This paper studies sample average approximation (SAA) in solving convex or strongly convex stochastic programming (SP) problems. Under some common regularity conditions, we show -- perhaps for the first time -- that SAA's sample complexity can be completely free from any quantification of metric entropy (such as the logarithm of the covering number), leading to a significantly more efficient rate with dimensionality $d$ than most existing results. From the newly established complexity bounds, an important revelation is that SAA and the canonical stochastic mirror descent (SMD) method, two mainstream solution approaches to SP, entail almost identical rates of sample efficiency, rectifying a persistent theoretical discrepancy of SAA from SMD by the order of $O(d)$. Furthermore, this paper explores non-Lipschitzian scenarios where SAA maintains provable efficacy but the corresponding results for SMD remain mostly unexplored, indicating the potential of SAA's better applicability in some irregular settings.

翻译：本文研究在求解凸或强凸随机规划问题时使用的样本平均逼近方法。在一些常见的正则性条件下，我们证明了——或许是首次——SAA的样本复杂度可以完全独立于度量熵的任何量化（如覆盖数的对数），从而得到比现有大多数结果在维度$d$上显著更高效的收敛速率。从新建立的复杂度界中，一个重要发现是：作为随机规划的两大主流求解方法，SAA与经典随机镜像下降法具有几乎相同的样本效率速率，从而修正了二者之间长期存在的$O(d)$量级理论差异。此外，本文探讨了非Lipschitz连续的情形，其中SAA仍保持可证明的有效性，而SMD的相应结果大多尚未被探索，这表明SAA在某些非规则设置中可能具有更好的适用性。