1. Introduction to Reinforcement Learning
E0397 Lecture
Slides by Richard Sutton
(With small changes)

2. Reinforcement Learning: Basic Idea
Learn to take correct actions over time by experience
Similar to how humans learn: “trial and error”
Try an action –
“see” what happens
“remember” what happens
Use this to change choice of action next time you are in the same situation
“Converge” to learning correct actions
Focus on long term return, not immediate benefit
Some action may seem not beneficial for short term, but its long term return will be good.

3. RL in Cloud Computing Why are we talking about RL in this class?
Which problems due you think can be solved by a learning approach?
Is focus on long-term benefit appropriate?

4. RL Agent Object is to affect the environment
Environment is stochastic and uncertain
Environment
action
state
reward
Agent

5. What is Reinforcement Learning?
An approach to Artificial Intelligence
Learning from interaction
Goal-oriented learning
Learning about, from, and while interacting with an external environment
Learning what to do—how to map situations to actions—so as to maximize a numerical reward signal

6. Key Features of RL Learner is not told which actions to take
Trial-and-Error search
Possibility of delayed reward
Sacrifice short-term gains for greater long-term gains
The need to explore and exploit
Considers the whole problem of a goal-directed agent interacting with an uncertain environment

7. Examples of Reinforcement Learning
Robocup Soccer Teams Stone & Veloso, Reidmiller et al.
World’s best player of simulated soccer, 1999; Runner-up 2000
Inventory Management Van Roy, Bertsekas, Lee & Tsitsiklis
10-15% improvement over industry standard methods
Dynamic Channel Assignment Singh & Bertsekas, Nie & Haykin
World's best assigner of radio channels to mobile telephone calls
Elevator Control Crites & Barto
(Probably) world's best down-peak elevator controller
Many Robots
navigation, bi-pedal walking, grasping, switching between skills...
TD-Gammon and Jellyfish Tesauro, Dahl
World's best backgammon player

8. Supervised Learning Error = (target output – actual output)
Training Info = desired (target) outputs
Supervised Learning
System
Inputs
Outputs
Error = (target output – actual output)
Best examples: Classification/identification systems. E.g. fault classification, WiFi location determination, workload identification
E.g. Classification systems: give agent lots of data where data has been already classified. Agent can “train” itself using this knowledge.
Training phase is non trivial
(Normally) no learning happens in on-line phase

9. Reinforcement Learning
Training Info = evaluations (“rewards” / “penalties”) RL
System
Inputs
Outputs (“actions”)
Objective: get as much reward as possible
Absolutely no “already existing training data”
Agent learns ONLY by experience

10. Elements of RL State: state of the system
Actions: Suggested by RL agent, taken by actuators in the system
Actions change state (deterministically, or stochastically)
Reward: Immediate gain when taking action a in state s
Value: Long term benefit of taking action a in state s
Rewards gained over a long time horizon

11. The n-Armed Bandit Problem
Choose repeatedly from one of n actions; each choice is called a play
After each play , you get a reward , where
These are unknown action values
Distribution of depends only on
Objective is to maximize the reward in the long term, e.g., over 1000 plays
To solve the n-armed bandit problem,
you must explore a variety of actions
and exploit the best of them

12. The Exploration/Exploitation Dilemma
Suppose you form estimates
The greedy action at t is at
You can’t exploit all the time; you can’t explore all the time
You can never stop exploring; but you should always reduce exploring. Maybe.
action value estimates

13. Action-Value Methods
Methods that adapt action-value estimates and nothing else, e.g.: suppose by the t-th play, action had been chosen times, producing rewards then
“sample average”

14. e-Greedy Action Selection{
. . . the simplest way to balance exploration and exploitation

15. 10-Armed Testbed n = 10 possible actions
Each is chosen randomly from a normal distrib.:
each is also normal:
1000 plays
repeat the whole thing 2000 times and average the results

16. e-Greedy Methods on the 10-Armed Testbed

17. Incremental Implementation
Recall the sample average estimation method:
The average of the first k rewards is
(dropping the dependence on ):
Can we do this incrementally (without storing all the rewards)?
We could keep a running sum and count, or, equivalently:
This is a common form for update rules:
NewEstimate = OldEstimate + StepSize[Target – OldEstimate]

18. Tracking a Nonstationary Problem
Choosing to be a sample average is appropriate in a
stationary problem, i.e., when none of the
change over time,
But not in a nonstationary problem.
Better in the nonstationary case is:
exponential, recency-weighted average

19. Formal Description of RL

20. The Agent-Environment Interface
. . . s a r
t +1
t +2
t +3

21. The Agent Learns a Policy
Reinforcement learning methods specify how the agent changes its policy as a result of experience.
Roughly, the agent’s goal is to get as much reward as it can over the long run.

22. Getting the Degree of Abstraction Right
Time steps need not refer to fixed intervals of real time.
Actions can be low level (e.g., voltages to motors), or high level (e.g., accept a job offer), “mental” (e.g., shift in focus of attention), etc.
States can low-level “sensations”, or they can be abstract, symbolic, based on memory, or subjective (e.g., the state of being “surprised” or “lost”).
An RL agent is not like a whole animal or robot.
Reward computation is in the agent’s environment because the agent cannot change it arbitrarily.
The environment is not necessarily unknown to the agent, only incompletely controllable.

23. Goals and Rewards
Is a scalar reward signal an adequate notion of a goal?—maybe not, but it is surprisingly flexible.
A goal should specify what we want to achieve, not how we want to achieve it.
A goal must be outside the agent’s direct control—thus outside the agent.
The agent must be able to measure success:
explicitly;
frequently during its lifespan.

24. Returns
Episodic tasks: interaction breaks naturally into episodes, e.g., plays of a game, trips through a maze.
where T is a final time step at which a terminal state is reached, ending an episode.

25. Returns for Continuing Tasks
Continuing tasks: interaction does not have natural episodes.
Discounted return:

26. An Example Avoid failure: the pole falling beyond
a critical angle or the cart hitting end of
track.
As an episodic task where episode ends upon failure:
As a continuing task with discounted return:
In either case, return is maximized by
avoiding failure for as long as possible.

27. A Unified Notation
In episodic tasks, we number the time steps of each episode starting from zero.
We usually do not have to distinguish between episodes, so we write instead of for the state at step t of episode j.
Think of each episode as ending in an absorbing state that always produces reward of zero:
We can cover all cases by writing

28. The Markov Property
“the state” at step t, means whatever information is available to the agent at step t about its environment.
The state can include immediate “sensations,” highly processed sensations, and structures built up over time from sequences of sensations.
Ideally, a state should summarize past sensations so as to retain all “essential” information, i.e., it should have the Markov Property:

29. Markov Decision Processes
If a reinforcement learning task has the Markov Property, it is basically a Markov Decision Process (MDP).
If state and action sets are finite, it is a finite MDP.
To define a finite MDP, you need to give:
state and action sets
one-step “dynamics” defined by transition probabilities:
Expected values of reward:

30. An Example Finite MDP Recycling Robot
At each step, robot has to decide whether it should (1) actively search for a can, (2) wait for someone to bring it a can, or (3) go to home base and recharge.
Searching is better but runs down the battery; if runs out of power while searching, has to be rescued (which is bad).
Decisions made on basis of current energy level: high, low.
Reward = number of cans collected

31. Recycling Robot MDP

32. Value Functions
The value of a state is the expected return starting from that state; depends on the agent’s policy:
The value of taking an action in a state under policy p is the expected return starting from that state, taking that action, and thereafter following p :

33. Bellman Equation for a Policy p
The basic idea:
So:
Or, without the expectation operator:

34. More on the Bellman Equation
This is a set of equations (in fact, linear), one for each state.
The value function for p is its unique solution.
Backup diagrams:

35. Optimal Value Functions
For finite MDPs, policies can be partially ordered:
There are always one or more policies that are better than or equal to all the others. These are the optimal policies. We denote them all p *.
Optimal policies share the same optimal state-value function:
Optimal policies also share the same optimal action-value function:
This is the expected return for taking action a in state s and thereafter following an optimal policy.

36. Bellman Optimality Equation for V*
The value of a state under an optimal policy must equal
the expected return for the best action from that state:
The relevant backup diagram:
is the unique solution of this system of nonlinear equations.

37. Bellman Optimality Equation for Q*
The relevant backup diagram:
is the unique solution of this system of nonlinear equations.

38. Gridworld Actions: north, south, east, west; deterministic.
If would take agent off the grid: no move but reward = –1
Other actions produce reward = 0, except actions that move agent out of special states A and B as shown.
State-value function
for equiprobable
random policy;
g = 0.9

39. Why Optimal State-Value Functions are Useful
Any policy that is greedy with respect to is an optimal policy.
Therefore, given , one-step-ahead search produces the
long-term optimal actions.
E.g., back to the gridworld: *

40. What About Optimal Action-Value Functions?
Given , the agent does not even
have to do a one-step-ahead search:

41. Solving the Bellman Optimality Equation
Finding an optimal policy by solving the Bellman Optimality Equation requires the following:
accurate knowledge of environment dynamics;
we have enough space and time to do the computation;
the Markov Property.
How much space and time do we need?
polynomial in number of states (via dynamic programming methods; Chapter 4),
BUT, number of states is often huge (e.g., backgammon has about 1020 states).
We usually have to settle for approximations.
Many RL methods can be understood as approximately solving the Bellman Optimality Equation.

42. Solving MDPs
The goal in MDP is to find the policy that maximizes long term reward (value)
If all the information is there, this can be done by explicitly solving, or by iterative methods

43. Policy Iteration
policy evaluation
policy improvement
“greedification”

44. Policy Iteration

45. From MDPs to reinforcement learning

46. MDPs vs RL MDPs are great if we know transition probabilities
And expected immediate rewards
But in real life tasks – this is precisely what we do NOT know!
This is what turns the task into a learning problem
Simplest method to learn: Monte Carlo

47. 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

48. 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

49. First-visit Monte Carlo policy evaluation

50. 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
Also converges asymptotically if every state-action pair is visited
Exploring starts: Every state-action pair has a non-zero probability of being the starting pair

51. Monte Carlo Control
MC policy iteration: Policy evaluation using MC methods followed by policy improvement
Policy improvement step: greedify with respect to value (or action-value) function

52. Monte Carlo Exploring Starts
Fixed point is optimal policy p*
Now proven (almost)
Problem with Monte Carlo – learning happens only after end of entire episode

53. TD Prediction Policy Evaluation (the prediction problem):
for a given policy p, compute the state-value function
Recall:
target: the actual return after time t
target: an estimate of the return

54. Simple Monte Carlo T T T T T T T T T T T

55. Simplest TD Method T T T T T T T T T T T

56. cf. Dynamic ProgrammingT T T T T T T T T T T T T

57. TD methods bootstrap and sample
Bootstrapping: update involves an estimate
MC does not bootstrap
DP bootstraps
TD bootstraps
Sampling: update does not involve an expected value
MC samples
DP does not sample
TD samples

58. Example: Driving Home
(5)
(15)
(10)
(10)
(3)

59. Driving Home Changes recommended by Monte Carlo methods (a=1)
by TD methods (a=1)

60. Advantages of TD Learning
TD methods do not require a model of the environment, only experience
TD, but not MC, methods can be fully incremental
You can learn before knowing the final outcome
Less memory
Less peak computation
You can learn without the final outcome
From incomplete sequences
Both MC and TD converge (under certain assumptions to be detailed later), but which is faster?

61. Random Walk Example Values learned by TD(0) after
various numbers of episodes

62. TD and MC on the Random Walk
Data averaged over
100 sequences of episodes

63. Optimality of TD(0)
Batch Updating: train completely on a finite amount of data,
e.g., train repeatedly on 10 episodes until convergence.
Compute updates according to TD(0), but only update
estimates after each complete pass through the data.
For any finite Markov prediction task, under batch updating,
TD(0) converges for sufficiently small a.
Constant-a MC also converges under these conditions, but to
a difference answer!

64. Random Walk under Batch Updating
After each new episode, all previous episodes were treated as a batch, and algorithm was trained until convergence. All repeated 100 times.

65. You are the Predictor Suppose you observe the following 8 episodes:
A, 0, B, 0
B, 1
B, 0

66. You are the Predictor

67. You are the Predictor
The prediction that best matches the training data is V(A)=0
This minimizes the mean-square-error on the training set
This is what a batch Monte Carlo method gets
If we consider the sequentiality of the problem, then we would set V(A)=.75
This is correct for the maximum likelihood estimate of a Markov model generating the data
i.e, if we do a best fit Markov model, and assume it is exactly correct, and then compute what it predicts (how?)
This is called the certainty-equivalence estimate
This is what TD(0) gets

68. Learning An Action-Value Function

69. Sarsa: On-Policy TD Control
Turn this into a control method by always updating the
policy to be greedy with respect to the current estimate:

70. Windy Gridworld
undiscounted, episodic, reward = –1 until goal

71. Results of Sarsa on the Windy Gridworld

72. Q-Learning: Off-Policy TD Control

73. Planning and Learning Use of environment models
Integration of planning and learning methods

74. The Original Idea
Sutton, 1990

75. The Original Idea
Sutton, 1990

76. Models
Model: anything the agent can use to predict how the environment will respond to its actions
Distribution model: description of all possibilities and their probabilities
e.g.,
Sample model: produces sample experiences
e.g., a simulation model
Both types of models can be used to produce simulated experience
Often sample models are much easier to come by

77. Planning
Planning: any computational process that uses a model to create or improve a policy
Planning in AI:
state-space planning
plan-space planning (e.g., partial-order planner)
We take the following (unusual) view:
all state-space planning methods involve computing value functions, either explicitly or implicitly
they all apply backups to simulated experience

78. Planning Cont. Classical DP methods are state-space planning methods
Heuristic search methods are state-space planning methods
A planning method based on Q-learning:
Random-Sample One-Step Tabular Q-Planning

79. Learning, Planning, and Acting
Two uses of real experience:
model learning: to improve the model
direct RL: to directly improve the value function and policy
Improving value function and/or policy via a model is sometimes called indirect RL or model-based RL. Here, we call it planning.

80. Direct vs. Indirect RL
Indirect methods:
make fuller use of experience: get better policy with fewer environment interactions
Direct methods
simpler
not affected by bad models
But they are very closely related and can be usefully combined:
planning, acting, model learning, and direct RL can occur simultaneously and in parallel

81. The Dyna Architecture (Sutton 1990)

82. The Dyna-Q Algorithm
direct RL
model learning
planning

83. Dyna-Q on a Simple Maze
rewards = 0 until goal, when =1

84. Value Prediction with FA
As usual: Policy Evaluation (the prediction problem):
for a given policy p, compute the state-value function
In earlier chapters, value functions were stored in lookup tables.

85. Adapt Supervised Learning Algorithms
Training Info = desired (target) outputs
Supervised Learning
System
Inputs
Outputs
Training example = {input, target output}
Error = (target output – actual output)