1. Reinforcement Learning (1)
Bob Durrant
School of Computer Science
University of Birmingham
(Slides: Dr Ata Kabán)

2. Reinforcement Learning (1)
Learning by reinforcement
State-action rewards
Markov Decision Process
Policies and Value functions
Q-learning

3. Learning by reinforcement
Examples:
Learning to play Backgammon
Robot learning to dock on battery charger
Characteristics:
No direct training examples – delayed rewards instead
Need for exploration & exploitation
The environment is stochastic and unknown
The actions of the learner affect future rewards

4. Reinforcement Learning
Supervised Learning
Reinforcement Learning

5. Brief history & successes
Minsky’s PhD thesis (1954): Stochastic Neural-Analog Reinforcement Computer
Analogies with animal learning and psychology
TD-Gammon (Tesauro, 1992) – big success story
Job-shop scheduling for NASA space missions (Zhang and Dietterich, 1997)
Robotic soccer (Stone and Veloso, 1998) – part of the world-champion approach
‘An approximate solution to a complex problem can be better than a perfect solution to a simplified problem’

6. The RL problem States Actions Immediate rewards Eventual reward
Discount factor
from any starting state

7. Markov Decision Process (MDP)
MDP is a formal model of the RL problem
At each discrete time point
Agent observes state st and chooses action at
Receives reward rt from the environment and the state changes to st+1
Markov assumption:
rt=r(st,at) st+1=(st,at)
i.e. rt and st+1 depend only on the current state and action
In general, the functions r and  may not be deterministic and are not necessarily known to the agent

8. Agent’s Learning Task
Execute actions in environment, observe results and
Learn action policy that maximises
from any starting state in S.
Here is the discount factor for future rewards
Note:
Target function is
There are no training examples of the form (s,a) but only of the form ((s,a),r)

9. Example: TD-Gammon Immediate reward:
+100 if win
-100 if lose
0 for all other states
Trained by playing 1.5 million games against itself
Now approximately equal to the best human player

10. Example: Mountain-Car
States: position and velocity
Actions: accelerate forward, accelerate backward, coast
Rewards
Reward=-1for every step, until the car reaches the top
Reward=1 at the top, 0 otherwise, <1
The eventual reward will be maximised by minimising the number of steps to the top of the hill

11. Value function We will consider deterministic worlds first
Given a policy (adopted by the agent), define an evaluation function over states:
Property:

12. Example Grid world environment Six possible states
Arrows represent possible actions
G: goal state
One optimal policy – denoted *
What is the best thing to do when in each state?
Compute the values of the states for this policy – denoted V*

13. Restated, the task is to learn the optimal policy
r(s,a) (immediate reward) values: V*(s) values, with =0.9
one optimal policy:
V*(s6) = *0 = 100
V*(s5) = *100 = 90
V*(s4) = *90 = 81
Restated, the task is to learn the optimal policy

14. The task, revisited
We might try to have the agent learn the evaluation function V*
It could then do a look ahead search to chose the best action from any state using
… yes, if we knew both the transition function  and the reward function r. In general these are unknown to the agent, so it cannot choose actions this way 
BUT: there is a way to do it!! 

15. Q function Define a new function very similar to V*
What difference does this make?
If the agent learns Q, then it can choose the optimal actions even without knowing 
Let us see how.

16. Rewrite things replacing this new definition:
Now, let denote the agent’s current approximation to Q. Consider the iterative update rule. Under some assumptions (<s,a> visited infinitely often), this will converge to the true Q:

17. Q Learning algorithm (in deterministic worlds)
For each (s,a) initialise table entry
Observe current state s
Do forever:
Select an action a and execute it
Receive immediate reward r
Observe new state s’
Update table entry as follows:
s:=s’

18. Example updating Q
given the Q values from a previous iteration on the arrows

19. Sketch of the convergence proof of Q-learning
Consider the case of deterministic world, where each (s,a) is visited infinitely often.
Define a full interval as an interval during which each (s,a) is visited. It can be easily shown, that during any such interval, the absolute value of the largest error in table is reduced by a factor of .
Consequently, as <1, then after infinitely many updates, the largest error converges to zero.
Go through the details from [Mitchell, sec. 13.3,6.]

20. An Observation
As a consequence of the convergence proof, Q-learning need not train on optimal action sequences in order to converge to the optimal policy. It can learn the Q function (and hence the optimal policy) while training from actions chosen at random as long as the resulting training sequence visits every ordered (state, action) pair infinitely often.

21. Exploration versus Exploitation
The Q-learning algorithm doesn’t say how we could choose an action
If we choose an action that maximises our estimate of Q we could end up not exploring better alternatives
To converge on the true Q values we must favour higher estimated Q values but still have a chance of choosing worse estimated Q values for exploration (see the convergence proof of the Q-learning algorithm in [Mitchell, sec ]). An action selection function of the following form may employed, where k>0:

22. Nondeterministic case
What if the reward and the state transition are not deterministic? – e.g. in Backgammon learning and playing depends on rolls of dice!
Then V and Q need to be redefined using their expected values:
Similar reasoning and convergent update iteration will apply
Will continue next week.

23. Summary
Reinforcement learning is suitable for learning in uncertain environments where rewards may be delayed and subject to chance
The goal of a reinforcement learning program is to maximise the eventual reward
Q-learning is a form of reinforcement learning that doesn’t require that the learner has prior knowledge of how its actions affect the environment