Why are Target Networks used in Deep Q-Learning as opposed to the Expected Value equation?

Question

I understand we use a target network because it helps resolve issues regarding stability, however, that's not what I'm here to ask.
What I would like to understand is why a target network is used as a measure of ground truth as opposed to the expectation equation.
To clarify, here is what I mean. This is the process used for DQN:

In DQN, we begin with a state $S$
We then pass this state through a neural network which outputs Q values for each action in the action space
A policy e.g. epsilon-greedy is used to take an action
This subsequently produces the next state $S_{t+1}$
$S_{t+1}$ is then passed through a target neural network to produce target Q values
These target Q values are then injected into the Bellman equation which ultimately produces a target Q value via the Q-learning update rule equation
MSE is used on 6 and 2 to compute the loss
This is then back-propagated to update the parameters for the neural network in 2
The target neural network has its parameters updated every X epochs to match the parameters in 2

Why do we use a target neural network to output Q values instead of using statistics. Statistics seems like a more accurate way to represent this. By statistics, I mean this:
Q values are the expected return, given the state and action under policy π.
$Q(S_{t+1},a) = V^π(S_{t+1})$ = $mathbb{E}(r_{t+1}+ γr_{t+2}+ (γ^2)_{t+3} + ...    mid S_{t+1}) = {E}(∑γ^kr_{t+k+1}mid S_{t+1})$
We can then take the above and inject it into the Bellman equation to update our target Q value:
$Q(S_{t},a_t) + α*(r_t+γ*max(Q(S_{t+1},a))-Q(S_{t},a))$
So, why don't we set the target to the sum of diminishing returns? Surely a target network is very inaccurate, especially since the parameters in the first few epochs for the target network are completely random.

Neil Slater · Answer

Why are Target Networks used in Deep Q-Learning as opposed to the Expected Value equation?

In short, because for many problems, this learns more efficiently.
It is the difference between Monte Carlo (MC) methods and Temporal Difference (TD) learning.
You can use MC estimates for expected return in deep RL. They are slower for two reasons:

It takes far more experience to collect enough data to train a neural network, because to fully sample a return you need a whole episode. You cannot just use one episode at a time because that presents the neural network with correlated data. You would need to collect multiple episodes and fill a large experience table.

As an aside, you would also need to discard all the experience after each update, because sampled full returns are on-policy data. Or you could implement importance sampling for off-policy Monte Carlo control and re-calculate the correct updates when the policy starts to improve, which is added complexity.

Samples of full returns have a higher variance, so the sampled data is noisier.

In comparison, TD learning starts with biased samples. This bias reduces over time as estimates become better, but it is the reason why a target network is used (otherwise the bias would cause runaway feed back).
So you have a bias/variance trade off with TD representing high bias and MC representing high variance.
It is not clear theoretically which is better in general, because it depends on the nature of MDPs that you are solving with each method. In practice, on the types of problems Deep RL has been tried on, single-step TD learning appears to do better than MC sampling of returns, in terms of goals such as sample efficiency and learning time.
You can compromise between TD and MC using eligibility traces, resulting in TD($lambda$). However, this is awkward to implement in Deep RL due to the experience replay table. A simpler compromise is to use $n$-step returns e.g. $r_{t+1} + gamma r_{t+2} + gamma^2 r_{t+3} + gamma^3 text{max}_a(Q(s_{t+4},a))$, which was one of the refinements used in the "Rainbow" DQN paper - note that even though strictly in their case, this handled off-policy incorrectly (it should use importance sampling, but they didn't bother), it still worked well enough for low $n$ on the Atari problems.

Why are Target Networks used in Deep Q-Learning as opposed to the Expected Value equation?

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