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I've implemented the reinforcement learning algorithm for an agent to play snappy bird (a shameless cheap ripoff of flappy bird) utilizing a q-table for storing the history for future lookups. It works and eventually achieves perfect convergence after enough training.

Is it possible to implement a neural network to do function approximation in order to accomplish the purpose of the q-table? Obviously, storage is a concern with the q-table, but it doesn't seem to ever train with the neural net alone. Perhaps training the NN on an existing q-table would work, but I would like to not use a q-table at all if possible.

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Andrej Karpathy's blog has a tutorial on getting a neural network to learn pong with reinforcement learning. His commentary on the current state of the field is interesting.

He also provides a whole bunch of links (David Silver's course catches my eye). Here is a working link to the lecture videos.

Here are demos of DeepMinds game playing. Get links to the papers at Andrej Karpathy's blog above - rat fps

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Yes, it is possible. The field of deep reinforcement learning is all about using deep neural networks (that is, neural networks with at least one hidden layer) to approximate value functions (such as the $Q$ function) or policies.

Have a look at the paper A Brief Survey of Deep Reinforcement Learning that gives a brief survey of the field.

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