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I am looking for a result that shows the convergence of semi-gradient TD(0) algorithm with non-linear function approximation for on-policy prediction. Specifically, the update equation is given by (borrowing notation from Sutton and Barto (2018))

$$\mathbf w \leftarrow \mathbf w +\alpha [R + \gamma \hat v(S', \mathbf w) - \hat v(S, \mathbf w)] \nabla \hat v(S, \mathbf w)$$

where $\hat v(S, \mathbf w)$ is the approximate value function parameterized by $\mathbf w$.

Sutton and Barto (2018) mention that the above update equation converges when $\hat v$ is linear in $\mathbf w$. But I couldn't find a similar result for non-linear function approximation. Any help would be greatly appreciated.

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Apparently there is an example of non-convergence for semi-gradient sarsa, according to Rich Sutton (check slide 35). I guess TD(0) is not so different. So, probably your approximator will need to satisfy certain conditions to proof convergence.

Maybe this paper will be useful for you. It seems that they show that constraining your network to have relu activation functions allow you to show some convergence properties.

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