Optimal Oracle Inequalities for Projected Fixed-Point Equations, with Applications to Policy Evaluation
研究了在希尔伯特空间中利用随机观测求解线性不动点方程的方法，证明了基于Polyak-Ruppert平均的线性随机逼近方案的均方误差上界，并建立了信息论下界，表明该误差项在实例依赖意义下不可改进。结果精确刻画了线性函数逼近下时序差分学习方法在策略评估问题中的误差，并证明了其最优性。
Linear fixed-point equations in Hilbert spaces arise in a variety of settings, including reinforcement learning, and computational methods for solving differential and integral equations. We study methods that use a collection of random observations to compute approximate solutions by searching over a known low-dimensional subspace of the Hilbert space. First, we prove an instance-dependent upper bound on the mean-squared error for a linear stochastic approximation scheme that exploits Polyak–Ruppert averaging. This bound consists of two terms: an approximation error term with an instance-dependent approximation factor and a statistical error term that captures the instance-specific complexity of the noise when projected onto the low-dimensional subspace. Using information-theoretic methods, we also establish lower bounds showing that both of these terms cannot be improved, again in an instance-dependent sense. A concrete consequence of our characterization is that the optimal approximation factor in this problem can be much larger than a universal constant. We show how our results precisely characterize the error of a class of temporal difference learning methods for the policy evaluation problem with linear function approximation, establishing their optimality. Funding: This work was partially supported by grants from the Office of Naval Research [Grant DOD-ONRN00014-18-1-2640] and the National Science Foundation (NSF) [NSF-IIS Grant 1909365, NSF-DMS Grant 2015454, and NSF-CCF Grant 1955450] to M. J. Wainwright. Part of this work was performed when A. Pananjady was visiting the Simons Institute for the Theory of Computing, where he was supported by a Swiss Re Research Fellowship. Supplemental Material: The online supplementary file is available at https://doi.org/10.1287/moor.2022.1341 .