Stochastic approximation (SA) with multiple coupled sequences has found broad applications in machine learning such as bilevel learning and reinforcement learning (RL). In this paper, we study the finite-time convergence of nonlinear SA with multiple coupled sequences. Different from existing multi-timescale analysis, we seek for scenarios where a fine-grained analysis can provide the tight performance guarantee for multi-sequence single-timescale SA (STSA). At the heart of our analysis is the smoothness property of the fixed points in multi-sequence SA that holds in many applications. When all sequences have strongly monotone increments, we establish the iteration complexity of $\mathcal{O}(\epsilon^{-1})$ to achieve $\epsilon$-accuracy, which improves the existing $\mathcal{O}(\epsilon^{-1.5})$ complexity for two coupled sequences. When all but the main sequence have strongly monotone increments, we establish the iteration complexity of $\mathcal{O}(\epsilon^{-2})$. The merit of our results lies in that applying them to stochastic bilevel and compositional optimization problems, as well as RL problems leads to either relaxed assumptions or improvements over their existing performance guarantees.

翻译：具有多个相加序列的沙粒近似(SA) 在双级学习和强化学习(RL) 等机器学习中发现了广泛的应用。 在本文中, 我们研究了非线性SA与多个相加序列的有限时间趋同。 与现有的多时间尺度分析不同, 我们寻求的情景是, 细微分析能够为多个序列的单时标SA( STSA) 提供严格的性能保障。 我们分析的核心是, 多序列中固定点的平稳性能属性, 在许多应用程序中都存在多序列 SA 。 当所有序列都有强烈的单质递增时, 我们建立 $\ mathcal{ O} (\\ silon ⁇ -1}) 的循环复杂性, 以达到 $\ epslon$- 准确性能, 这可以改善现有的 $macal{ O} (\ epslon ⁇ - -15} 。 当所有主要序列都有强烈的单质递增时, 我们确定 $\ mathcal{ O} (epslon_ 2} (regilate) rogide mabilate magistress max mailde lade max orticilde ortical max max ortical max max max max max max ortide orticilticiltialtial ortialtial ortialtialtial ortialtial orticild ortial orticild subild subild subild subild subild subild subild subild subil subild sub mas mas mas mas mas mas mas mas subild mas mas subild subild mas mas) mas mas mas mas mas sub subild su mas mas mas mas ma