Reinforcement Learning 1 in explore-then-commit, epsilon-greedy, Boltzmann exploration

R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 1
Reinforcement Learning
Slides from
R.S. Sutton and A.G. Barto
Reinforcement Learning: An Introduction
http://www.cs.ualberta.ca/~sutton/book/the-book.html
http://rlai.cs.ualberta.ca/RLAI/RLAIcourse/RLAIcourse.html

The Agent-Environment Interface

Agent and environment interact at discrete time steps : t 0,1, 2, 
Agent observes state at step t: st S
produces action at step t : at  A(st )
gets resulting reward : rt 1 
and resulting next state : st1
t
. . .
st a
rt +1 st +1
t +1
a
rt +2 st +2
t +2
a
rt +3 st +3
. . .
t +3
a

Policy at step t, t :
a mapping from states to action probabilities
t (s, a)  probability that at a when st s
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.

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 be 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, which
consist of many RL agents as well as other components.
 The environment is not necessarily unknown to the agent,
only incompletely controllable.
 Reward computation is in the agent’s environment because
the agent cannot change it arbitrarily.

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.

Returns

Suppose the sequence of rewards after step t is :
rt 1, rt2 , rt 3, 
What do we want to maximize?
In general,
we want to maximize the expected return, E Rt
 , for each step t.
Episodic tasks: interaction breaks naturally into
episodes, e.g., plays of a game, trips through a maze.

Rt rt1  rt2   rT ,
where T is a final time step at which a terminal state is reached,
ending an episode.

Returns for Continuing Tasks
Continuing tasks: interaction does not have natural episodes.
Discounted return:

Rt rt1 rt2  2
rt 3   k
rtk1,
k 
0


where , 0  1, is the discount rate.
shortsighted 0    1 farsighted

An Example
Avoid failure: the pole falling beyond
a critical angle or the cart hitting end of
track.
reward 1 for each step before failure
 return  number of steps before failure
As an episodic task where episode ends upon failure:
As a continuing task with discounted return:
reward  1 upon failure; 0 otherwise
 return   k
, for k steps before failure
In either case, return is maximized by
avoiding failure for as long as possible.

Another Example
Get to the top of the hill
as quickly as possible.
reward  1 for each step where not at top of hill
 return   number of steps before reaching top of hill
Return is maximized by minimizing
number of steps reach the top of the hill.

A Unified Notation
 In episodic tasks, we number the time steps of each
episode starting from zero.
 We usually do not have 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
st st, j
Rt  k
rt k 1,
k 
0


where can be 1 only if a zero reward absorbing state is always reached.

The Markov Property
 By “the state” at step t, the book 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:

Pr st 1  
s ,rt1 r st ,at ,rt , st  1,at  1,,r1,s0 ,a0
 
Pr st 1  
s ,rt 1 r st ,at
 
for all 
s , r, and histories st ,at,rt , st  1,at 1,,r1, s0 ,a0.

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:

reward probabilities:
Ps 
s
a
Pr st 1  
s st s,at a
  for all s, 
s S, a A(s).
Rs 
s
a
E rt1 st s,at a,st1  
s
  for all s, 
s S, a A(s).

Recycling Robot
An Example Finite MDP
 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

Value Functions
State- value function for policy  :
V

(s) E Rt st s
 E k
rt k 1 st s
k 
0








Action - value function for policy  :
Q

(s, a) E Rt st s, at a
 E k
rt k1 st s,at a
k0








 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 
is the expected return starting from that state, taking that
action, and thereafter following  :

Bellman Equation for a Policy 

Rt rt1  rt 2 2
rt3 3
rt4 
rt1   rt2  rt 3  2
rt4 
 
rt1  Rt1
The basic idea:
So: V
(s) E Rt st s
 
E rt1  V st 1
 st s
 
Or, without the expectation operator:
V

(s)  (s,a) Ps 
s
a
Rs 
s
a
 V

( 
s )
 

s

a


More on the Bellman Equation
V

(s)  (s,a) Ps 
s
a
Rs 
s
a
 V

( 
s )
 

s

a

This is a set of equations (in fact, linear), one for each state.
The value function for  is its unique solution.
Backup diagrams:
for V
for Q

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;
= 0.9

  
 if and only if V
(s) V 

(s) for all s S
Optimal Value Functions
 For finite MDPs, policies can be partially ordered:
 There is always at least one (and possibly many) policies that
is better than or equal to all the others. This is an optimal
policy. We denote them all *.
 Optimal policies share the same optimal state-value function:
 Optimal policies also share the same optimal action-value
function:
V

(s) max

V

(s) for all s S
Q
(s,a) max

Q
(s, a) for all s S and a A(s)
This is the expected return for taking action a in state s
and thereafter following an optimal policy.

Bellman Optimality Equation for V*
V
(s) max
aA(s)
Q 
(s,a)
max
aA(s)
E rt 1  V

(st 1) st s, at a
 
max
aA(s)
Ps 
s
a

s
 Rs 
s
a
 V
( 
s )
 
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.
V


Bellman Optimality Equation for Q*
Q
(s,a) E rt1   max

a
Q
(st1, 
a ) st s,at a
 
 Ps 
s
a
Rs 
s
a
 max

a
Q
( 
s , 
a )
 

s

The relevant backup diagram:
is the unique solution of this system of nonlinear equations.
Q*

Why Optimal State-Value Functions are Useful
V

V
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:

What About Optimal Action-Value Functions?
Given , the agent does not even
have to do a one-step-ahead search:
Q*

(s) arg max
aA(s)
Q
(s,a)

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 an 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 10**20 states).
 We usually have to settle for approximations.
 Many RL methods can be understood as approximately solving the
Bellman Optimality Equation.

TD Prediction
Policy Evaluation (the prediction problem):
for a given policy , compute the state-value function V
The simplest TD method, TD(0):
V(st )  V(st )  rt 1  V(st1 )  V(st )
 
target: an estimate of the return

Simplest TD Method
T T T T
T
T T T T T
st1
rt1
st
V(st )  V(st )  rt 1  V(st1 )  V(st )
 
T
T
T
T
T
T T T T T

Example: Driving Home
State Elapsed Time
(minutes)
Predicted
Time to Go
Predicted
Total Time
leaving office 0 30 30
reach car,
raining
5 35 40
exit highway 20 15 35
behind truck 30 10 40
home street 40 3 43
arrive home 43 0 43

Driving Home
Changes recommended by
Monte Carlo methods =1)
Changes recommended
by TD methods (=1)

Advantages of TD Learning
 TD methods do not require a model of the environment,
only experience
 TD 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

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

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

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 .
Constant- MC also converges under these conditions, but to
a difference answer!

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.

Learning An Action-Value Function
Estimate Q
for the current behavior policy .
After every transition from a nonterminal state st, do this :
Q st , at
  Q st, at
   rt1  Q st 1,at 1
  Q st ,at
 
 
If st1 is terminal, then Q(st 1, at 1 ) 0.

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:

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

Results of Sarsa on the Windy Gridworld

Cliffwalking
greedy= 0.1

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