Single-Timescale Multi-Sequence Stochastic Approximation Without Fixed Point Smoothness: Theories and Applications
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Stochastic approximation (SA) that involves multiple coupled sequences, 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. This paper establishes 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. In addition, when all involved operators are strongly monotone except for the main one, MSSA converges at a rate of \(\mathcal{O}(K^{-\frac{1}{2}})\). These theoretical findings align with those established for single-sequence SA. Applying these 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.