1. Chapter 5: Monte Carlo Methods
Monte Carlo methods learn from complete sample returns
Only defined for episodic tasks
Monte Carlo methods learn directly from experience
On-line: No model necessary and still attains optimality
Simulated: No need for a full model
R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction

2. Monte Carlo Policy Evaluation
Goal: learn Vp(s)
Given: some number of episodes under p which contain s
Idea: Average returns observed after visits to s 1 2 3 4 5
Every-Visit MC: average returns for every time s is visited in an episode
First-visit MC: average returns only for first time s is visited in an episode
Both converge asymptotically
R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction

3. First-visit Monte Carlo Policy Evaluation
Initialize:
p policy to be evaluated
V an arbitrary state-value function
Returns(s) empty list, for all seS
Repeat forever:
Generate an episode using p
For each state s appearing in the episode
R return following the first occurrence of s
Append R to Return(s)
V(s) average(Returns(s))
R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction

4. Blackjack example
Object: Have your card sum be greater than the dealers without exceeding 21.
States (200 of them):
current sum (12-21)
dealer’s showing card (ace-10)
do I have a useable ace?
Reward: +1 for winning, 0 for a draw, -1 for losing
Actions: stick (stop receiving cards), hit (receive another card)
Policy: Stick if my sum is 20 or 21, else hit
R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction

5. Blackjack Value Functions
After many MC state visit evaluations the state-value function is well approximated
Dynamic Programming parameters difficult to formulate here!
For instance with a given hand and a decision to stay what would be the expected return value?
R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction

6. Backup Diagram for Monte Carlo
Entire episode is included while in DP only one step transitions
Only one choice at each state (unlike DP which uses all possible transitions in one step)
Estimates for all states are independent so MC does not bootstrap (build on other estimates)
Time required to estimate one state does not depend on the total number of states
R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction

7. The Power of Monte Carlo
Example - Elastic Membrane (Dirichlet Problem)
How do we compute the shape of the membrane or bubble attached to a fixed frame?
R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction

8. Two Approaches Relaxation Kakutani’s algorithm, 1945
Iterate on the grid
and compute averages
(like DP iterations)
Kakutani’s algorithm, 1945
Use many random walks and
average the boundary points values
(like MC approach)
R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction

9. Monte Carlo Estimation of Action Values (Q)
Monte Carlo is most useful when a model is not available
We want to learn Q*
Qp(s,a) - average return starting from state s and action a following p
Converges asymptotically if every state-action pair is visited
To assure this we must maintain exploration to visit many state-action pairs
Exploring starts: Every state-action pair has a non-zero probability of being the starting pair
R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction

10. Monte Carlo Control (to approximate an optimal policy)
Generalized Policy
Iteration GPI
MC policy iteration: Policy evaluation by approximating Qp using MC methods, followed by policy improvement
Policy improvement step: greedy with respect to Qp (action-value) function – no model needed to construct greedy policy
R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction

11. Convergence of MC Control
Policy improvement theorem tells us:
This assumes exploring starts and infinite number of episodes for MC policy evaluation
To solve the latter:
update only to a given level of performance
alternate between evaluation and improvement per episode
R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction

12. Monte Carlo Exploring Starts
Fixed point is optimal policy p*
Proof is one of the fundamental questions of RL
R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction

13. Blackjack Example continued
Exploring starts – easy to enforce by generating state-action pairs randomly
Initial policy as described before (sticks only at 20 or 21)
Initial action-value function equal to zero
R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction

14. On-policy Monte Carlo Control
On-policy: learn or improve the policy currently executing
How do we get rid of exploring starts?
Need soft policies: p(s,a) > 0 for all s and a
e.g. e-soft policy
probability of action selection:
greedy action
non-max actions
Similar to GPI: move policy towards greedy policy (i.e. e-soft)
Converges to best e-soft policy
R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction

15. On-policy MC Control
R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction

16. Learning about p while following another
R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction

17. Off-policy Monte Carlo control
On-policy estimates the policy value while following it
so the policy to generate behavior and estimate its result is identical
Off-policy assumes separate behavior and estimation policies
Behavior policy generates behavior in environment
may be randomized to sample all actions
Estimation policy is policy being learned about
may be deterministic (greedy)
Off-policy estimates one policy while following another
Average returns from behavior policy by using nonzero probabilities of selecting all actions for the estimation policy
May be slow to find improvement after nongreedy action
R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction

18. Off-policy MC control
R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction

19. Incremental Implementation
MC can be implemented incrementally
saves memory
Compute the weighted average of each return
non-incremental
incremental equivalent
R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction

20. Racetrack Exercise
States: grid squares, velocity horizontal and vertical
Rewards: -1 on track, -5 off track
Only the right turns allowed
Actions: +1, -1, 0 to velocity
0 < Velocity < 5
Stochastic: 50% of the time it moves 1 extra square up or right
R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction

21. Summary MC has several advantages over DP:
Can learn directly from interaction with environment
No need for full models
No need to learn about ALL states
Less harm by Markovian violations (later in book)
MC methods provide an alternate policy evaluation process
One issue to watch for: maintaining sufficient exploration
exploring starts, soft policies
No bootstrapping (as opposed to DP)
R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction