AI Insight
This paper proves that temporal difference (TD) learning in reinforcement learning automatically combines the benefits of both TD and Monte Carlo methods without requiring complete state space coverage. The authors establish an oracle inequality showing that TD's estimation error is bounded by a newly defined "subgraph Bellman variance" for any sufficiently visited subset of states, meaning the algorithm adapts to partial coverage without needing to explicitly select which states to focus on. The theoretical results are supported by a minimax lower bound and experimental validation examining how performance transitions between different variance regimes.
Why it matters
This theoretical advance explains why TD methods work well in practice even when agents cannot visit all states uniformly, which is common in complex real-world reinforcement learning applications. The findings provide rigorous justification for using TD in scenarios with incomplete exploration and could inform the design of more sample-efficient learning algorithms.
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⚠️ Preprint – Noch nicht peer-reviewed
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Abstract: Temporal difference (TD) and Monte Carlo (MC) are two fundamental principles for value function estimation in reinforcement learning. TD is asymptotically efficient for value estimation, whereas MC requires only adequate target visitation. We show that these advantages are automatically combined by ordinary TD in the tabular setting. We identify the subgraph Bellman variance, arising from bootstrapping within a subgraph and rollouts outside it, as a fundamental quantity connecting the MC and efficient TD benchmarks. With independent trajectories, we prove a non-asymptotic oracle inequality: the mean squared error of full TD is bounded by this variance divided by the sample size for every sufficiently covered subgraph containing the target, up to a higher-order remainder whose occupancy dependence is confined to that subgraph. No subgraph is supplied to the algorithm. TD thus adapts without selecting a subgraph or requiring global coverage. A finite-sample minimax lower bound shows that the asymptotic variance of TD need not be attainable before rare states are adequately observed. Experimental studies examine the transition between these variance scales and its dependence on visitation structure.
Source: Temporal difference learning adapts to incomplete coverage: an oracle inequality over subgraphs