In the machine learning and optimization community, there are two main approaches for the convex risk minimization problem, namely the Stochastic Approximation (SA) and the Sample Average Approximation (SAA). In terms of the oracle complexity (required number of stochastic gradient evaluations), both approaches are considered equivalent on average (up to a logarithmic factor). The total complexity depends on a specific problem, however, starting from the work [A. Nemirovski, A. Juditsky, G. Lan, and A. Shapiro, Robust stochastic approximation approach to stochastic programming, SIAM. J. Opt. 19 (2009), pp. 1574–1609] it was generally accepted that the SA is better than the SAA. We show that for the Wasserstein barycenter problem, this superiority can be inverted. We provide a detailed comparison by stating the complexity bounds for the SA and SAA implementations calculating barycenters defined with respect to optimal transport distances and entropy-regularized optimal transport distances. As a byproduct, we also construct confidence intervals for the barycenter defined with respect to entropy-regularized optimal transport distances in the ${\ell }_2$-norm. The preliminary results are derived for a general convex optimization problem given by the expectation to have other applications besides the Wasserstein barycenter problem.

在机器学习和优化社区中，凸风险最小化问题有两种主要方法，即随机近似（SA）和样本平均近似（SAA）。就预言机复杂性（所需的随机梯度评估数量）而言，两种方法平均而言被认为是等效的（最多为对数因子）。然而，总的复杂性取决于一个特定的问题，从工作开始 [A. Nemirovski、A. Juditsky、G. Lan 和 A. Shapiro，随机规划的稳健随机近似方法，暹罗。J. 选择。19 (2009), pp. 1574–1609] 人们普遍认为 SA 优于 SAA。我们表明，对于 Wasserstein 重心问题，这种优势可以颠倒。我们通过说明 SA 和 SAA 实现计算重心的复杂性界限来提供详细的比较，这些重心是根据最佳传输距离和熵正则化最佳传输距离定义的。作为副产品，我们还为根据熵正则化最佳传输距离定义的重心构建置信区间${\ell }_2$-规范。初步结果是针对一般凸优化问题推导出来的，该问题预期除 Wasserstein 重心问题外还有其他应用。