Two-Timescale Linear Stochastic Approximation: Constant Stepsizes Go a Long Way

• URL: http://arxiv.org/abs/2410.13067v1
• Date: Wed, 16 Oct 2024 21:49:27 GMT
• Title: Two-Timescale Linear Stochastic Approximation: Constant Stepsizes Go a Long Way
• Authors: Jeongyeol Kwon, Luke Dotson, Yudong Chen, Qiaomin Xie,
• Abstract summary: We investigate it constant stpesize schemes through the lens of Markov processes. We derive explicit geometric and non-asymptotic convergence rates, as well as the variance and bias introduced by constant stepsizes.
• Score: 12.331596909999764
• License:
• Abstract: Previous studies on two-timescale stochastic approximation (SA) mainly focused on bounding mean-squared errors under diminishing stepsize schemes. In this work, we investigate {\it constant} stpesize schemes through the lens of Markov processes, proving that the iterates of both timescales converge to a unique joint stationary distribution in Wasserstein metric. We derive explicit geometric and non-asymptotic convergence rates, as well as the variance and bias introduced by constant stepsizes in the presence of Markovian noise. Specifically, with two constant stepsizes $\alpha < \beta$, we show that the biases scale linearly with both stepsizes as $\Theta(\alpha)+\Theta(\beta)$ up to higher-order terms, while the variance of the slower iterate (resp., faster iterate) scales only with its own stepsize as $O(\alpha)$ (resp., $O(\beta)$). Unlike previous work, our results require no additional assumptions such as $\beta^2 \ll \alpha$ nor extra dependence on dimensions. These fine-grained characterizations allow tail-averaging and extrapolation techniques to reduce variance and bias, improving mean-squared error bound to $O(\beta^4 + \frac{1}{t})$ for both iterates.

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