1. Reinforcement Learning
Known
Unknown
Assumed
Current state
Available actions
Experienced rewards
Transition model
Reward structure
Markov transitions
Fixed reward for (s,a,s’)
Problem: Find values for fixed policy  (policy evaluation)
Model-based learning: Learn the model, solve for values
Model-free learning: Solve for values directly (by sampling)

2. The Story So Far: MDPs and RL
Things we know how to do:
Techniques:
If we know the MDP
Compute V*, Q*, * exactly
Evaluate a fixed policy 
If we don’t know the MDP
We can estimate the MDP then solve
We can estimate V for a fixed policy 
We can estimate Q*(s,a) for the optimal policy while executing an exploration policy
Model-based DPs
Value and policy Iteration
Policy evaluation
Model-based RL
Model-free RL:
Value learning
Q-learning

3. Recap: Optimal Utilities
The utility of a state s:
V*(s) = expected utility starting in s and acting optimally
The utility of a q-state (s,a):
Q*(s,a) = expected utility starting in s, taking action a and thereafter acting optimally
The optimal policy:
*(s) = optimal action from state s
s is a state s a
(s, a) is a q-state
s, a
(s,a,s’) is a transition
s,a,s’ s’

4. Model-Free Learning Model-free (temporal difference) learning s
Experience world through episodes
Update estimates each transition
Over time, updates will mimic Bellman updates a
s, a r s’
Q-Value Iteration (model-based, requires known MDP)
Q-Learning (model-free, requires only experienced transitions)

5. Temporal-Difference Learning
Big idea: learn from every experience!
Update V(s) each time we experience (s,a,s’,r)
Likely s’ will contribute updates more often
Temporal difference learning
Policy still fixed!
Move values toward value of whatever successor occurs: running average! s
(s)
s, (s) s’
Sample of V(s):
Update to V(s):
Same update:

6. Exponential Moving Average
Makes recent samples more important
Forgets about the past (distant past values were wrong anyway)
Easy to compute from the running average
Decreasing learning rate can give converging averages

7. Example: TD Policy Evaluation
(1,1) up -1
(1,2) up -1
(1,3) right -1
(2,3) right -1
(3,3) right -1
(3,2) up -1
(4,3) exit +100
(done)
(1,1) up -1
(1,2) up -1
(1,3) right -1
(2,3) right -1
(3,3) right -1
(3,2) up -1
(4,2) exit -100
(done)
Take  = 1,  = 0.5

8. Problems with TD Value Learning
TD value leaning is a model-free way to do policy evaluation
However, if we want to turn values into a (new) policy, we’re sunk:
Idea: learn Q-values directly
Makes action selection model-free too! a s
s, a
s,a,s’ s’

9. Active Learning Full reinforcement learning In this case:
You don’t know the transitions T(s,a,s’)
You don’t know the rewards R(s,a,s’)
You can choose any actions you like
Goal: learn the optimal policy
… what value iteration did!
In this case:
Learner makes choices!
Fundamental tradeoff: exploration vs. exploitation
This is NOT offline planning! You actually take actions in the world and find out what happens…

10. Model-Based Active Learning
In general, want to learn the optimal policy, not evaluate a fixed policy
Idea: adaptive dynamic programming
Learn an initial model of the environment:
Solve for the optimal policy for this model (value or policy iteration)
Refine model through experience and repeat
Crucial: we have to make sure we actually learn about all of the model

11. Example: Greedy ADP
Imagine we find the lower path to the good exit first
Some states will never be visited following this policy from (1,1)
We’ll keep re-using this policy because following it never collects the regions of the model we need to learn the optimal policy ? ?

12. What Went Wrong?
Problem with following optimal policy for current model:
Never learn about better regions of the space if current policy neglects them
Fundamental tradeoff: exploration vs. exploitation
Exploration: must take actions with suboptimal estimates to discover new rewards and increase eventual utility
Exploitation: once the true optimal policy is learned, exploration reduces utility
Systems must explore in the beginning and exploit in the limit ? ?

13. Detour: Q-Value Iteration
Value iteration: find successive approx optimal values
Start with V0*(s) = 0, which we know is right (why?)
Given Vi*, calculate the values for all states for depth i+1:
But Q-values are more useful!
Start with Q0*(s,a) = 0, which we know is right (why?)
Given Qi*, calculate the q-values for all q-states for depth i+1:

14. Q-Learning We’d like to do Q-value updates to each Q-state:
But can’t compute this update without knowing T, R
Instead, compute average as we go
Receive a sample transition (s,a,r,s’)
This sample suggests
But we want to average over results from (s,a) (Why?)
So keep a running average

15. Q-Learning Properties
Will converge to optimal policy
If you explore enough (i.e. visit each q-state many times)
If you make the learning rate small enough
Basically doesn’t matter how you select actions (!)
Off-policy learning: learns optimal q-values, not the values of the policy you are following S E S E

16. Exploration / Exploitation
Several schemes for forcing exploration
Simplest: random actions ( greedy)
Every time step, flip a coin
With probability , act randomly
With probability 1-, act according to current policy
Regret: expected gap between rewards during learning and rewards from optimal action
Q-learning with random actions will converge to optimal values, but possibly very slowly, and will get low rewards on the way
Results will be optimal but regret will be large
How to make regret small?

17. Exploration Functions
When to explore
Random actions: explore a fixed amount
Better ideas: explore areas whose badness is not (yet) established, explore less over time
One way: exploration function
Takes a value estimate and a count, and returns an optimistic utility, e.g (exact form not important)

18. Q-Learning
Q-learning produces tables of q-values:

19. Q-Learning
In realistic situations, we cannot possibly learn about every single state!
Too many states to visit them all in training
Too many states to hold the q-tables in memory
Instead, we want to generalize:
Learn about some small number of training states from experience
Generalize that experience to new, similar states
This is a fundamental idea in machine learning, and we’ll see it over and over again

20. Example: Pacman
Let’s say we discover through experience that this state is bad:
In naïve q learning, we know nothing about this state or its q states:
Or even this one!

21. Feature-Based Representations
Solution: describe a state using a vector of features (properties)
Features are functions from states to real numbers (often 0/1) that capture important properties of the state
Example features:
Distance to closest ghost
Distance to closest dot
Number of ghosts
1 / (dist to dot)2
Is Pacman in a tunnel? (0/1)
…… etc.
Is it the exact state on this slide?
Can also describe a q-state (s, a) with features (e.g. action moves closer to food)

22. Linear Feature Functions
Using a feature representation, we can write a q function (or value function) for any state using a few weights:
Advantage: our experience is summed up in a few powerful numbers
Disadvantage: states may share features but actually be very different in value!

23. Function Approximation
Q-learning with linear q-functions:
Intuitive interpretation:
Adjust weights of active features
E.g. if something unexpectedly bad happens, disprefer all states with that state’s features
Formal justification: online least squares
Exact Q’s
Approximate Q’s

24. Example: Q-Pacman

25. Linear Regression Prediction Prediction10 20 30 40 22 24 26 40 20 20
Figure 1: scatter(1:20,10+(1:20)+2*randn(1,20),'k','filled'); a=axis; a(3)=0; axis(a);
Prediction
Prediction

26. Ordinary Least Squares (OLS)
Error or “residual”
Observation
Prediction
Figure 1: scatter(1:20,10+(1:20)+2*randn(1,20),'k','filled'); a=axis; a(3)=0; axis(a); 20

27. Minimizing Error Imagine we had only one point x with features f(x):
Approximate q update explained:
“target”
“prediction”

28. Overfitting Degree 15 polynomial 30 25 20 15 10 5 -5 -10 -15 2 4 6 8-5
-10
-15 2 4 6 8 10 12 14 16 18 20

29. Policy Search

30. Policy Search
Problem: often the feature-based policies that work well aren’t the ones that approximate V / Q best
E.g. your value functions from project 2 were probably horrible estimates of future rewards, but they still produced good decisions
We’ll see this distinction between modeling and prediction again later in the course
Solution: learn the policy that maximizes rewards rather than the value that predicts rewards
This is the idea behind policy search, such as what controlled the upside-down helicopter

31. Policy Search Simplest policy search: Problems:
Start with an initial linear value function or q-function
Nudge each feature weight up and down and see if your policy is better than before
Problems:
How do we tell the policy got better?
Need to run many sample episodes!
If there are a lot of features, this can be impractical

32. Policy Search* Advanced policy search:
Write a stochastic (soft) policy:
Turns out you can efficiently approximate the derivative of the returns with respect to the parameters w (details in the book, optional material)
Take uphill steps, recalculate derivatives, etc.

33. Take a Deep Breath… We’re done with search and planning!
Next, we’ll look at how to reason with probabilities
Diagnosis
Tracking objects
Speech recognition
Robot mapping
… lots more!
Last part of course: machine learning