数据驱动算子分裂方法的有限样本保证：基于鞅不等式

中文摘要 AI 辅助

算子分裂方法是求解优化、控制和博弈论中出现的结构化单调包含问题的一类基础算法。我们考虑随机环境中常见的情形：其中一个组成算子的前向评估要么没有闭式解，要么计算成本高昂，因此使用有限数量的含噪oracle样本进行近似。我们为数据驱动的Davis-Yin分裂算法产生的输出质量建立了无分布假设的有限样本证书。与以往工作不同，我们的分析直接通过鞅不等式控制残差误差，而不是依赖针对特定替代损失的算法稳定性论证，从而首次提供了统计超额误差随样本量可证明消失的先验证书。我们进一步证明，在Davis-Yin分裂算法线性收敛的情况下，我们的界对迭代次数的依赖性从线性增长改善为指数衰减。我们在一个具有不确定资产回报的随机投资组合优化问题上验证了我们的理论结果。

英文摘要

Operator splitting methods are a fundamental class of algorithms for solving structured monotone inclusion problems arising in optimization, control, and game theory. We consider the case, common in stochastic regimes, where the forward evaluation of one of the constituent operators is either unavailable in closed form or computationally expensive to evaluate, and is therefore approximated using a finite number of noisy oracle samples. We establish distribution-free, finite-sample certificates for the quality of the output produced by data-driven Davis-Yin splitting algorithms. Unlike previous works, our analysis directly controls the residual error via martingale inequalities instead of relying on algorithmic stability arguments for a tailored surrogate loss, yielding the first a priori certificates whose statistical excess provably vanishes with the sample size. We further show that, under linear convergence of the Davis-Yin splitting algorithm, the dependence of our bounds on the iteration count improves from linear growth to exponential decay. We validate our theoretical results on a stochastic portfolio optimization problem with uncertain asset returns.

发表机构

• KTH Royal Institute of Technology（皇家理工学院）
• IMT School for Advanced Studies Lucca（卢卡高等研究院）

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