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Finite-Sample Analysis of Off-Policy TD-Learning via Generalized Bellman Operators
Zaiwei Chen · Siva Theja Maguluri · Sanjay Shakkottai · Karthikeyan Shanmugam

Thu Dec 09 04:30 PM -- 06:00 PM (PST) @
In TD-learning, off-policy sampling is known to be more practical than on-policy sampling, and by decoupling learning from data collection, it enables data reuse. It is known that policy evaluation has the interpretation of solving a generalized Bellman equation. In this paper, we derive finite-sample bounds for any general off-policy TD-like stochastic approximation algorithm that solves for the fixed-point of this generalized Bellman operator. Our key step is to show that the generalized Bellman operator is simultaneously a contraction mapping with respect to a weighted $\ell_p$-norm for each $p$ in $[1,\infty)$, with a common contraction factor. Off-policy TD-learning is known to suffer from high variance due to the product of importance sampling ratios. A number of algorithms (e.g. $Q^\pi(\lambda)$, Tree-Backup$(\lambda)$, Retrace$(\lambda)$, and $Q$-trace) have been proposed in the literature to address this issue. Our results immediately imply finite-sample bounds of these algorithms. In particular, we provide first-known finite-sample guarantees for $Q^\pi(\lambda)$, Tree-Backup$(\lambda)$, and Retrace$(\lambda)$, and improve the best known bounds of $Q$-trace in \citep{chen2021finite}. Moreover, we show the bias-variance trade-offs in each of these algorithms.

Author Information

Zaiwei Chen (Georgia Institute of Technology)
Siva Theja Maguluri (Georgia Institute of Technology)
Sanjay Shakkottai (University of Texas at Austin)
Karthikeyan Shanmugam (IBM Research, NY)

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