Single-Timescale Multi-Sequence Stochastic Approximation Without Fixed Point Smoothness: Theories and Applications

Stochastic approximation (SA) that involves multiple coupled sequences, also known as multiple-sequence SA (MSSA), finds diverse applications in the fields of signal processing and machine learning. However, existing theoretical understandings of MSSA are limited: the multi-timescale analysis implies a slow convergence rate, whereas the single-timescale analysis relies on a stringent fixed point smoothness assumption. In this paper, we establish tighter single-timescale analysis for MSSA, without assuming smoothness of the fixed points. Our theoretical findings reveal that, when all involved operators are strongly monotone, MSSA converges at a rate of $\tilde{\mathcal{O}}(K^{-1})$, where $K$ denotes the total number of iterations. When all involved operators are strongly monotone except for the main one, MSSA converges at a rate of $O(K^{-\frac{1}{2}})$. These rates align with the ones established for single-sequence SA. Applying our novel theoretical findings to bilevel optimization and communication-efficient distributed learning offers relaxed assumptions and/or simpler algorithms with performance guarantees, as validated by numerical experiments.