(log(T/T ), by running SGD for T iterations and returning the average point. How- ever, recent results showed that using a different algorithm, one can get an optimal O(1/T ) rate. This might lead one to believe that standard SGD is suboptimal, and maybe should even be replaced as a method of choice. In this paper, we investigate the optimality of SGD in a stochastic setting. We show that for smooth problems, the algorithm attains the optimal O(1/T) rate. However, for non-smooth problems, the convergence rate with averaging might really be Ω (log(T)/T ), and this is not just an artifact of the analysis. On the flip side, we show that a simple modification of the averaging step success to recover the O(1/T ) rate, and no other change of the algorithm is necessary. We also present experimental results which support our findings, and point out open problems.
Rakhlin, A., Shamir, O., & Sridharan, K. (2012). Making Gradient Descent Optimal for Strongly Convex Stochastic Optimization. Proceedings of the 29th International Conference on Machine Learning, Retrieved from https://repository.upenn.edu/statistics_papers/102
Date Posted: 27 November 2017