This paper studies sample average approximation (SAA) in solving convex or strongly convex stochastic programming (SP) problems. In estimating SAA's sample efficiency, the state-of-the-art sample complexity bounds entail metric entropy terms (such as the logarithm of the feasible region's covering number), which often grow polynomially with problem dimensionality. While it has been shown that metric entropy-free complexity rates are attainable under a uniform Lipschitz condition, such an assumption can be overly critical for many important SP problem settings. In response, this paper presents perhaps the first set of metric entropy-free sample complexity bounds for the SAA under standard SP assumptions -- in the absence of the uniform Lipschitz condition. The new results often lead to an $O(d)$-improvement in the complexity rate than the state-of-the-art. 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 also 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. Our numerical experiment results on SAA for solving a simulated SP problem align with our theoretical findings.

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近似方法。在评估SAA的样本效率时，现有最优的样本复杂度界包含度量熵项（如可行域覆盖数的对数），这些项通常随问题维度多项式增长。尽管已有研究表明，在一致Lipschitz条件下可以获得无度量熵的复杂度率，但该假设对许多重要的随机规划问题设置可能过于严苛。为此，本文提出了一套可能在标准随机规划假设下（无需一致Lipschitz条件）首个无度量熵的SAA样本复杂度界。新结果通常使复杂度率较现有最优水平获得$O(d)$量级的改进。基于新建立的复杂度界，一个重要发现是：作为随机规划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率，从而将SAA与SMD的理论差异也提升了$O(d)$量级。此外，本文探索了非Lipschitz场景，其中SAA保持可证明的有效性，而SMD的相应结果大多尚未得到研究，这表明SAA在某些非规则设置中可能具有更好的适用性。我们在求解模拟随机规划问题时对SAA进行的数值实验结果与理论发现一致。