New Convergence Guarantee Achieved for Unprojected Temporal Difference Learning
A new paper on arXiv demonstrates that unprojected Temporal Difference (TD) learning with linear function approximation achieves a convergence rate of Õ(1/√T) in expectation, even under Markovian noise. This resolves an open problem posed by Bhandari et al. at COLT 2018, who conjectured that removing the artificial projection-onto-bounded-set assumption would require additional regularity conditions. The result simplifies a foundational algorithm in reinforcement learning theory while providing stronger, more practical guarantees.
Temporal Difference learning with linear function approximation is a central algorithm in reinforcement learning, but prior convergence guarantees in the 'robust' setting — where results do not depend on the minimal curvature of the potential function — have relied on the artificial assumption that iterates are projected onto a bounded set. Bhandari et al. (COLT 2018) identified removing this projection as an open problem, hypothesizing that extra regularity conditions would be necessary. The new paper by Wei-Cheng Lee and collaborators shows that simple unprojected TD(0) achieves a rate of Õ(‖θ*‖²/√T) in expectation without any such regularity condition. The key technical contribution is the identification of a novel self-bounding property of TD updates, which is exploited to guarantee that iterates remain bounded without explicit projection. Only a minor polylogarithmic correction to the learning rate schedule is required. The result holds under Markovian noise, which reflects realistic sampling conditions in reinforcement learning environments. The work advances the theoretical foundations of TD learning by closing a long-standing gap between practical algorithm use and formal convergence theory.
What's missing
It is unclear whether the Õ(‖θ*‖²/√T) rate is tight or whether matching lower bounds exist. The practical impact of the required polylogarithmic learning rate correction on empirical performance is not discussed.
What different sources said
- arXiv cs.LGCenter
A Robust $\widetilde{\mathcal{O}}(1/\sqrt{T})$ Rate for Unprojected TD Learning with Linear Function Approximation
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