A Finite-Time Analysis of TD Learning with Linear Function Approximation without Projections nor Strong Convexity

We investigate the finite-time convergence properties of Temporal Difference (TD) learning with linear function approximation, a cornerstone algorithm in reinforcement learning. While prior work has established convergence guarantees, these results typically rely on the assumption that each iterate is projected onto a bounded set or that the learning rate is set according to the unknown strong convexity constant -- conditions that are both artificial and do not match the current practice. In this paper, we challenge the necessity of such assumptions and present a refined analysis of TD learning. We show that the simple projection-free variant converges with a rate of $\tilde{\mathcal{O}}(\frac{||θ^*||^2_2}{\sqrt{T}})$, even in the presence of Markovian noise. Our analysis reveals a novel self-bounding property of the TD updates and exploits it to guarantee bounded iterates.