# Concrete Example for Q Learning

I am not sure if I understood the q learning algorithms correctly. Therefore I would give a concrete example and ask if someone can tell me how to update the q value correctly.

First I initialized a Neural Network with random weights. It shall henceforth evaluate the Q Value for all possible actions(4) given a State S.

Then the following happens. The agent is playing and is exploring. For 3 steps the Q Values evaluated were: (0,-1,-5,0), (0,-1,0,0), (0,-.6,0,0)

The reward given was: 0,0,1 The action took were: (1.,1.,1.) In the random walk example (same reward given), it was: (1.,2.,3.)

So what are the new Q - Values, assuming a discount factor of 0.99 and the learning rate 0.1?

The States for Simplicity are only one number: 1,1.3,2.4 Where 2.4 is the state who ends the game...

The same example holds for exploiting. Is the algorithm the same here?

Here you see my last implementation:

    public void rlearn(ArrayList<Tuple> tupels, double learningrate, double discountfactor) {

//newQ = sum of all rewards you have got through
for(int i = tupels.size()-1; i > 0; i--) {
MLData in = new BasicMLData(45);
MLData out = new BasicMLData(5);

int index = 0;
for(double w : tupels.get(i).statefirst.elements) {
}

//Now start updating Q - Values
double qnew = 0;
if(i <= tupels.size()-2){
qnew = tupels.get(i).rewardafter + discountfactor*qMax(tupels.get(i+1));
} else {
qnew = tupels.get(i).rewardafter;
}

tupels.get(i).qactions.elements[tupels.get(i).actionTaken] = qnew;
index = 0;
for(double w : tupels.get(i).qactions.elements) {
}
}
}


Edit: This is the qMax - function:

    private double qMax(Tuple tuple) {
double max = Double.MIN_VALUE;
for(double w : tuple.qactions.elements) {
if(w > max) {
max = w;
}
}
return max;
}

• Commented Jan 4, 2019 at 15:32
• Your example needs to give the action taken on each step that generated those sampled rewards. You should include at least one step where the non-maximising action was taken. For a full explanation, you should give the example data in the form of state_label, predicted_rewards, action_taken, actual_reward, next_state_label, end_flag - these don't all need to be in vector/numeric form, although it would help if the Q values and rewards are (as you have already done), plus the action id needs to be numeric in order to find what the predicted Q value was Commented Jan 4, 2019 at 16:17
• Your right, I edited the Acgtions taken, What are endflag and nextstatelabel? Commented Jan 4, 2019 at 16:30
• state_label and next_state_label identify the states in the trajectory - it is implied in your question but not stated that your neural network estimates $(q(s, a_0), q(s, a_1), q(s, a_2), q(s, a_3) )$, and we need to know $s$. It is $q(s,a)$ that you revise, using $\text{max}_{a'} q(s',a')$ to improve the estimate, so you need to identify $s$ (state_label), $s'$ (next_state_label) and $a$. The end_flag is boolean - whether the transition ends an episode - that is critical information on how you learn Q values, because $\text{max}_{a'} q(s',a')$ is by definition $0$ in that case Commented Jan 4, 2019 at 16:54
• I understood the end_flag. But I think I do not understand the state label. For what do you need the states again? Just to recalculate the error of the net. (see above) Commented Jan 4, 2019 at 16:57

Most Deep Q-learning implementations I have read are based on Deep Q-Networks (DQN). In DQN, the q-value network maps an input state to a vector of q-values, one for each action:

$$Q(s, \mathbf{w}) \to \mathbf{v}$$

where $$s$$ is the input state from the environment, $$\mathbf{w}$$ are the parameters of the neural network, and $$\mathbf{v}$$ is a vector of q-values, where $$v_i$$ is the estimated q-value of the ith action. In the Sutton and Barto book, the q-value function is written as $$Q(s, a, \mathbf{w})$$, which corresponds to the network output for action $$a$$.

Unlike tabular Q-learning, Deep Q-learning updates the parameters of the the neural network according to the gradients of the loss function with respect to the parameters. DQN uses the loss function

$$L(\mathbf{w}) = [(r + \gamma max_{a'} Q(s', a', \mathbf{w^-})) - Q(s, a, \mathbf{w})]^2$$

where $$\gamma$$ is the discount rate, $$a$$ is the selected action (either greedily or randomly for an $$epsilon$$-greedy behavior policy), $$s'$$ is the next state, $$a'$$ is the argmax action for the next state, and $$\mathbf{w^-}$$ is an older version of the network weights $$\mathbf{w}$$ that is used to help stabilize training.

In deep Q-learning, training directly updates parameters, not q-values. Parameters are updated by taking a small step in the direction of the gradient of the loss function

$$\mathbf{w} \gets \mathbf{w} + \alpha [(r + \gamma max_{a'} Q(s', a', \mathbf{w^-})) - Q(s, a, \mathbf{w})] \nabla_w Q(s, a, \mathbf{w})$$

where $$\alpha$$ is the learning rate.

In frameworks like tensorflow or pytorch the derivative is calculated automatically by giving the loss function and model parameters directly to an optimizer class which uses some variation of mini-batch gradient descent. In eagerly executed tensorflow updating the parameters for a mini-batch might look something like

batch = buffer.sample(batch_size)
observations, actions, rewards, next_obervations = batch

qvalues = model(observations, training=True)
next_qvalues = target_model(next_obervations)
# r + max_{a'} Q(s', a') for the batch
target_qvalues = rewards + gamma * tf.reduce_max(next_qvalues, axis=-1)
# Q(s, a) for the batch
selected_qvalues = tf.reduce_sum(tf.one_hot(actions, depth=qvalues.shape[-1]) * qvalues, axis=-1)
loss = tf.reduce_mean((target_qvalues - selected_qvalues)**2)



Though I am not familiar with the Encog neural network framework you are using, based on the example Brain.java file from your Github repo and Chapter 5 of the Encog User Manual and the Encog neural network examples on Github it looks like weights are updated as follows:

1. A training set is constructed from pairs of input and target output.
2. A Propagation instance, train, is constructed with a network and training set. Different subclasses of Propagation use different loss functions to update the network parameters.
3. The method train.iterate() is called to run the network on the inputs, calculate the loss between the network outputs and target outputs, and update the weights according to the loss.

For DQN, a training set is constructed from a random sample from the experience replay buffer to help stabilize training. A training set could also be the trajectory of an episode, which is what the tupels argument in the example code of the question appears to be.

The input would be the statefirst member of each element of tupels. Since the network produces a vector of q-values, the target output must also be a vector of q-values.

The target output element for the selected action is $$r + \gamma max_{a'} Q(s', a', \mathbf{w^-})$$, In the example code of the question, this is

double qnew = 0;
if(i <= tupels.size()-2){
qnew = tupels.get(i).rewardafter + discountfactor*qMax(tupels.get(i+1));
} else {
qnew = tupels.get(i).rewardafter;
}
tupels.get(i).qactions.elements[tupels.get(i).actionTaken] = qnew


The target output elements for actions that were not selected should be $$Q(s, b, \mathbf{w})$$, where $$b$$ is one of the non-selected actions. This should have the effect of ignoring the q-values of non-selected actions by making the network output equal to the target output.

So what are the new Q - Values, assuming a discount factor of 0.99 and the learning rate 0.1?

Assuming you mean target outputs by the new Q - Values, and given the trajectory of actions, (1, 1, 1), and q-value vectors from the question, the concrete target outputs are (0, 0 + 0.99 * 0, -5, 0), (0, 0 + 0.99 * 0, 0, 0), and (0, 1 + 0, 0, 0).

• $$w←w+α[(r+γmaxa′Q(s′,a′,w−))−Q(s,a,w)]∇wQ(s,a,w)$$ Does this mean I have to hold always an older version of the network? I do not understand this. I understand what the loss function does. It is the normal loss function for most BP-algorithms. But how to calculate: $$maxa′Q(s′,a′,w−))$$ Commented Jan 5, 2019 at 9:55
• Yes, the DQN method uses an older version of the model which gets replaced every N time steps by the current model. According to the paper this improves training. Commented Jan 6, 2019 at 7:06
• You could calculate $max_{a'} Q(s', a', w^-)$ by running the state s' through an older copy of the model (i.e. the target network) and choosing the max of the outputs. Commented Jan 6, 2019 at 7:17
• Unfortunately, this still does not work. Is there anything else to consider. I must have done an obvious mistake. Commented Jan 6, 2019 at 11:56