We consider maximization of stochastic monotone continuous submodular functions (CSF) with a diminishing return property. Existing algorithms only guarantee the performance \textit{in expectation}, and do not bound the probability of getting a bad solution. This implies that for a particular run of the algorithms, the solution may be much worse than the provided guarantee in expectation. In this paper, we first empirically verify that this is indeed the case. Then, we provide the first \textit{high-probability} analysis of the existing methods for stochastic CSF maximization, namely PGA, boosted PGA, SCG, and SCG++. Finally, we provide an improved high-probability bound for SCG, under slightly stronger assumptions, with a better convergence rate than that of the expected solution. Through extensive experiments on non-concave quadratic programming (NQP) and optimal budget allocation, we confirm the validity of our bounds and show that even in the worst-case, PGA converges to $OPT/2$, and boosted PGA, SCG, SCG++ converge to $(1 - 1/e)OPT$, but at a slower rate than that of the expected solution.

翻译：我们考虑具有递减回报性质的随机单调连续次模函数（CSF）的最大化问题。现有算法仅保证其性能的\textit{期望值}，并未对获得劣质解的概率给出界。这意味着对于算法的某次特定运行，其解可能远差于期望保证。本文首先通过实验验证了这一现象确实存在。随后，我们首次对现有随机CSF最大化方法（即PGA、增强PGA、SCG和SCG++）进行了\textit{高概率}分析。最后，在稍强假设条件下，我们改进了SCG的高概率界，使其收敛速度快于期望解的收敛速度。通过在非凹二次规划（NQP）和最优预算分配问题上的大量实验，我们确认了所提出界的有效性，并表明即使在最坏情况下，PGA收敛至$OPT/2$，而增强PGA、SCG和SCG++收敛至$(1 - 1/e)OPT$，但收敛速度慢于期望解。