Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch
1
Organizing Principles for Learning
in the Brain
Associative Learning:
Hebb rule and variations, self

organizing maps
Adaptive Hedonism:
Brain seeks pleasure and avoids pain: conditioning and
reinforcement learning
Imitation:
Brain specially set up to learn from other brains?
Imitation learning approaches
Supervised Learning:
Of course brain has no explicit teacher, but timing of development
may lead to some circuits being trained by others
Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch
2
Classical Conditioning and
Reinforcement Learning
Outline:
1.
classical conditioning and its variations
2.
Rescorla Wagner rule
3.
instrumental conditioning
4.
Markov decision processes
5.
reinforcement learning
Note: this presentation follows a chapter of “Theoretical Neuroscience” by Dayan&Abbott
Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch
3
Example of embodied models of reward based learning:
Skinnerbots in Touretzky’s lab at CMU:
http://www

2.cs.cmu.edu/~dst/Skinnerbots/index.html
Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch
4
Project Goals
We are developing computational theories of operant conditioning.
While classical (Pavlovian) conditioning has a well

developed theory,
implemented in the Rescorla

Wagner model and its descendants (work
by Sutton & Barto, Grossberg, Klopf, Gallistel, and others), there is at
present no comprehensive theory of operant conditioning. Our work
has four components:
1. Develop computationally explicit models of operant conditioning that
reproduce classical animal learning experiments with rats, dogs, pigeons,
etc.
2. Demonstrate the workability of these models by implementing them on
mobile robots, which then become
trainable robots
(Skinnerbots). We
originally used
Amelia
, a B21 robot manufactured by
Real World Interface
,
as our implementation platform. We are moving to the Sony AIBO.
3. Map our computational theories onto neuroanatomical structures known
to be involved in animal learning, such as the hippocampus, amygdala, and
striatum.
4. Explore issues in human

robot interaction that arise when non

scientists
try to train robots as if they were animals.
also at: http://www

2.cs.cmu.edu/~dst/Skinnerbots/index.html
Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch
5
Classical Conditioning
Pavlov’s classic finding:
(classical conditioning)
Initially, sight of food leads to dog salivating:
food salivating
unconditioned stimulus, US unconditioned response, UR
(reward)
Sound of bell consistently precedes food. Afterwards, bell leads
to salivating:
bell salivating
conditioned stimulus, CS conditioned response, CR
(expectation of reward)
Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch
6
Variations of Conditioning 1
Extinction:
Stimulus (bell) repeatedly shown without reward (food):
conditioned response (salivating) reduced.
Partial reinforcement:
Stimulus only sometimes preceding reward:
conditioned response weaker than in classical case.
Blocking (2 stimuli):
First: stimulus S1 associated with reward: classical conditioning.
Then: stimulus S1 and S2 shown together followed by reward:
Association between S2 and reward
not
learned.
Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch
7
Variations of Conditioning 2
Inhibitory Conditioning (2 stimuli):
Alternate 2 types of trials:
1. S1 followed by reward.
2. S1+S2 followed by absence of reward.
Result: S2 becomes predictor of absence of reward.
To show this use for example the following 2 methods:
A. train animal to predict reward based on S2.
Result: learning slowed
B. train animal to predict reward based on S3, then show S2+S3.
Result: conditioned response weaker than for S3 alone.
Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch
8
Variations of Conditioning 3
Overshadowing (2 stimuli):
Repeatedly present S1+S2 followed by reward.
Result: often, reward prediction shared unequally between stimuli.
Example (made up):
red light + high pitch beep precede pigeon food.
Result: red light more effective in predicting the food than
high pitch beep.
Secondary Conditioning:
S1 preceding reward (classical case). Then, S2 preceding S1.
Result: S2 leads to prediction of reward.
But: if S1 following S2 showed too often: extinction will occur
Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch
9
Summary of Conditioning Findings
(incomplete, has been studied extensively for decades,
many books on topic)
figure taken from Dayan&Abbott
Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch
10
Modeling Conditioning
The Rescorla Wagner rule
(1972):
Consider
stimulus variable
u
representing presence (
u
=1) or
absence (
u
=0) of stimulus. Correspondingly,
reward variable
r
represents presence or absence of reward.
The
expected reward
v
is modeled as “stimulus
x
weight”:
v
=
wu
Learning is done by adjusting the weight to minimize error
between predicted reward and actual reward.
Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch
11
Rescorla Wagner Rule
Denote the
prediction error
by
δ
(delta):
δ
=
r

v
Learning rule:
w
:=
w
+
ε
δ
u ,
where
ε
is a learning rate.
Q: Why is this useful?
A: This rule does
stochastic gradient descent
to minimize the
expected squared error (
r

v
)
2
, w converges to <r>. R.W. rule
is variant of the “delta rule” in neural networks.
Note: in psychological terms the learning rate is measure
of
associability
of stimulus with reward.
u
u
wu
r
wu
r
w
v
r
w
w
2
)
)(
(
2
)
(
)
(
2
2
2
Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch
12
Rescorla Wagner Rule Example
prediction error
δ
=
r

v;
learning rule:
w
:=
w
+
ε
δ
u
figure taken from Dayan&Abbott
Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch
13
Multiple Stimuli
Essentially the same idea/learning rule:
In case of multiple stimuli:
v
=
w∙u
(predicted reward = dot product of stimulus vector and weight vector)
Prediction error
:
δ
=
r

v
Learning rule:
w
:=
w
+
ε
δ
u
i
j
j
j
i
i
i
i
u
u
w
w
r
r
w
v
r
w
w
2
)
(
2
)
(
)
(
2
2
2
u
w
u
w
Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch
14
In how far does Rescorla Wagner rule account for variants
of classical conditioning?
(prediction:
v
=
w∙u
; error:
δ
=
r

v; learning:
w
:=
w
+
ε
δ
u
)
figure taken from Dayan&Abbott
Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch
15
(prediction:
v
=
w∙u
; error:
δ
=
r

v; learning:
w
:=
w
+
ε
δ
u
)
Extinction, Partial Reinforcement:
o.k., since w converges to <r>
Blocking:
during pre

training, w
1
converges to r. During training
v=w
1
u
1
+w
2
u
2
=r, hence
δ
=0 and w
2
does not grow.
Inhibitory Conditioning:
on S1 only trials, w
1
gets positive value.
on S1+S2 trials, v=w
1
+w
2
must converge to zero, hence w
2
becoming negative.
Overshadow:
v=w
1
+w
2
goes to r, but w
1
and w
2
may become
different if there are different learning rates
ε
i
for them.
Secondary Conditioning:
R.

W.

rule predicts negative S2 weight!
Rescorla Wagner rule qualitatively accounts for wide range
of conditioning phenomena but not secondary conditioning.
Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch
16
Temporal Difference Learning
Motivation
: need to keep track of time within a trial
Idea
: (Sutton&Barto, 1990)
Try to predict the total future reward expected from time t onward
to the time T of end of trial. Assume time is in discrete steps.
Predicted
total future reward from time t (one stimulus case):
t
T
t
r
t
R
0
)
(
)
(
t
t
u
w
t
v
0
)
(
)
(
)
(
Problem
: how to adjust the weight? Would like to adjust w(
τ
)
to make v(t) approximate the true total future reward R(t)
(reward that is yet to come) but this is unknown since lying in future.
Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch
17
TD Learning cont’d.
t
T
t
r
t
R
0
)
(
)
(
t
t
u
w
t
v
0
)
(
)
(
)
(
Solution
: (Temporal Difference Learning Rule)
)
(
)
(
)
(
)
(
t
u
t
w
w
, with
)
(
)
1
(
)
(
)
(
t
v
t
v
t
r
t
temporal
difference
To see why this makes sense:
1
0
0
)
1
(
)
(
)
(
t
T
t
T
t
r
t
r
t
r
We want
v
(
t
) to approximate left hand side but also:
v
(
t
+1) should
approximate 2
nd
term of right hand side. Hence:
)
1
(
)
(
)
(
)
(
0
t
v
t
r
t
r
t
v
t
T
or
)
1
(
)
(
)
(
t
v
t
v
t
r
Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch
18
TD Learning Rule Example
figure taken from Dayan&Abbott
)
(
)
(
)
(
)
(
t
u
t
w
w
)
(
)
1
(
)
(
)
(
t
v
t
v
t
r
t
;
Note:
temporal difference
learning rule can also
account for secondary
conditioning
(sorry, no example)
reward and time course
of reward correctly
predicted!
Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch
19
Dopamine and Reward Prediction
figure taken from Dayan&Abbott
(VTA=
ventral
tegmental
area
(midbrain))
VTA neurons
fire for unex

pected reward:
seem to re

present the
prediction
error
δ
Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch
20
Instrumental Conditioning
So far
: only concerned with prediction of reward.
Didn’t consider agent’s
actions
. Reward usually depends on what
you do!
Skinner
boxes
, etc.
Distinguish two scenarios:
A.
Rewards follow actions immediately (
Static Action Choice
)
Example: n

armed bandit (slot machine)
B. Rewards may be delayed (
Sequential Action Choice
)
Example: playing chess
Goal:
choose actions to maximize rewards
Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch
21
Static Action Choice
Consider bee foraging:
Bee can choose to fly to blue or yellow flowers,
wants to maximize nectar volume.
Bees learn to fly to “better” flower in single session (~40 flowers)
Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch
22
Simple model of bee foraging
When bee chooses blue: reward ~p(r
b
)
or yellow: reward ~p(r
y
)
Assume model bee has
stochastic policy
:
chooses to fly to blue or yellow flower with p(b) or p(y) respectively.
A “convenient” assumption: p(b), p(y) follow
softmax decision rule
:
Notes:
p(b)+p(y)=1; m
b
, m
y
are
action values
to be adjusted;
β
: inverse temperature: big
β
deterministic behavior
)
exp(
)
exp(
)
exp(
)
(
y
b
b
m
m
m
b
p
)
exp(
)
exp(
)
exp(
)
(
y
b
y
m
m
m
y
p
))
(
(
)
(
y
b
m
m
b
p
)
exp(
1
/
1
)
(
x
x
, where
Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch
23
Exploration

Exploitation dilemma
Why use softmax action selection?
Idea
: bee could also choose “better” action all the time.
But
: bee can’t be sure that better action is really better action.
Bee needs to test and continuously verify which action leads
to higher rewards.
This is the famous
exploration

exploitation dilemma
of
reinforcement learning:
Need to
explore
to know what’s good.
Need to
exploit
what you know is good to maximize reward.
Generalization of softmax
to many possible actions:
a
N
a
a
a
m
m
a
p
1
'
'
)
exp(
)
exp(
)
(
Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch
24
The Indirect Actor
Question:
how to adjust the action values
m
a
?
Idea
: have action values adapt to average reward for that action:
m
b
= <
r
b
> and
m
y
= <
r
y
>
This can be achieved with simple delta rule:
m
b
m
b
+
εδ
, where
δ
=
r
b

m
b
This is
indirect
actor because action choice is mediated
indirectly by expected amounts of rewards.
Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch
25
Indirect Actor Example
figure taken from
Dayan&Abbott
Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch
26
The Direct Actor
figure taken from Dayan&Abbott
Idea
: choose action values directly to maximize
expected reward
Maximize this expected reward by stochastic gradient ascent:
y
b
r
y
p
r
b
p
r
)
(
)
(
)
)
(
)
(
)
(
)
(
(
y
b
b
r
b
p
y
p
r
y
p
b
p
m
r
)
))(
(
(
r
r
b
p
m
m
a
ab
b
b
This leads to the following learning rule:
where
δ
ab
is the “Kronecker delta” and
r
is a parameter often chosen
to be an estimate of the average reward per time.
¯
Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch
27
Direct Actor Example
figure taken from
Dayan&Abbott
again: nectar
volumes reversed
after first 100 visits
Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch
28
Sequential Action Choice
So far:
immediate reward after each action (n

armed bandit problem)
Now:
delayed rewards
, can be in
different states
Example:
Maze Task
figure taken from
Dayan&Abbott
Amount of reward after decision at second intersection depends on
action taken at first intersection.
Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch
29
Policy Iteration
Big body of research on how to solve this and more complicated
tasks, easily filling an entire course by itself. Here we just consider
one example method:
policy iteration
.
Assumption:
state is
fully observable
(in contrast to only
partially
observable
), i.e. the rat knows exactly where it is at any time.
Idea:
maintain and improve a
stochastic policy
, determining actions
at each decision point (A,B,C) using action
values and softmax decision.
Two elements:
critic:
use temporal difference learning to predict
future rewards from A,B,C if current policy is followed
actor:
maintain and improve the policy
figure taken from Dayan&Abbott
Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch
30
Policy Iteration cont’d.
How to formalize this idea?
Introduce
state variable
u
to describe whether rat is at A,B,C.
Also introduce
action value vector
m
(
u
) describing the policy.
(softmax rule assigns probability of action
a
based on action values)
Immediate reward for taking action
a
in state
u
:
r
a
(
u
)
Expected future reward for starting in state
u
and following
current policy:
v
(
u
) (
state value
).
The rat’s estimate for this is denoted by
w
(
u
).
Policy Evaluation
(critic): estimate w(u) using
temporal difference learning.
Policy Improvement
(actor): improve action
values
m
(
u
)
based on estimated state values.
figure taken from Dayan&Abbott
Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch
31
Policy Evaluation
Initially, assume all action values are 0, i.e.
left/right equally likely everywhere.
True value of each state can be found
by inspection:
v(B) = ½(5+0)=2.5; v(C) = ½(2+0)=1;
v(A) = ½(v(B)+v(C))=1.75.
These values can be learned with temporal difference learning rule:
)
(
)
(
u
w
u
w
with
)
(
)
'
(
)
(
u
v
u
v
u
r
a
where
u’
is the state that results from taking action
a
in state
u
.
figure taken from Dayan&Abbott
Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch
32
Policy Evaluation Example
)
(
)
(
u
w
u
w
with
)
(
)
'
(
)
(
u
v
u
v
u
r
a
figures taken from
Dayan&Abbott
Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch
33
Policy Improvement
)
(
)
'
(
)
(
u
v
u
v
u
r
a
figures taken from
Dayan&Abbott
))
;
'
(
(
)
(
)
(
'
'
'
u
a
p
u
m
u
m
aa
a
a
How to adjust action values?
where
and p(
a
’;
u
) is the softmax probability of chosing action
a’
in
state
u
as determined by
m
a’
(u)
.
Example: consider starting out from random policy and assume
state value estimates
w
(u) are accurate. Consider
u
=A, leads to
75
.
0
)
(
)
(
0
A
v
B
v
75
.
0
)
(
)
(
0
A
v
C
v
for left turn
for right turn
rat will increase
probability of
going left in A
Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch
34
Policy Improvement Example
figures taken from
Dayan&Abbott
Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch
35
Some Extensions

Introduction of a
state vector
u

discounting
of future rewards: put more
emphasis on rewards in the near future
than rewards that are far away.
Note: reinforcement learning is big subfield
of machine learning. There is a good
introductory textbook by Sutton and Barto.
Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch
36
Questions to discuss/think about
1.
Even at one level of abstraction there are many different
“Hebbian”, or Reinforcement learning rules;
is it important
which one you use? What is the right one?
2.
The applications we discussed in Hebbian and Reinforcement
learning considered networks passively receiving simple
sensory input and learning to code it or behave “well”;
how
can we model learning through
interaction
with complex
environments? Why might it be important to do so?
3.
The problems we considered so far are very “low

level”, no
hint of “complex behaviors” yet.
How can we bridge this huge
divide? How can we “scale up”? Why is it difficult?
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