A Single-timescale Analysis for Stochastic Approximation with Multiple Coupled Sequences

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 scenarios where a fine-grained analysis can provide a tight performance guarantee for single-timescale multi-sequence 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 the main sequence does not have a strongly monotone increment, we establish the iteration complexity of $\mathcal{O}(\epsilon^{-2})$. We showcase the power of our result by applying it to stochastic bilevel and compositional optimization problems, as well as RL problems, all of which recover the best known or lead to improvements over their existing guarantees.