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 metric entropy-free sample complexity bounds for the SAA under standard SP assumptions} -- in the absence of the uniform Lipschitz condition. For a $d$-dimensional problem, the new results often lead to an $O(d)$-improvement in the complexity rate compared with 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 a factor 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. The results of our numerical experiments 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样本平均近法的应用。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条件下可获得度量熵无关的复杂度率，但该假设对许多重要的随机规划问题设定可能过于严苛。为此，本文在标准随机规划假设下——即不依赖一致Lipschitz条件——提出了SAA的度量熵无关样本复杂度界。对于$d$维问题，新结果相较于现有最优方法通常带来$O(d)$量级的复杂度率改进。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率，从而将SAA与SMD的理论差异也缩小了$O(d)$倍。此外，本文探讨了非Lipschitz情形下SAA仍保持可证明有效性，而SMD的相应结果大多尚未探索，这表明SAA在某些非规则设定中可能具有更好的适用潜力。数值实验结果与我们的理论发现相一致。