In previous posts (here and here), deep Q reinforcement learning was introduced. In these posts, examples were presented where neural networks were used to train an agent to act within an environment to maximize rewards. The neural network was trained using something called Q-learning. However, deep Q learning (DQN) has a flaw – it can be unstable due to biased estimates of future rewards, and this slows learning. In this post, I’ll introduce Double Q learning which can solve this bias problem and produce better Q-learning outcomes. We’ll be running a Double Q network on a modified version of the Cartpole reinforcement learning environment. We’ll also be developing the network in TensorFlow 2 – at the time of writing, TensorFlow 2 is in beta and installation instructions can be found here. The code examined in this post can be found here.

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# A recap of deep Q learning

As mentioned above, you can go here and here to review deep Q learning. However, a quick recap is in order. The goal of the neural network in deep Q learning is to learn the function $Q(s_t, a_t; \theta_t)$. At a given time in the game / episode, the agent will be in a state $s_t$. This state is fed into the network, and various Q values will be returned for each of the possible actions $a_t$ from state $s_t$. The $\theta_t$ refers to the parameters of the neural network (i.e. all the weight and bias values).

The agent chooses an action based on an epsilon-greedy policy $\pi$. This policy is a combination of randomly selected actions combined with the output of the deep Q neural network – with the probability of a randomly selected action decreasing over the training time. When the deep Q network is used to select an action, it does so by taking the maximum Q value returned over all the actions, for state $s_t$. For example, if an agent is in state 1, and this state has 4 possible actions which the agent can perform, it will output 4 Q values. The action which has the highest Q value is the action which will be selected. This can be expressed as:

$$a = argmax Q(s_t, a; \theta_t)$$

Where the *argmax* is performed over all the actions / output nodes of the neural network. That’s how actions are chosen in deep Q learning. How does training occur? It occurs by utilising the Q-learning / Bellman equation. The equation looks like this:

$$Q_{target} = r_{t+1} + \gamma \max_{{a}}Q(s_{t+1}, a;\theta_t)$$

How does this read? For a given action *a* from state $s_{t}$, we want to train the network to predict the following:

- The immediate reward for taking this action $r_{t+1}$, plus
- The discounted reward for the best possible action in the subsequent state ($s_{t+1}$)

If we are successful in training the network to predict these values, the agent will consistently chose the action which gives the best immediate reward ($r_{t+1}$) plus the discounted future rewards of future states $\gamma \max_{{a}}Q(s_{t+1}, a;\theta_t)$. The $\gamma$ term is the discount term, which places less value on future reward than present rewards (but usually only marginally).

In deep Q learning, the game is repeatedly played and the states, actions and rewards are stored in memory as a list of tuples or an array – ($s_t$, $a$, $r_t$, $s_{t+1}$). Then, for each training step, a random batch of these tuples is extracted from memory and the $Q_{target}(s_t, a_t)$ is calculated and compared to the value produced from the current network $Q(s_t, a_t)$ – the mean squared difference between these two values is used as the loss function to train the neural network.

That’s a fairly brief recap of deep Q learning – for a more extended treatment see here and here. The next section will explain the problems with standard deep Q learning.

# The problem with deep Q learning

The problem of deep Q learning has to do with the way it sets the target values:

$$Q_{target} = r_{t+1} + \gamma \max_{{a}}Q(s_{t+1}, a;\theta_t)$$

Namely, the issue is with the $\max$ value. This part of the equation is supposed to estimate the value of the rewards for future actions if action *a *is taken from the current state $s_t$. That’s a bit of a mouthful, but just consider it as trying to estimate the optimal future rewards $r_future$ if action *a* is taken.

The problem is that in many environments, there is random noise. Therefore, as an agent explores an environment, it is not directly observing *r *or $r_future$, but something like $r + \epsilon$, where $\epsilon$ is the noise. In such an environment, after repeated playing of the game, we would hope that the network would learn to make unbiased estimates of the expected value of the rewards – so *E[r]. *If it can do this, we are in a good spot – the network should pick out the best actions for current and future rewards, despite the presence of noise.

This is where the $\max$ operation is a problem – it produces *biased* estimates of the future rewards, not the unbiased estimates we require for optimal results. An example will help explain this better. Consider the environment below. The agent starts in state *A* and at each state can move left or right. The states C, D and F are terminal states – the game ends once these points are reached. The *r* values are the rewards the agent receives when transitioning from state to state.

All the rewards are deterministic except for the rewards when transitioning from states B to C and B to D. The rewards for these transitions are randomly drawn from a normal distribution with a mean of 1 and a standard deviation of 4.

We know the expected rewards, *E[r]* from taking either action (B to C or B to D) is 1 – however, there is a lot of noise associated with these rewards. Regardless, on average, the agent should ideally learn to always move to the left from A, towards E and finally F where *r *always equals 2.

Let’s consider the $Q_{target}$ expression for these cases. Let’s set $\gamma$ to be 0.95. The $Q_target$ expression to move to the left from *A *is: $Q_{target} = 0 + 0.95 * \max([0, 2]) = 1.9$. The two action options from E are to either move right (r = 0) or left (r = 2). The maximum of these is obviously 2, and hence we get the result 1.9.

What about in the opposite direction, moving right from *A*? In this case, it is $Q_{target} = 0 + 0.95 * \max([N(1, 4), N(1, 4)])$. We can explore the long term value of this “moving right” action by using the following code snippet:

import numpy as np Ra = np.zeros((10000,)) Rc = np.random.normal(1, 4, 10000) Rd = np.random.normal(1, 4, 10000) comb = np.vstack((Ra, Rc, Rd)).transpose() max_result = np.max(comb, axis=1) print(np.mean(Rc)) print(np.mean(Rd)) print(np.mean(max_result))

Here a 10,000 iteration trial is created of what the $\max$ term will yield in the long term of running a deep Q agent in this environment. *Ra* is the reward for moving back to the left towards A (always zero, hence *np.zeros()*). *Rc *and *Rd* are both normal distributions, with mean 1 and standard deviation of 4. Combining all these options together and taking the maximum for each trial gives us what the trial-by-trial $\max$ term will be (*max_result*). Finally, the expected values (i.e. the means) of each quantity are printed. As expected, the mean of *Rc *and *Rd* are approximately equal to 1 – the mean which we set for their distributions. However, the expected value / mean from the $\max$ term is actually around 3!

You can see the problem here. Because the $\max$ term is always taking the maximum value from the random draws of the rewards, it tends to be positively biased and does not give a true indication of the expected values of the rewards for a move in this direction (i.e. 1). As such, an agent using the deep Q learning methodology will not chose the optimal action from *A *(i.e. move left) but will rather tend to move right!

Therefore, in noisy environments, it can be seen that deep Q learning will tend to overestimate rewards. Eventually, deep Q learning will converge to a reasonable solution, but it is potentially much slower than it needs to be. A further problem occurs in deep Q learning which can cause instability in the training process. Consider that in deep Q learning the same network both choses the best action *and *determines the value of choosing said actions. There is a feedback loop here which can exacerbate the previously mentioned reward overestimation problem, and further slow down the learning process. This is clearly not ideal, and this is why Double Q learning was developed.

# An introduction to Double Q reinforcement learning

The paper that introduced Double Q learning initially proposed the creation of two separate networks which predicted $Q^A$ and $Q^B$ respectively. These networks were trained on the same environment / problem, but were each randomly updated. So, say, 50% of the time, $Q^A$ was updated based on a certain random set of training tuples, and 50% of the time $Q^B$ was updated on a different random set of training tuples. Importantly, the update or target equation for network A had an estimate of the future rewards from network B – not itself. This new approach does two things:

- The A and B networks are trained on different training samples – this acts to remove the overestimation bias, as, on average, if network A sees a high noisy reward for a certain action, it is likely that network B will see a lower reward – hence the noise effects will cancel
- There is a decoupling between the
*choice*of the best action and the*evaluation*of the best action

The algorithm from the original paper is as follows:

As can be observed, first an action is chosen from either $Q^A(s_t,.)$ or $Q^B(s_t,.)$ and the rewards, next state, action etc. are stored in the memory. Then either UPDATE(A) or UPDATE(B) is chosen randomly. Next, for the state $s_{t+1}$ (or s’ in the above) the predicted Q value for all actions from this state are taken from network A or B, and the action with the highest predicted Q value is chosen, *a**. Note that, within UPDATE(A), this action is chosen from the output of the $Q^A$ network.

Next, you’ll see something interesting. Consider the update equation for $Q^A$ above – I’ll represent it in more familiar, neural network based notation below:

$$Q^A_{target} = r_{t+1} + \gamma Q^B(s_{t+1}, a*)$$

Notice that, while the best action a* from the next state ($s_{t+1}$) is chosen from network A, the discounted reward for taking that future action is *extracted from network B*. This removes any bias associated with the $argmax$ from network A, and also decouples the choice of actions from the evaluation of the value of such actions (i.e. breaks the feedback loop). This is the heart of the Double Q reinforcement learning.

## The Double DQN network

The same author of the original Double Q algorithm shown above proposed an update of the algorithm in this paper. This updated algorithm can still legitimately be called a Double Q algorithm, but the author called it Double DQN (or DDQN) to disambiguate. The main difference in this algorithm is the removal of the randomized back-propagation based updating of two networks A and B. There are still two networks involved, but instead of training both of them, only a primary network is actually trained via back-propagation. The other network, often called the target network, is periodically copied from the primary network. The update operation for the primary network in the Double DQN network looks like the following:

$$Q_{target} = r_{t+1} + \gamma Q(s_{t+1}, argmax Q(s_{t+1}, a; \theta_t); \theta^-_t)$$

Alternatively, keeping in line with the previous representation:

$$a* = argmax Q(s_{t+1}, a; \theta_t)$$

$$Q_{target} = r_{t+1} + \gamma Q(s_{t+1}, a*); \theta^-_t)$$

Notice that, as per the previous algorithm, the action *a** with the highest Q value from the next state ($s_{t+1}$) is extracted from the primary network, which has weights $\theta_t$. This primary network is also often called the “online” network – it is the network from which action decisions are taken. However, notice that, when determining $Q_{target}$, the discounted Q value is taken from the *target* network with weights $\theta^-_t$. Therefore, the *actions* for the agent to take are extracted from the online network, but the *evaluation* of the future rewards are taken from the target network. So far, this is similar to the UPDATE(A) step shown in the previous Double Q algorithm.

The difference in this algorithm is that the target network weights ($\theta^-_t$) are not trained via back-propagation – rather they are periodically copied from the online network. This reduces the computational overhead of training two networks by back-propagation. This copying can either be a periodic “hard copy”, where the weights are copied from the online network to the target network with no modification, or a more frequent “soft copy” can occur, where the existing target weight values and the online network values are blended. In the example which will soon follow, soft copying will be performed every training iteration, under the following rule:

$$\theta^- = \theta^- (1-\tau) + \theta \tau$$

With $\tau$ being a small constant (i.e. 0.05).

This DDQN algorithm achieves both decoupling between the action choice and evaluation, and it has been shown to remove the bias of deep Q learning. In the next section, I’ll present a code walkthrough of a training algorithm which contains options for both standard deep Q networks and Double DQNs.

## A Double Q network example in TensorFlow 2

In this example, I’ll present code which trains a double Q network on the Cartpole reinforcement learning environment. This environment is implemented in OpenAI gym, so you’ll need to have that package installed before attempting to run or replicate. The code for this example can be found on this site’s Github repo.

First, we declare some constants and create the environment:

STORE_PATH = '/Users/andrewthomas/Adventures in ML/TensorFlowBook/TensorBoard' MAX_EPSILON = 1 MIN_EPSILON = 0.01 LAMBDA = 0.0005 GAMMA = 0.95 BATCH_SIZE = 32 TAU = 0.08 RANDOM_REWARD_STD = 1.0 env = gym.make("CartPole-v0") state_size = 4 num_actions = env.action_space.n

Notice the epsilon greedy policy parameters (MIN_EPSILON, MAX_EPSILON, LAMBDA) which dictate how long the exploration period of the training should last. GAMMA is the discount rate of future rewards. The final constant RANDOM_REWARD_STD will be explained later in more detail.

It can be observed that the CartPole environment has a state size of 4, and the number of actions available are extracted directly from the environment (there are only 2 of them). Next the primary (or online) network and the target network are created using the Keras Sequential API:

primary_network = keras.Sequential([ keras.layers.Dense(30, activation='relu', kernel_initializer=keras.initializers.he_normal()), keras.layers.Dense(30, activation='relu', kernel_initializer=keras.initializers.he_normal()), keras.layers.Dense(num_actions) ]) target_network = keras.Sequential([ keras.layers.Dense(30, activation='relu', kernel_initializer=keras.initializers.he_normal()), keras.layers.Dense(30, activation='relu', kernel_initializer=keras.initializers.he_normal()), keras.layers.Dense(num_actions) ]) primary_network.compile(optimizer=keras.optimizers.Adam(), loss='mse')

The code above is fairly standard Keras model definitions, with dense layers and ReLU activations, and He normal initializations (for further information, see these posts: Keras, ReLU activations and initialization). Notice that only the primary network is compiled, as this is the only network which will be trained via the Adam optimizer.

class Memory: def __init__(self, max_memory): self._max_memory = max_memory self._samples = [] def add_sample(self, sample): self._samples.append(sample) if len(self._samples) > self._max_memory: self._samples.pop(0) def sample(self, no_samples): if no_samples > len(self._samples): return random.sample(self._samples, len(self._samples)) else: return random.sample(self._samples, no_samples) @property def num_samples(self): return len(self._samples) memory = Memory(50000)

Next a generic Memory class object is created. This holds all the ($s_t$, a, $r_t$, $s_{t+1}$) tuples which are stored during training, and includes functionality to extract random samples for training. In this example, we’ll be using a Memory instance with a maximum sample buffer of 50,000 rows.

def choose_action(state, primary_network, eps): if random.random() < eps: return random.randint(0, num_actions - 1) else: return np.argmax(primary_network(state.reshape(1, -1)))

The function above executes the epsilon greedy action policy. As explained in previous posts on deep Q learning, the epsilon value is slowly reduced and the action selection moves from the random selection of actions to actions selected from the primary network. A final training function needs to be reviewed, but first we’ll examine the main training loop:

num_episodes = 1000 eps = MAX_EPSILON render = False train_writer = tf.summary.create_file_writer(STORE_PATH + f"/DoubleQ_{dt.datetime.now().strftime('%d%m%Y%H%M')}") double_q = False steps = 0 for i in range(num_episodes): state = env.reset() cnt = 0 avg_loss = 0 while True: if render: env.render() action = choose_action(state, primary_network, eps) next_state, reward, done, info = env.step(action) reward = np.random.normal(1.0, RANDOM_REWARD_STD) if done: next_state = None # store in memory memory.add_sample((state, action, reward, next_state)) loss = train(primary_network, memory, target_network if double_q else None) avg_loss += loss state = next_state # exponentially decay the eps value steps += 1 eps = MIN_EPSILON + (MAX_EPSILON - MIN_EPSILON) * math.exp(-LAMBDA * steps) if done: avg_loss /= cnt print(f"Episode: {i}, Reward: {cnt}, avg loss: {avg_loss:.3f}, eps: {eps:.3f}") with train_writer.as_default(): tf.summary.scalar('reward', cnt, step=i) tf.summary.scalar('avg loss', avg_loss, step=i) break cnt += 1

Starting from the *num_episodes *loop, we can observe that first the environment is reset, and the current state of the agent returned. A *while True* loop is then entered into, which is only exited when the environment returns the signal that the episode has been completed. The code will render the Cartpole environment if the relevant variable has been set to True.

The next line shows the action selection, where the primary network is fed into the previously examined *choose_network* function, along with the current state and the epsilon value. This action is then fed into the environment by calling the *env.step()* command. This command returns the next state that the agent has entered ($s_{t+1}$), the reward ($r_{t+1}$) and the *done* Boolean which signifies if the episode has been completed.

The Cartpole environment is completely deterministic, with no randomness involved except in the initialization of the environment. Because Double Q learning is superior to deep Q learning especially when there is randomness in the environment, the Cartpole environment has been externally transformed into a stochastic environment on the next line. Normally, the reward from the Cartpole environment is a deterministic value of 1.0 for every step the pole stays upright. Here, however, the reward is replaced with a sample from a normal distribution, with mean 1.0 and standard deviation equal to the constant RANDOM_REWARD_STD.

In the first pass – RANDOM_REWARD_STD is set to 0.0 to transform the environment back to a deterministic case, but this will be changed in the next example run.

After this, the memory is added to and the primary network is trained.

Notice that the *target_network* is only passed to the training function if the *double_q* variable is set to True. If *double_q *is set to False, the training function defaults to standard deep Q learning. Finally the state is updated, and if the environment has signalled the episode has ended, some logging is performed and the *while *loop is exited.

It is now time to review the *train* function, which is where most of the work takes place:

def train(primary_network, memory, target_network=None): if memory.num_samples < BATCH_SIZE * 3: return 0 batch = memory.sample(BATCH_SIZE) states = np.array([val[0] for val in batch]) actions = np.array([val[1] for val in batch]) rewards = np.array([val[2] for val in batch]) next_states = np.array([(np.zeros(state_size) if val[3] is None else val[3]) for val in batch]) # predict Q(s,a) given the batch of states prim_qt = primary_network(states) # predict Q(s',a') from the evaluation network prim_qtp1 = primary_network(next_states) # copy the prim_qt into the target_q tensor - we then will update one index corresponding to the max action target_q = prim_qt.numpy() updates = rewards valid_idxs = np.array(next_states).sum(axis=1) != 0 batch_idxs = np.arange(BATCH_SIZE) if target_network is None: updates[valid_idxs] += GAMMA * np.amax(prim_qtp1.numpy()[valid_idxs, :], axis=1) else: prim_action_tp1 = np.argmax(prim_qtp1.numpy(), axis=1) q_from_target = target_network(next_states) updates[valid_idxs] += GAMMA * q_from_target.numpy()[batch_idxs[valid_idxs], prim_action_tp1[valid_idxs]] target_q[batch_idxs, actions] = updates loss = primary_network.train_on_batch(states, target_q) if target_network is not None: # update target network parameters slowly from primary network for t, e in zip(target_network.trainable_variables, primary_network.trainable_variables): t.assign(t * (1 - TAU) + e * TAU) return loss

The first line is a bypass of this function if the memory does not contain more than 3 x the batch size – this is to ensure no training of the primary network takes place until there is a reasonable amount of samples within the memory.

Next, a batch is extracted from the memory – this is a list of tuples. The individual state, actions and reward values are then extracted and converted to numpy arrays using Python list comprehensions. Note that the *next_state* values are set to zeros if the raw *next_state* values are None – this only happens when the episode has terminated.

Next the sampled states ($s_t$) are passed through the network – this returns the values $Q(s_t, a; \theta_t)$. The next line extracts the Q values from the primary network for the next states ($s_{t+1}$). Next, we want to start constructing our *target_q *values ($Q_target$). These are the “labels” which will be supplied to the primary network to train towards.

Note that the *target_q* values are the same as the *prim_qt* ($Q(s_t, a; \theta_t)$) values except for the index corresponding to the action chosen. So, for instance, let’s say a single sample of the *prim_qt* values are [0.5, -0.5] – but the action chosen from $s_t$ was 0. We only want to update the 0.5 value while training, the remaining values in *target_q *remain equal to *prim_qt* (i.e. [update, -0.5]). Therefore, in the next line, we create *target_q* by simply converting *prim_qt* from a tensor into its numpy equivalent. This is basically a copy of the values from *prim_qt *t0 *target_q*. We convert to numpy also, as it is easier to deal with indexing in numpy than TensorFlow at this stage.

To affect these updates, we create a new variable *updates*. The first step is to set the update values to the sampled rewards – the $r_{t+1}$ values are the same regardless of whether we are performing deep Q learning or Double Q learning. In the following lines, these update values will be added to in order to capture the discounted future reward terms. The next line creates an array called *valid_idxs*. This array is to hold all those samples in the batch which don’t include a case where *next_state* is zero. When *next_state* is zero, this means that the episode has terminated. In those cases, only the first term of the equation below remains ($r_{t+1}$):

$$Q_{target} = r_{t+1} + \gamma Q(s_{t+1}, a*); \theta^-_t)$$

Seeing as *update* already includes the first term, any further additions to update need to exclude these indexes.

The next line, *batch_idxs, *is simply a numpy *arange* which counts out the number of samples within the batch. This is included to ensure that the numpy indexing / broadcasting to follow works properly.

The next line switches depending on whether Double Q learning has been enabled or not. If *target_network* is None, then standard deep Q learning ensures. In such a case, the following term is calculated and added to updates (which already includes the reward term):

$$\gamma \max Q(s_{t+1}, a; \theta)$$

Alternatively, if *target_network* is not None, then Double Q learning is performed. The first line:

prim_action_tp1 = np.argmax(prim_qtp1.numpy(), axis=1)

calculates the following equation shown earlier:

$$a* = argmax Q(s_{t+1}, a; \theta_t)$$

The next line extracts the Q values from the *target* network for state $s_{t+1}$ and assigns this to variable *q_from_target*. Finally, the update term has the following added to it:

$$\gamma Q(s_{t+1}, a*); \theta^-_t)$$

Notice, that the numpy indexing extracts from *q_from_target *all the valid batch samples, and within those samples, all the highest Q actions drawn from the *primary *network (i.e. a*).

Finally, the *target_q* values corresponding to the actions *a *from state $s_t$ are updated with the *update* array.

Following this, the primary network is trained on this batch of data using the Keras *train_on_batch*. The last step in the function involves copying the primary or online network values into the target network. This can be varied so that this step only occurs every X amount of training steps (especially when one is doing a “hard copy”). However, as stated previously, in this example we’ll be doing a “soft copy” and therefore every training step involves the target network weights being moved slightly towards the primary network weights. As can be observed, for every trainable variable in both the primary and target networks, the target network trainable variables are assigned new values updated via the previously presented formula:

$$\theta^- = \theta^- (1-\tau) + \theta \tau$$

That (rather lengthy) explanation concludes the discussion of how Double Q learning can be implemented in TensorFlow 2. Now it is time to examine the results of the training.

## Double Q results for a deterministic case

In the first case, we are going to examine the deterministic training case when RANDOM_REWARD_STD is set to 0.0. The TensorBoard graph below shows the results:

As can be observed, in both the Double Q and deep Q training cases, the networks converge on “correctly” solving the Cartpole problem – with eventual consistent rewards of 180-200 per episode (a total reward of 200 is the maximum available per episode in the Cartpole environment). The Double Q case shows slightly better performance in reaching the “solved” state than the deep Q network implementation. This is likely due to better stability in decoupling the choice and evaluation of the actions, but it is not a conclusive result in this rather simple deterministic environment.

However, what happens when we increase the randomness by elevated RANDOM_REWARD_STD > 0?

## Double Q results for a stochastic case

The results below show the case when RANDOM_REWARD_STD is increased to 1.0 – in this case, the rewards are drawn from a random normal distribution of mean 1.0 and standard deviation of 1.0:

As can be seen, in this case, the Double Q network significantly outperforms the deep Q training methodology. This demonstrates the effect of biasing in the deep Q training methodology, and the advantages of using Double Q learning in your reinforcement learning tasks.

I hope this post was helpful in increasing your understanding of both deep Q and Double Q reinforcement learning. Keep an eye out for future posts on reinforcement learning.

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