  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.
