High-Order Error Bounds for Markovian LSA with Richardson-Romberg Extrapolation
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909728235323392 |
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| author | Levin, Ilya Naumov, Alexey Samsonov, Sergey |
| author_facet | Levin, Ilya Naumov, Alexey Samsonov, Sergey |
| contents | In this paper, we study the bias and high-order error bounds of the Linear Stochastic Approximation (LSA) algorithm with Polyak-Ruppert (PR) averaging under Markovian noise. We focus on the version of the algorithm with constant step size $α$ and propose a novel decomposition of the bias via a linearization technique. We analyze the structure of the bias and show that the leading-order term is linear in $α$ and cannot be eliminated by PR averaging. To address this, we apply the Richardson-Romberg (RR) extrapolation procedure, which effectively cancels the leading bias term. We derive high-order moment bounds for the RR iterates and show that the leading error term aligns with the asymptotically optimal covariance matrix of the vanilla averaged LSA iterates. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_05570 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | High-Order Error Bounds for Markovian LSA with Richardson-Romberg Extrapolation Levin, Ilya Naumov, Alexey Samsonov, Sergey Machine Learning Optimization and Control Statistics Theory 62L20 In this paper, we study the bias and high-order error bounds of the Linear Stochastic Approximation (LSA) algorithm with Polyak-Ruppert (PR) averaging under Markovian noise. We focus on the version of the algorithm with constant step size $α$ and propose a novel decomposition of the bias via a linearization technique. We analyze the structure of the bias and show that the leading-order term is linear in $α$ and cannot be eliminated by PR averaging. To address this, we apply the Richardson-Romberg (RR) extrapolation procedure, which effectively cancels the leading bias term. We derive high-order moment bounds for the RR iterates and show that the leading error term aligns with the asymptotically optimal covariance matrix of the vanilla averaged LSA iterates. |
| title | High-Order Error Bounds for Markovian LSA with Richardson-Romberg Extrapolation |
| topic | Machine Learning Optimization and Control Statistics Theory 62L20 |
| url | https://arxiv.org/abs/2508.05570 |