# Understanding the equation of TD(0) in the paper “Learning to predict by the methods of temporal differences”

In the paper Learning to predict by the methods of temporal differences (p. 15), the weights in the temporal difference learning are updated as given by the equation $$\Delta w_t = \alpha \left(P_{t+1} - P_t\right) \sum_{k=1}^{t}{\lambda^{t-k} \nabla_w P_k} \tag{4} \,.$$ When $$\lambda = 0$$, as in TD(0), how does the method learn? As it appears, with $$\lambda = 0$$, there will never be a change in weight and hence no learning.

Am I missing anything?

I think the detail that you're missing is that one of the terms in the sum (the final "iteration" of the sum, the case where $$k = t$$) has $$\lambda$$ raised to the power $$0$$, and anything raised to the power $$0$$ (even $$0$$) is equal to $$1$$. So, for $$\lambda = 0$$, your update equation becomes
$$\Delta w_t = \alpha \left( P_{t+1} - P_t \right) \nabla_w P_t,$$
• At page $16$ of the same paper Learning to Predict by the Methods of Temporal Differences (1988), Sutton actually states that $\Delta w_t = \alpha \left( P_{t+1} - P_t \right) \nabla_w P_t$ is the learning rule when $\lambda = 0$. – nbro Jun 1 at 16:51
• He starts with the supervised setting and then derives the Widrow-Hoff (or delta) rule. The TD rule is then a special case of the delta rule, where the errors $z - P_t$ are replaced with a summation of the successive temporal-difference predictions. However, how is that specific 1-step TD learning rule exactly related to the usual learning rules of (tabular) temporal difference methods, where apparently no gradient is needed? – nbro Jun 1 at 16:58
• @nbro You can view tabular methods as methods using linear function "approximation", where there is a single binary feature for every possible state-action pair. Then there would be a gradient needed, but the gradient would simply be $1$ for the "binary feature" corresponding to the state-action pair, and $0$ everywhere else. – Dennis Soemers Jun 1 at 17:33