$1.$ Q$-$Learning $[35$ Points$]$

This time, although the Gridworld looks similar, it is not an MDP anymore. That means, the only information you get from the game object is game.get$\_$actions$($state: State$),$ which returns a set of Actions. All other methods have been removed.

You will also see that the GUI adds a new episode label. Each iteration is a single update, and each episode is a collections of iterations such that the agent starts from the starting state at the beginning of an episode and reaches a terminal state at the end of an episode.

A stub of a Q$-$learner is specified in QLearningAgent. A QLearningAgent takes in

game, an object to get a set of available actions for a state ss$.$

discount, the discount factor $\backslash$gamma $\backslash$gamma $.$

learning$\_$rate, the learning rate $\backslash$alpha $\backslash$alpha $.$

explore$\_$prob, the probability of exploration $$ for each iteration.

Note that the states in Gridworld aren$$t given as well. Your code should be able to deal with a new state and a new $($state$,$ action$)$ pair.

Then you need to implement the following methods for this part:

agent.get$\_$q$\_$value$($state$,$ action$)$ returns the Q$-$value for $($state$,$ action$)$ pair. Q$($s$,$a$)$Q$($s$,$a$)$ from Q$-$table. For a never seen pair, the Q$-$value should be $00.$

agent.get$\_$value$($state$)$ returns the value of the state V$($s$)$V$($s$).$

agent.get$\_$best$\_$policy$($state$)$ returns the best policy of the state, $\backslash$pi $($s$)\backslash$pi $($s$).$

agent.update$($state$,$ action, next$\_$state, reward$)$ updates the Q$-$value for $($state$,$ action$)$ pair, based on the value of the next$\_$state and reward given.

Note: For get$\_$best$\_$policy, you should break ties randomly for better behavior. The random.choice$()$ function will help.

For more instructions, refer to the lecture slides and comments in the skeleton file.

Important: Make sure that in your get$\_$value and get$\_$best$\_$policy functions, you only access Q$-$values by calling get$\_$q$\_$value. This abstraction will be useful for the Approximate Q$-$learning you will implement later when you override get$\_$q$\_$value to use features of state$-$action pairs rather than state$-$action pairs directly.

With the Q$-$learning update in place, you can watch your Q$-$learner learn under manual control, using the keyboard arrow keys:

python gridworld.py

Hint: to help

# template

class QLearningAgent:

$"""$Implement Q Reinforcement Learning Agent using Q$-$table."""

def $\_\_$init$\_\_($self$,$ game, discount, learning$\_$rate, explore$\_$prob$)$:

$"""$Store any needed parameters into the agent object.

Initialize Q$-$table.

$"""$

$...$ # TODO

def get$\_$q$\_$value$($self$,$ state, action$)$:

$"""$Retrieve Q$-$value from Q$-$table.

For an never seen $($s$,$a$)$ pair, the Q$-$value is by default $0.$

$"""$

return $0$ # TODO

def get$\_$value$($self$,$ state$)$:

$"""$Compute state value from Q$-$values using Bellman Equation.

V$($s$)=$ max$\_$a Q$($s$,$a$)$

$"""$

return $0$ # TODO

def get$\_$best$\_$policy$($self$,$ state$)$:

$"""$Compute the best action to take in the state using Policy Extraction.

$\backslash$pi $($s$)=$ argmax$\_$a Q$($s$,$a$)$

If there are ties, return a random one for better performance.

Hint: use random.choice$().$

$"""$

return None # TODO

def update$($self$,$ state, action, next$\_$state, reward$)$:

$"""$Update Q$-$values using running average.

Q$($s$,$a$)=(1-\backslash$alpha $)$ Q$($s$,$a$)+\backslash$alpha $($R $+\backslash$gamma V$($s$\text{'}))$

Where $\backslash$alpha is the learning rate, and $\backslash$gamma is the discount.

Note: You should not call this function in your code.

$"""$

$...$ # TODO

# $2.$ Epsilon Greedy

def get$\_$action$($self$,$ state$)$:

$"""$Compute the action to take for the agent, incorporating exploration.

That is$,$ with probability $\backslash$epsi $,$ act randomly.

Otherwise, act according to the best policy.

Hint: use random.random$()<\backslash$epsi to check if exploration is needed.

$"""$

return None # TODO

# $3.$ Bridge Crossing Revisited

def question$3()$:

epsilon $=...$

learning$\_$rate $=...$

return epsilon, learning$\_$rate

# If not possible, return 'NOT POSSIBLE'

# $5.$ Approximate Q$-$Learning

class ApproximateQAgent$($QLearningAgent$)$:

$"""$Implement Approximate Q Learning Agent using weights."""

def $\_\_$init$\_\_($self$,*$args$,$ extractor$)$:

$"""$Initialize parameters and store the feature extractor.

Initialize weights table."""

super$().\_\_$init$\_\_(*$args$)$

$...$ # TODO

def get$\_$weight$($self$,$ feature$)$:

$"""$Get weight of a feature.

Never seen feature should have a weight of $0.$

$"""$

return $0$ # TODO

def get$\_$q$\_$value$($self$,$ state, action$)$:

$"""$Compute Q value based on the dot product of feature components and weights.

Q$($s$,$a$)=$ w$\_1*$ f$\_1($s$,$a$)+$ w$\_2*$ f$\_2($s$,$a$)+...+$ w$\_$n $*$ f$\_$n$($s$,$a$)$

$"""$

return $0$ # TODO

def update$($self$,$ state, action, next$\_$state, reward$)$:

$"""$Update weights using least$-$squares approximation.

$\backslash$Delta $=$ R $+\backslash$gamma V$($s$\text{'})-$ Q$($s$,$a$)$

Then update weights: w$\_$i $=$ w$\_$i $+\backslash$alpha $*\backslash$Delta $*$ f$\_$i$($s$,$ a$)$

$"""$

$...$ # TODO