Baird counterexample was proposed by Leemon Baird in 1995, first used to show that the Temporal Difference (TD(0)) algorithm diverges on this example. Since then, it is often used to test and compare off-policy learning algorithms. Gradient TD algorithms solved the divergence issue of TD on Baird counterexample. However, their convergence on this example is still very slow, and the nature of the slowness is not well understood, e.g., see (Sutton and Barto 2018). This note is to understand in particular, why TDC is slow on this example, and provide a debugging analysis to understand this behavior. Our debugging technique can be used to study the convergence behavior of two-time-scale stochastic approximation algorithms. We also provide empirical results of the recent Impression GTD algorithm on this example, showing the convergence is very fast, in fact, in a linear rate. We conclude that Baird counterexample is solved, by an algorithm with the convergence guarantee to the TD solution in general, and a fast convergence rate.

Gradient Temporal Difference (GTD) algorithms (Sutton et al., 2008, 2009) are the first $O(d)$ ($d$ is the number features) algorithms that have convergence guarantees for off-policy learning with linear function approximation. Liu et al. (2015) and Dalal et. al. (2018) proved the convergence rates of GTD, GTD2 and TDC are $O(t^{-\alpha/2})$ for some $\alpha \in (0,1)$. This bound is tight (Dalal et al., 2020), and slower than $O(1/\sqrt{t})$. GTD algorithms also have two step-size parameters, which are difficult to tune. In literature, there is a "single-time-scale" formulation of GTD. However, this formulation still has two step-size parameters. This paper presents a truly single-time-scale GTD algorithm for minimizing the Norm of Expected td Update (NEU) objective, and it has only one step-size parameter. We prove that the new algorithm, called Impression GTD, converges at least as fast as $O(1/t)$. Furthermore, based on a generalization of the expected smoothness (Gower et al. 2019), called $L$-$\lambda$ smoothness, we are able to prove that the new GTD converges even faster, in fact, with a linear rate. Our rate actually also improves Gower et al.'s result with a tighter bound under a weaker assumption. Besides Impression GTD, we also prove the rates of three other GTD algorithms, one by Yao and Liu (2008), another called A-transpose-TD (Sutton et al., 2008), and a counterpart of A-transpose-TD. The convergence rates of all the four GTD algorithms are proved in a single generic GTD framework to which $L$-$\lambda$ smoothness applies. Empirical results on Random walks, Boyan chain, and Baird counterexample show that Impression GTD converges much faster than existing GTD algorithms for both on-policy and off-policy learning problems, with well-performing step-sizes in a big range.