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, lifting a 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样本平均逼近方法的应用。在一些常见的正则性条件下，我们首次证明SAA的样本复杂度可以完全独立于度量熵的任何量化指标（如覆盖数的对数），从而获得比现有大多数结果更高效的维度$d$相关收敛速率。新建立的复杂度界限揭示了一个重要发现：作为随机规划问题的两种主流求解方法，SAA与经典随机镜像下降法具有几乎相同的样本效率收敛速率，这将SAA与SMD之间的理论差异缩小了$O(d)$量级。此外，本文还探讨了非Lipschitz连续场景下SAA仍保持可证明有效性的情况，而SMD在相应场景下的理论结果尚待探索，这表明SAA在某些非规则设置中可能具有更好的适用性。