The problem we have been solving in the previous lesson might seem like a toy problem, not really applicable for real life scenarios. This is not the case, because many real world problems also share this scenario - including playing Chess or Go. They are similar, because we also have a board with given rules and a **discrete state**.
In this lesson we will apply the same principles of Q-Learning to a problem with **continuous state**, i.e. a state that is given by one or more real numbers. We will deal with the following problem:
> **Problem**: If Peter wants to escape from the wolf, he needs to be able to move faster. We will see how Peter can learn to skate, in particular, to keep balance, using Q-Learning.
@ -10,11 +14,10 @@ We will use a simplified version of balancing known as a **CartPole** problem. I
In this lesson, we will be using a library called **OpenAI Gym** to simulate different **environments**. You can run this lesson's code locally (eg. from Visual Studio Code), in which case the simulation will open in a new window. When running the code online, you may need to make some tweaks to the code, as described [here](https://towardsdatascience.com/rendering-openai-gym-envs-on-binder-and-google-colab-536f99391cc7).
## OpenAI Gym
In the previous lesson, the rules of the game and the state were given by the `Board` class which we defined ourselves. Here we will use a special **simulation environment**, which will simulate the physics behind the balancing pole. One of the most popular simulation environments for training reinforcement learning algorithms is called a [Gym](https://gym.openai.com/), which is maintained by [OpenAI](https://openai.com/). By using this gym we can create difference **environments** from a cartpole simulation to Atari games.
@ -33,76 +36,84 @@ import numpy as np
import random
```
## A cartpole environment
## Exercise - initialize a cartpole environment
To work with a cartpole balancing problem, we need to initialize corresponding environment. Each environment is associated with an:
* **Observation space** that defines the structure of information that we receive from the environment. For cartpole problem, we receive position of the pole, velocity and some other values.
* **Action space** that defines possible actions. In our case the action space is discrete, and consists of two actions - **left** and **right**. (code block 2)
```python
env = gym.make("CartPole-v1")
print(env.action_space)
print(env.observation_space)
print(env.action_space.sample())
```
To see how the environment works, let's run a short simulation for 100 steps. At each step, we provide one of the actions to be taken - in this simulation we just randomly select an action from `action_space`. Run the code below and see what it leads to.
> **Note**: Remember that it is preferred to run this code on local Python installation! (code block 3)
- **Observation space** that defines the structure of information that we receive from the environment. For cartpole problem, we receive position of the pole, velocity and some other values.
```python
env.reset()
for i in range(100):
env.render()
env.step(env.action_space.sample())
env.close()
```
- **Action space** that defines possible actions. In our case the action space is discrete, and consists of two actions - **left** and **right**. (code block 2)
You should be seeing something similar to this one:
During simulation, we need to get observations in order to decide how to act. In fact, `step` function returns us back current observations, reward function, and the `done` flag that indicates whether it makes sense to continue the simulation or not: (code block 4)
To see how the environment works, let's run a short simulation for 100 steps. At each step, we provide one of the actions to be taken - in this simulation we just randomly select an action from `action_space`.
```python
env.reset()
1. Run the code below and see what it leads to.
done = False
while not done:
env.render()
obs, rew, done, info = env.step(env.action_space.sample())
print(f"{obs} -> {rew}")
env.close()
```
You will end up seeing something like this in the notebook output:
You may also notice that reward value on each simulation step is always 1. This is because our goal is to survive as long as possible, i.e. keep the pole to a reasonably vertical position for the longest period of time.
1. During simulation, we need to get observations in order to decide how to act. In fact, the step function returns current observations, a reward function, and the done flag that indicates whether it makes sense to continue the simulation or not: (code block 4)
> In fact, CartPole simulation is considered solved if we manage to get the average reward of 195 over 100 consecutive trials.
```python
env.reset()
done = False
while not done:
env.render()
obs, rew, done, info = env.step(env.action_space.sample())
print(f"{obs} -> {rew}")
env.close()
```
You will end up seeing something like this in the notebook output:
The observation vector that is returned at each step of the simulation contains the following values:
- Position of cart
- Velocity of cart
- Angle of pole
- Rotation rate of pole
1. Get min and max value of those numbers: (code block 5)
```python
print(env.observation_space.low)
print(env.observation_space.high)
```
You may also notice that reward value on each simulation step is always 1. This is because our goal is to survive as long as possible, i.e. keep the pole to a reasonably vertical position for the longest period of time.
✅ In fact, the CartPole simulation is considered solved if we manage to get the average reward of 195 over 100 consecutive trials.
## State discretization
@ -110,50 +121,52 @@ In Q=Learning, we need to build Q-Table that defines what to do at each state. T
There are a few ways we can do this:
* If we know the interval of a certain value, we can divide this interval into a number of **bins**, and then replace the value by the bin number that it belongs to. This can be done using the numpy [`digitize`](https://numpy.org/doc/stable/reference/generated/numpy.digitize.html) method. In this case, we will precisely know the state size, because it will depend on the number of bins we select for digitalization.
- **Divide into bins**. If we know the interval of a certain value, we can divide this interval into a number of **bins**, and then replace the value by the bin number that it belongs to. This can be done using the numpy [`digitize`](https://numpy.org/doc/stable/reference/generated/numpy.digitize.html) method. In this case, we will precisely know the state size, because it will depend on the number of bins we select for digitalization.
✅ We can use linear interpolation to bring values to some finite interval (say, from -20 to 20), and then convert numbers to integers by rounding them. This gives us a bit less control on the size of the state, especially if we do not know the exact ranges of input values. For example, in our case 2 out of 4 values do not have upper/lower bounds on their values, which may result in the infinite number of states.
In our example, we will go with the second approach. As you may notice later, despite undefined upper/lower bounds, those value rarely take values outside of certain finite intervals, thus those states with extreme values will be very rare.
Here is the function that will take the observation from our model, and produces a tuple of 4 integer values: (code block 6)
1. Here is the function that will take the observation from our model and produce a tuple of 4 integer values: (code block 6)
1. Let's also explore another discretization method using bins: (code block 7)
print("Sample bins for interval (-5,5) with 10 bins\n",create_bins((-5,5),10))
```python
def create_bins(i,num):
return np.arange(num+1)*(i[1]-i[0])/num+i[0]
print("Sample bins for interval (-5,5) with 10 bins\n",create_bins((-5,5),10))
ints = [(-5,5),(-2,2),(-0.5,0.5),(-2,2)] # intervals of values for each parameter
nbins = [20,20,10,10] # number of bins for each parameter
bins = [create_bins(ints[i],nbins[i]) for i in range(4)]
def discretize_bins(x):
return tuple(np.digitize(x[i],bins[i]) for i in range(4))
```
ints = [(-5,5),(-2,2),(-0.5,0.5),(-2,2)] # intervals of values for each parameter
nbins = [20,20,10,10] # number of bins for each parameter
bins = [create_bins(ints[i],nbins[i]) for i in range(4)]
1. Let's now run a short simulation and observe those discrete environment values. Feel free to try both `discretize` and `discretize_bins` and see if there is a difference.
def discretize_bins(x):
return tuple(np.digitize(x[i],bins[i]) for i in range(4))
```
✅ discretize_bins returns the bin number, which is 0-based. Thus for values of input variable around 0 it returns the number from the middle of the interval (10). In discretize, we did not care about the range of output values, allowing them to be negative, thus the state values are not shifted, and 0 corresponds to 0. (code block 8)
Let's now run a short simulation and observe those discrete environment values. Feel free to try both `discretize` and `discretize_bins` and see if there is a difference.
> **Note**: `discretize_bins` returns the bin number, which is 0-based, thus for values of input variable around 0 it returns the number from the middle of the interval (10). In `discretize`, we did not care about the range of output values, allowing them to be negative, thus the state values are not shifted, and 0 corresponds to 0. (code block 8)
```python
env.reset()
```python
env.reset()
done = False
while not done:
#env.render()
obs, rew, done, info = env.step(env.action_space.sample())
#print(discretize_bins(obs))
print(discretize(obs))
env.close()
```
done = False
while not done:
#env.render()
obs, rew, done, info = env.step(env.action_space.sample())
#print(discretize_bins(obs))
print(discretize(obs))
env.close()
```
> **Note**: Uncomment the line starting with `env.render` if you want to see how the environment executes. Otherwise you can execute it in the background, which is faster. We will use this "invisible" execution during our Q-Learning process.
✅ Uncomment the line starting with env.render if you want to see how the environment executes. Otherwise you can execute it in the background, which is faster. We will use this "invisible" execution during our Q-Learning process.
## The Q-Table structure
@ -161,91 +174,95 @@ In our previous lesson, the state was a simple pair of numbers from 0 to 8, and
However, sometimes precise dimensions of the observation space are not known. In case of the `discretize` function, we may never be sure that our state stays within certain limits, because some of the original values are not bound. Thus, we will use a slightly different approach and represent Q-Table by a dictionary.
We will use the pair *(state,action)* as the dictionary key, and the value would correspond to Q-Table entry value. (code block 9)
1. Use the pair *(state,action)* as the dictionary key, and the value would correspond to Q-Table entry value. (code block 9)
```python
Q = {}
actions = (0,1)
```python
Q = {}
actions = (0,1)
def qvalues(state):
return [Q.get((state,a),0) for a in actions]
```
def qvalues(state):
return [Q.get((state,a),0) for a in actions]
```
Here we also define a function `qvalues()`, which returns a list of Q-Table values for a given state that corresponds to all possible actions. If the entry is not present in the Q-Table, we will return 0 as the default.
Here we also define a function `qvalues`, which returns a list of Q-Table values for a given state that corresponds to all possible actions. If the entry is not present in the Q-Table, we will return 0 as the default.
## Let's start Q-Learning
## Let's start Q-Learning!
Now we are ready to teach Peter to balance!
Now we are ready to teach Peter to balance! First, let's set some hyperparameters: (code block 10)
1. First, let's set some hyperparameters: (code block 10)
```python
# hyperparameters
alpha = 0.3
gamma = 0.9
epsilon = 0.90
```
```python
# hyperparameters
alpha = 0.3
gamma = 0.9
epsilon = 0.90
```
Here, `alpha` is the **learning rate** that defines to which extent we should adjust the current values of Q-Table at each step. In the previous lesson we started with 1, and then decreased `alpha` to lower values during training. In this example we will keep it constant just for simplicity, and you can experiment with adjusting `alpha` values later.
Here, `alpha` is the **learning rate** that defines to which extent we should adjust the current values of Q-Table at each step. In the previous lesson we started with 1, and then decreased `alpha` to lower values during training. In this example we will keep it constant just for simplicity, and you can experiment with adjusting `alpha` values later.
`gamma` is the **discount factor** that shows to which extent we should prioritize future reward over current reward.
`gamma` is the **discount factor** that shows to which extent we should prioritize future reward over current reward.
`epsilon` is the **exploration/exploitation factor** that determines whether we should prefer exploration to exploitation or vice versa. In our algorithm, we will in `epsilon` percent of the cases select the next action according to Q-Table values, and in the remaining number of cases we will execute a random action. This will allow us to explore areas of the search space that we have never seen before.
`epsilon` is the **exploration/exploitation factor** that determines whether we should prefer exploration to exploitation or vice versa. In our algorithm, we will in `epsilon` percent of the cases select the next action according to Q-Table values, and in the remaining number of cases we will execute a random action. This will allow us to explore areas of the search space that we have never seen before.
✅ In terms of balancing - choosing random action (exploration) would act as a random punch in the wrong direction, and the pole would have to learn how to recover the balance from those "mistakes"
✅ In terms of balancing - choosing random action (exploration) would act as a random punch in the wrong direction, and the pole would have to learn how to recover the balance from those "mistakes"
### Improve the algorithm
We can also make two improvements to our algorithm from the previous lesson:
* Calculating average cumulative reward over a number of simulations. We will print the progress each 5000 iterations, and we will average out our cumulative reward over that period of time. It means that if we get more than 195 point - we can consider the problem solved, with even higher quality than required.
- **Calculate average cumulative reward**, over a number of simulations. We will print the progress each 5000 iterations, and we will average out our cumulative reward over that period of time. It means that if we get more than 195 point - we can consider the problem solved, with even higher quality than required.
* We will calculate maximum average cumulative result`Qmax`, and we will store the Q-Table corresponding to that result. When you run the training you will notice that sometimes the average cumulative result starts to drop, and we want to keep the values of Q-Table that correspond to the best model observed during training.
We will also collect all cumulative rewards at each simulation at `rewards` vector for further plotting. (code block 11)
```python
def probs(v,eps=1e-4):
v = v-v.min()+eps
v = v/v.sum()
return v
Qmax = 0
cum_rewards = []
rewards = []
for epoch in range(100000):
obs = env.reset()
done = False
cum_reward=0
# == do the simulation ==
while not done:
s = discretize(obs)
if random.random()<epsilon:
# exploitation - chose the action according to Q-Table probabilities
- **Calculate maximum average cumulative result**,`Qmax`, and we will store the Q-Table corresponding to that result. When you run the training you will notice that sometimes the average cumulative result starts to drop, and we want to keep the values of Q-Table that correspond to the best model observed during training.
1. Collect all cumulative rewards at each simulation at `rewards` vector for further plotting. (code block 11)
```python
def probs(v,eps=1e-4):
v = v-v.min()+eps
v = v/v.sum()
return v
Qmax = 0
cum_rewards = []
rewards = []
for epoch in range(100000):
obs = env.reset()
done = False
cum_reward=0
# == do the simulation ==
while not done:
s = discretize(obs)
if random.random()<epsilon:
# exploitation - chose the action according to Q-Table probabilities
* We are very close to achieving the goal of getting 195 cumulative rewards over 100+ consecutive runs of the simulation, or we may have actually achieved it! Even if we get smaller numbers, we still do not know, because we average over 5000 runs, and only 100 runs is required in the formal criteria.
- **Close to our goal**. We are very close to achieving the goal of getting 195 cumulative rewards over 100+ consecutive runs of the simulation, or we may have actually achieved it! Even if we get smaller numbers, we still do not know, because we average over 5000 runs, and only 100 runs is required in the formal criteria.
* Sometimes the reward start to drop, which means that we can "destroy" already learnt values in the Q-Table with the ones that make the situation worse.
- **Reward starts to drop**. Sometimes the reward start to drop, which means that we can "destroy" already learnt values in the Q-Table with the ones that make the situation worse.
This is more clearly visible if we plot training progress.
This observation is more clearly visible if we plot training progress.
To make learning more stable, it makes sense to adjust some of our hyperparameters during training. In particular:
* For **learning rate**, `alpha`, we may start with values close to 1, and then keep decreasing the parameter. With time, we will be getting good probability values in the Q-Table, and thus we should be adjusting them slightly, and not overwriting completely with new values.
- **For learning rate**, `alpha`, we may start with values close to 1, and then keep decreasing the parameter. With time, we will be getting good probability values in the Q-Table, and thus we should be adjusting them slightly, and not overwriting completely with new values.
* We may want to increase the `epsilon` slowly, in order to explore less and exploit more. It probably makes sense to start with lower value of `epsilon`, and move up to almost 1.
- **Increase epsilon**. We may want to increase the `epsilon` slowly, in order to explore less and exploit more. It probably makes sense to start with lower value of `epsilon`, and move up to almost 1.
> **Task 1**: Play with hyperparameter values and see if you can achieve higher cumulative reward. Are you getting above 195?
@ -301,6 +318,7 @@ You should see something like this:
> **Task 3**: Here, we were using the final copy of Q-Table, which may not be the best one. Remember that we have stored the best-performing Q-Table into `Qbest` variable! Try the same example with the best-performing Q-Table by copying `Qbest` over to `Q` and see if you notice the difference.
@ -308,6 +326,7 @@ You should see something like this:
> **Task 4**: Here we were not selecting the best action on each step, but rather sampling with corresponding probability distribution. Would it make more sense to always select the best action, with the highest Q-Table value? This can be done by using `np.argmax` function to find out the action number corresponding to highers Q-Table value. Implement this strategy and see if it improves the balancing.
chris
4 years ago
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