overview over different methods. different types/classes of learning unsupervised learning...

Machine Learning Classical Conditioning Synaptic Plasticity

Dynamic Prog.(Bellman Eq.)

REINFORCEMENT LEARNING UN-SUPERVISED LEARNINGexample based correlation based

d-Rule

Monte CarloControl

Q-Learning

TD( )often =0

ll

TD(1) TD(0)

Rescorla/Wagner

Neur.TD-Models(“Critic”)

Neur.TD-formalism

DifferentialHebb-Rule

(”fast”)

STDP-Modelsbiophysical & network

EVALUATIVE FEEDBACK (Rewards)

NON-EVALUATIVE FEEDBACK (Correlations)

SARSA

Correlationbased Control

(non-evaluative)

ISO-Learning

ISO-Modelof STDP

Actor/Critictechnical & Basal Gangl.

Eligibility Traces

Hebb-Rule


(”slow”)

supervised L.

Anticipatory Control of Actions and Prediction of Values Correlation of Signals

=

=

=

Neuronal Reward Systems(Basal Ganglia)

Biophys. of Syn. PlasticityDopamine Glutamate

STDP

LTP(LTD=anti)

ISO-Control

Overview over different methods

Different Types/Classes of Learning

Unsupervised Learning (non-evaluative feedback)

• Trial and Error Learning.

• No Error Signal.

• No influence from a Teacher, Correlation evaluation only.

Reinforcement Learning (evaluative feedback)

• (Classic. & Instrumental) Conditioning, Reward-based Lng.

• “Good-Bad” Error Signals.

• Teacher defines what is good and what is bad.

Supervised Learning (evaluative error-signal feedback)

• Teaching, Coaching, Imitation Learning, Lng. from examples and more.

• Rigorous Error Signals.

• Direct influence from a teacher/teaching signal.

Basic Hebb-Rule: = ui v << 1di

dt

For Learning: One input, one output.

An unsupervised learning rule:

A supervised learning rule (Delta Rule):

! i ! ! i à ör ! iE

No input, No output, one Error Function Derivative,where the error function compares input- with output-examples.

A reinforcement learning rule (TD-learning):

One input, one output, one reward.

wi ! wi + ö[r(t + 1) + í v(t + 1) à v(t)]uà(t)

map

Self-organizing maps:unsupervised learning

Neighborhood relationships are usually preserved (+)

Absolute structure depends on initial condition and cannot be predicted (-)

input

C:\Program Files\VideoLAN\VLC\vlc.exe C:\Users\worgott\Documents\AAA_DriveD\Lehre\neurocomp\Biology\SOM\som1.avi


dt

For Learning: One input, one output



! i ! ! i à ör ! iE



One input, one output, one reward


I. Pawlow

Classical Conditioning


dt

For Learning: One input, one output



! i ! ! i à ör ! iE



One input, one output, one reward


Supervised Learning: Example OCR

The influence of the type of learning on speed and autonomy of the learner

Correlation based learning: No teacher

Reinforcement learning , indirect influence

Reinforcement learning, direct influence

Supervised Learning, Teacher

Programming

Learning Speed Autonomy

Hebbian learning

AB

A

B

t

When an axon of cell A excites cell B and repeatedly or persistently takes part in firing it, some growth processes or metabolic change takes place in one or both cells so that A‘s efficiency ... is increased.

Donald Hebb (1949)




d-Rule

Monte CarloControl

Q-Learning

TD( )often =0

ll

TD(1) TD(0)

Rescorla/Wagner


Neur.TD-formalism


(”fast”)




SARSA


(non-evaluative)

ISO-Learning

ISO-Modelof STDP


Eligibility Traces

Hebb-Rule


(”slow”)

supervised L.


=

=

=



STDP

LTP(LTD=anti)

ISO-Control


You are here !

Hebbian Learning

…Basic Hebb-Rule:

…correlates inputs with outputs by the…

= v u1 << 1d

dt

vu1

Vector Notation

Cell Activity: v = w . u

This is a dot product, where w is a weight vector and uthe input vector. Strictly we need to assume that weightchanges are slow, otherwise this turns into a differential eq.

= v u1 << 1d

dtSingle Input

= v u << 1dw

dtMany Inputs

As v is a single output, it is scalar.

Averaging Inputs= <v u> << 1

dw

dt

We can just average over all input patterns and approximate the weight change by this. Remember, this assumes that weight changes are slow.

If we replace v with w . u we can write:

= Q . w where Q = <uu> is the input correlation matrix

dw

dt

Note: Hebb yields an instable (always growing) weight vector!

Synaptic plasticity evoked artificially

Examples of Long term potentiation (LTP)and long term depression (LTD).

LTP First demonstrated by Bliss and Lomo in 1973. Since then induced in many different ways, usually in slice.

LTD, robustly shown by Dudek and Bear in 1992, in Hippocampal slice.

LTP will lead to new synaptic contacts

Conventional LTP = Hebbian Learning

Symmetrical Weight-change curve

Pre

tPre

Post

tPost

Synaptic change %

Pre

tPre

Post

tPost

The temporal order of input and output does not play any role

Spike timing dependent plasticity - STDP

Markram et. al. 1997

Pre follows Post:Long-term Depression

Pre

tPre

Post

tPost

Synaptic

change %

Spike Timing Dependent Plasticity: Temporal Hebbian Learning

Weight-change curve (Bi&Poo, 2001)

Pre

tPre

Post

tPost

Pre precedes Post:Long-term Potentiation

Aca

usal

Causal

(possibly)

= v u1 << 1d

dtSingle Input

= v u << 1dw

dtMany Inputs

As v is a single output, it is scalar.

Averaging Inputs= <v u> << 1

dw

dt

We can just average over all input patterns and approximate the weight change by this. Remember, this assumes that weight changes are slow.

If we replace v with w . u we can write:

= Q . w where Q = <uu> is the input correlation matrix

dw

dt

Note: Hebb yields an instable (always growing) weight vector!

Back to the Math. We had:

= (v - ) u << 1dw

dt

Covariance Rule(s)

Normally firing rates are only positive and plain Hebb would yield only LTP.Hence we introduce a threshold to also get LTD

Output threshold

= v (u - << 1dw

dtInput vector threshold

Many times one sets the threshold as the average activity of somereference time period (training period)

= <v> or = <u> together with v = w . u we get:

= C . w, where C is the covariance matrix of the input

dw

dthttp://en.wikipedia.org/wiki/Covariance_matrix

C = <(u-<u>)(u-<u>)> = <uu> - <u2> = <(u-<u>)u>

The covariance rule can produce LTP without (!) post-synaptic output.This is biologically unrealistic and the BCM rule (Bienenstock, Cooper,Munro) takes care of this.

BCM- Rule

= vu (v - ) << 1dw

dt

As such this rule is again unstable, but BCM introduces a sliding threshold

= (v2 - ) < 1d

dt

Note the rate of threshold change should be faster than then weight

changes (), but slower than the presentation of the individual inputpatterns. This way the weight growth will be over-dampened relative to the (weight – induced) activity increase.

Evidence for weight normalization:Reduced weight increase as soon as weights are already big(Bi and Poo, 1998, J. Neurosci.)

Problem: Hebbian Learning can lead to unlimited weight growth.

Solution: Weight normalizationa) subtractive (subtract the mean change of all weights from each individual weight).b) multiplicative (mult. each weight by a gradually decreasing factor).

Examples of Applications • Kohonen (1984). Speech recognition - a map of

phonemes in the Finish language• Goodhill (1993) proposed a model for the

development of retinotopy and ocular dominance, based on Kohonen Maps (SOM)

• Angeliol et al (1988) – travelling salesman problem (an optimization problem)

• Kohonen (1990) – learning vector quantization (pattern classification problem)

• Ritter & Kohonen (1989) – semantic maps

OD ORI

Differential Hebbian Learning of SequencesLearning to act in response to

sequences of sensor events




d-Rule

Monte CarloControl

Q-Learning

TD( )often =0

ll

TD(1) TD(0)

Rescorla/Wagner


Neur.TD-formalism


(”fast”)




SARSA


(non-evaluative)

ISO-Learning

ISO-Modelof STDP


Eligibility Traces

Hebb-Rule


(”slow”)

supervised L.


=

=

=



STDP

LTP(LTD=anti)

ISO-Control


You are here !

I. Pawlow

History of the Concept of TemporallyAsymmetrical Learning: Classical Conditioning

I. Pawlow


Correlating two stimuli which are shifted with respect to each other in time.

Pavlov’s Dog: “Bell comes earlier than Food”

This requires to remember the stimuli in the system.

Eligibility Trace: A synapse remains “eligible” for modification for some time after it was active (Hull 1938, then a still abstract concept).

0 = 1

1

Unconditioned Stimulus (Food)

Conditioned Stimulus (Bell)

Response

X

+Stimulus Trace E

The first stimulus needs to be “remembered” in the system

Classical Conditioning: Eligibility Traces

I. Pawlow


Eligibility Traces

Note: There are vastly different time-scales for (Pavlov’s) hehavioural experiments:

Typically up to 4 seconds

as compared to STDP at neurons:

Typically 40-60 milliseconds (max.)

Defining the TraceIn general there are many ways to do this, but usually one chooses a trace that looks biologically realistic and allows for some analytical calculations, too.

EPSP-like functions:-function:

Double exp.:

This one is most easy to handle analytically and, thus, often used.

DampenedSine wave:

Shows an oscillation.

h(t) =n

0 t<0hk(t) tõ 0

h(t) = teà atk

h(t) = b1sin(bt) eà at

k

h(t) = î1(eà at à eà bt)

k




d-Rule

Monte CarloControl

Q-Learning

TD( )often =0

ll

TD(1) TD(0)

Rescorla/Wagner


Neur.TD-formalism


(”fast”)




SARSA


(non-evaluative)

ISO-Learning

ISO-Modelof STDP


Eligibility Traces

Hebb-Rule


(”slow”)

supervised L.


=

=

=



STDP

LTP(LTD=anti)

ISO-Control


Mathematical formulation of learning rules is

similar but time-scales are much different.

Early: “Bell”

Late: “Food”

x

)( )( )( tytutdt

dii

Differential Hebb Learning Rule

Xi

X0

Simpler Notationx = Inputu = Traced Input

V

V’(t)

ui

u0

Convolution used to define the traced input,

Correlation used to calculate weight growth.

)()()()()()()( xfxgxgxfduuxgufxh

u

)()()()()()()( xgxfxfxgduxugufxh

w

Produces asymmetric weight change curve(if the filters h produce unimodal „humps“)

)(' )( )( tvtutdt

dii

Derivative of the Output

Filtered Input

)( )()( tuttv ii

Output

T

Differential Hebbian Learning

Conventional LTP

Symmetrical Weight-change curve

Pre

tPre

Post

tPost

Synaptic change %

Pre

tPre

Post

tPost

The temporal order of input and output does not play any role

Produces asymmetric weight change curve(if the filters h produce unimodal „humps“)

)(' )( )( tvtutdt

dii

Derivative of the Output

Filtered Input

)( )()( tuttv ii

Output

T

Differential Hebbian Learning

Weight-change curve

(Bi&Poo, 2001)

T=tPost - tPrems

Pre follows Post:Long-term Depression

Pre

tPre

Post

tPost

Synaptic change %Pre

tPre

Post

tPost

Pre precedes Post:Long-term Potentiation

Spike-timing-dependent plasticity(STDP): Some vague shape similarity




d-Rule

Monte CarloControl

Q-Learning

TD( )often =0

ll

TD(1) TD(0)

Rescorla/Wagner


Neur.TD-formalism


(”fast”)




SARSA


(non-evaluative)

ISO-Learning

ISO-Modelof STDP


Eligibility Traces

Hebb-Rule


(”slow”)

supervised L.


=

=

=



STDP

LTP(LTD=anti)

ISO-Control


You are here !

PlasticSynapse

NMDA/AMPA

Postsynaptic:Source of Depolarization

The biophysical equivalent of Hebb’s postulate

Presynaptic Signal(Glu)

Pre-Post Correlation,but why is this needed?

i n

o u t

i n

o u t

Plasticity is mainly mediated by so calledN-methyl-D-Aspartate (NMDA) channels.

These channels respond to Glutamate as their transmitter andthey are voltage depended:

Biophysical Model: Structure

x NMDA synapse

v

Hence NMDA-synapses (channels) do require a (hebbian) correlation between pre and post-synaptic activity!

Source of depolarization:

1) Any other drive (AMPA or NMDA)

2) Back-propagating spike

Local Events at the Synapse

Local

Current sources “under” the synapse:• Synaptic current

Isynaptic

GlobalIBP

• Influence of a Back-propagating spike

• Currents from all parts of the dendritic tree

IDendritic

u1

x1

v

Pre-syn. Spike

BP- or D-Spike

* 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 2 4 6 8 10

V*h

gNMDA

0 40 80 t [ms]

g [nS ]NM DA

0.1

On „Eligibility Traces“

Membrane potential:

Weight Synaptic input

Depolarization source

deprest

iii

ii IR

tVVVEttV

dt

dC

)(

))((g )()(

1

0

X

v

v’

ISO-Learning

h

h

x

x0

1

• Dendritic compartment

• Plastic synapse with NMDA channels Source of Ca2+ influx and coincidence detector

PlasticSynapse NMDA/AMPA

depi

ii IVEt~dt

dV ))((g

NMDA/AMPAgBP spike

Source of Depolarization

Dendritic spike

• Source of depolarization: 1. Back-propagating spike 2. Local dendritic spike

Model structure

Plasticity Rule(Differential Hebb)

NMDA synapse -Plastic synapse

depi

ii IVEtdt

dV ))((g ~

NMDA/AMPAg

NMDA/AMPA

Source of depolarization

Instantenous weight change:

)(' )( )( tFtctdt

dN

Presynaptic influence Glutamate effect on NMDA channels

Postsynaptic influence

0 40 80 t [ms]

g [nS ]NM DA

0.1

Normalized NMDA conductance:

NMDA channels are instrumental for LTP and LTD induction (Malenka and Nicoll, 1999; Dudek and Bear ,1992)

V

tt

N eMg

eec

][1 2

// 21

Pre-synaptic influence


depi

ii IVEtdt

dV ))((g ~

NMDA/AMPAg

NMDA/AMPA


)(' )( )( tFtctdt

dN

0 10

0

-40

-60

-20

20V [m V]

20 t [m s]

0 10

0

-40

-60

-20

20V [m V]

20 t [ms]

0 10

0

-40

-60

-20

20V [m V]

20 t [m s]

0 10

0

-40

-60

-20

20V [m V]

20 t [m s]

Dendriticspikes

Back-propagating spikes

(Larkum et al., 2001

Golding et al, 2002

Häusser and Mel, 2003)

(Stuart et al., 1997)

Depolarizing potentials in the dendritic tree


depi

ii IVEtdt

dV ))((g ~

NMDA/AMPAg

NMDA/AMPA


Postsyn. Influence

)(' )( )( tFtctdt

dN

For F we use a low-pass filtered („slow“) version of a back-propagating or a dendritic spike.

0 10

0

-40

-60

-20

20V [m V]

20 t [m s]

0 10

0

-40

-60

-20

20V [m V]

20 t [m s]

0 50 150 t [ms]100

0

-40

-60

-20

V [mV]

0 5 0 1 5 0 t [ m s ]1 0 0

0

- 4 0

- 6 0

- 2 0

V [ m V ]

0 20 80 t [ms]40 60

0

-40

-60

-20

V [mV]

0 20 80 t [ms]40 60

0

-40

-60

-20

V [mV]

0 10

0

-40

-60

-20

20V [m V]

20 t [ms]

0 10

0

-40

-60

-20

20V [m V]

20 t [m s]

BP and D-Spikes

0 10

0

-40

-60

-20

20V [m V]

20 t [m s]

0 10

0

-40

-60

-20

20V [m V]

20 t [m s]

0-20 40 T [ms]-40 20

-0.01

-0.03

-0.01

0.01

0-20 40 T [ms]-40 20

-0.01

-0.03

-0.01

0.01

Back-propagating spike

Weight change curve

T

NMDAr activation

Back-propagating spike

T=tPost – tPre

Weight Change Curves

Source of Depolarization: Back-Propagating Spikes

CLOSED LOOP LEARNING

• Learning to Act (to produce appropriate behavior)

• Instrumental (Operant) Conditioning

This is an open-loopsystem

Sensor 2

conditionedInput

Bell Food Salivation

Pavlov, 1927

Temporal Sequence

This is an Open Loop System !

Adaptable

Neuron

Env.

Closed loop

Sensing Behaving

Instrumental/Operant Conditioning

Behaviorism“All we need to know in order

to describe and explain behavior is this: actions

followed by good outcomes are likely to recur, and actions followed by bad

outcomes are less likely to recur.” (Skinner, 1953)

Skinner had invented the type of experiments called operant conditioning.

B.F. Skinner (1904-1990)

Operant behavior: occurs without an observable external stimulus. Operates on the organism’s environment. The behavior is instrumental in securing a stimulus more representative of everyday learning.

Skinner Box

OPERANT CONDITIONING TECHNIQUES

• POSITIVE REINFORCEMENT = increasing a behavior by administering a reward

• NEGATIVE REINFORCEMENT = increasing a behavior by removing an aversive stimulus when a behavior occurs

• PUNISHMENT = decreasing a behavior by administering an aversive stimulus following a behavior OR by removing a positive stimulus

• EXTINCTION = decreasing a behavior by not rewarding it




d-Rule

Monte CarloControl

Q-Learning

TD( )often =0

ll

TD(1) TD(0)

Rescorla/Wagner


Neur.TD-formalism


(”fast”)




SARSA


(non-evaluative)

ISO-Learning

ISO-Modelof STDP


Eligibility Traces

Hebb-Rule


(”slow”)

supervised L.


=

=

=



STDP

LTP(LTD=anti)

ISO-Control


You are here !

How to assure behavioral & learning convergence ??

This is achieved by starting with a stable reflex-like actionand learning to supercede it by an anticipatory action.

Remove beforebeing hit !

C o n tro lle rC o n tro lled

S ystemC o n tro lS ign a ls

F e edb ack

D is tu rba ncesS e t-P o in t

X 0

Reflex Only

(Compare to an electronic closed loop controller!)

This structure assures initial (behavioral) stability (“homeostasis”)

Think of a Thermostat !

Robot Application

x

Early: “Vision”

Late: “Bump”

Robot Application

Initially built-in behavior: Retraction reaction whenever an obstacle is touched.

Learning Goal:Correlate the vision signals with the touch signals and navigate without collisions.

Robot Example

ControllerControlled

SystemControlSignals

Feedback

DisturbancesSet-Point

X0X1early late

What has happened during learningto the system ?

The primary reflex re-action has effectively been eliminatedand replaced by an anticipatory action




d-Rule

Monte CarloControl

Q-Learning

TD( )often =0

ll

TD(1) TD(0)

Rescorla/Wagner


Neur.TD-formalism


(”fast”)




SARSA


(non-evaluative)

ISO-Learning

ISO-Modelof STDP


Eligibility Traces

Hebb-Rule


(”slow”)

supervised L.


=

=

=



STDP

LTP(LTD=anti)

ISO-Control

Overview over different methods – Supervised Learning

You are

her

e !

And many more

Supervised learningmethods are mostlynon-neuronal andwill therefore not

be discussedhere.

Reinforcement Learning (RL)

Learning from rewards (and punishments)

Learning to assess the value of states.

Learning goal directed behavior.

RL has been developed rather independently from two different fields:

1) Dynamic Programming and Machine Learning (Bellman Equation).

2) Psychology (Classical Conditioning) and later Neuroscience (Dopamine System in the brain)

I. Pawlow

Back to Classical Conditioning

U(C)S = Unconditioned Stimulus

U(C)R = Unconditioned Response

CS = Conditioned Stimulus

CR = Conditioned Response

Less “classical” but also Conditioning !(Example from a car advertisement)

Learning the association

CS → U(C)RPorsche → Good Feeling




d-Rule

Monte CarloControl

Q-Learning

TD( )often =0

ll

TD(1) TD(0)

Rescorla/Wagner


Neur.TD-formalism


(”fast”)




SARSA


(non-evaluative)

ISO-Learning

ISO-Modelof STDP


Eligibility Traces

Hebb-Rule


(”slow”)

supervised L.


=

=

=



STDP

LTP(LTD=anti)

ISO-Control

Overview over different methods – Reinforcement Learning

You are here !




d-Rule

Monte CarloControl

Q-Learning

TD( )often =0

ll

TD(1) TD(0)

Rescorla/Wagner


Neur.TD-formalism


(”fast”)




SARSA


(non-evaluative)

ISO-Learning

ISO-Modelof STDP


Eligibility Traces

Hebb-Rule


(”slow”)

supervised L.


=

=

=



STDP

LTP(LTD=anti)

ISO-Control


And later also here !

US = r,R = “Reward”

CS = s,u = Stimulus = “State1”

CR = v,V = (Strength of the) Expected Reward = “Value”

UR = --- (not required in mathematical formalisms of RL)

Weight = = weight used for calculating the value; e.g. v=u

Action = a = “Action”

Policy = = “Policy”

1 Note: The notion of a “state” really only makes sense as soon as there is more than one state.

Notation

A note on “Value” and “Reward Expectation”

If you are at a certain state then you would value this state according to how much reward you can expect when moving on from this state to the end-point of your trial.

Hence:

Value = Expected Reward !

More accurately:

Value = Expected cumulative future discounted reward.

(for this, see later!)

1) Rescorla-Wagner Rule: Allows for explaining several types of conditioning experiments.

2) TD-rule (TD-algorithm) allows measuring the value of states and allows accumulating rewards. Thereby it generalizes the Resc.-Wagner rule.

3) TD-algorithm can be extended to allow measuring the value of actions and thereby control behavior either by ways of

a) Q or SARSA learning or with

b) Actor-Critic Architectures

Types of Rules




d-Rule

Monte CarloControl

Q-Learning

TD( )often =0

ll

TD(1) TD(0)

Rescorla/Wagner


Neur.TD-formalism


(”fast”)




SARSA


(non-evaluative)

ISO-Learning

ISO-Modelof STDP


Eligibility Traces

Hebb-Rule


(”slow”)

supervised L.


=

=

=



STDP

LTP(LTD=anti)

ISO-Control


You are here !

Rescorla-Wagner Rule

Pavlovian:

Extinction:

Partial:

Train Result

u→r

u→r u→●

Pre-Train

u→r u→●

u→v=max

u→v=0

u→v<max

We define: v = u, with u=1 or u=0, binary and → + du with d = r - v

This learning rule minimizes the avg. squared error between actual reward r and the prediction v, hence min<(r-v)2>

We realize that d is the prediction error.

The associability between stimulus u and reward r is represented by the learning rate .

Pawlovian

Extinction

Partial

Stimulus u is paired with r=1 in 100% of the discrete “epochs” for Pawlovianand in 50% of the cases for Partial.

Rescorla-Wagner Rule, Vector Form for Multiple Stimuli

We define: v = w.u, and w → w + du with d = r – v

Where we minimize d

Blocking:

Train Result

u1+u2→r

Pre-Train

u1→v=max, u2→v=0u1→r

For Blocking: The association formed during pre-training leads to d=0. As 2 starts with zero the expected reward v=1u1+2u2 remains at r. This keeps d=0 and the new association with u2 cannot be learned.


Inhibitory:

Train ResultPre-Train

u1+u2→●, u1→r u1→v=max, u2→v<0

Inhibitory Conditioning: Presentation of one stimulus together with the reward and alternating presenting a pair of stimuli where the reward is missing. In this case the second stimulus actually predicts the ABSENCE of the reward (negative v).

Trials in which the first stimulus is presented together with the reward lead to 1>0.

In trials where both stimuli are present the net prediction will be v=1u1+2u2 = 0.

As u1,2=1 (or zero) and 1>0, we get 2<0 and, consequentially, v(u2)<0.


Overshadow:


u1+u2→r u1→v<max, u2→v<max

Overshadowing: Presenting always two stimuli together with the reward will lead to a “sharing” of the reward prediction between them. We get v= 1u1+2u2 = r. Using different learning rates will lead to differently strong growth of 1,2 and represents the often observed different saliency of the two stimuli.


Secondary:


u1→r u2→u1 u2→v=max

Secondary Conditioning reflect the “replacement” of one stimulus by a new one for the prediction of a reward.

As we have seen the Rescorla-Wagner Rule is very simple but still able to represent many of the basic findings of diverse conditioning experiments.

Secondary conditioning, however, CANNOT be captured.

Predicting Future Reward

Animals can predict to some degree such sequences and form the correct associations. For this we need algorithms that keep track of time.

Here we do this by ways of states that are subsequently visited and evaluated.

The Rescorla-Wagner Rule cannot deal with the sequentiallity of stimuli (required to deal with Secondary Conditioning). As a consequence it treats this case similar to Inhibitory Conditioning lead to negative 2.

Prediction and Control

The goal of RL is two-fold:

1) To predict the value of states (exploring the state space following a policy) – Prediction Problem.

2) Change the policy towards finding the optimal policy – Control Problem.

• State,• Action,• Reward,• Value,• Policy

Terminology (again):

Markov Decision Problems (MDPs)

1 2 3 4 5 6 7 8

9 10 11 12

13 14

15 16

r1 r2

a2 a15a14a1

s

terminal states

states

actions

rewards

If the future of the system depends always only on the current state and action then the system is said to be “Markovian”.

What does an RL-agent do ?An RL-agent explores the state space trying to

accumulate as much reward as possible. It follows a behavioral policy performing actions (which usually will lead the agent from one state to the next).

For the Prediction Problem: It updates the value of each given state by assessing how much future (!) reward can be obtained when moving onwards from this state (State Space). It does not change the policy, rather it evaluates it. (Policy Evaluation).

For the Control Problem: It updates the value of each given action at a given state and of by assessing how much future reward can be obtained when performing this action at that state (State-Action Space, which is larger

than the State Space). and all following actions at the following state moving onwards.Guess: Will we have to evaluate ALL states and actions onwards?

p(N) = 0.5p(S) = 0.125p(W) = 0.25p(E) = 0.125

Policy:

x x x x x

R R

0.0

value = 0.0everywherereward R=1

possible startlocations

0.9

0.9

0.8

0.1 0.1 0.1 0.1 0.1

etc

Policy Evaluationgive values of states

Exploration – Exploitation Dilemma: The agent wants to get as much cumulative reward (also often called return) as possible. For this it should always perform the most rewarding action “exploiting” its (learned) knowledge of the state space. This way it might however miss an action which leads (a bit further on) to a much more rewarding path. Hence the agent must also “explore” into unknown parts of the state space. The agent must, thus, balance its policy to include exploitation and exploration.

What does an RL-agent do ?

Policies1) Greedy Policy: The agent always exploits and

selects the most rewarding action. This is sub-optimal as the agent never finds better new paths.

Policies2 -Greedy Policy: With a small probability the

agent will choose a non-optimal action. *All non-optimal actions are chosen with equal probability.* This can take very long as it is not known how big should be. One can also “anneal” the system by gradually lowering to become more and more greedy.

3) Softmax Policy: -greedy can be problematic because of (*). Softmax ranks the actions according to their values and chooses roughly following the ranking using for example:

P

b=1

n

exp( TQb)

exp( TQa) where Qa is value of the

currently to be evaluated action a and T is a temperature parameter. For large T all actions have approx. equal probability to get selected.




d-Rule

Monte CarloControl

Q-Learning

TD( )often =0

ll

TD(1) TD(0)

Rescorla/Wagner


Neur.TD-formalism


(”fast”)




SARSA


(non-evaluative)

ISO-Learning

ISO-Modelof STDP


Eligibility Traces

Hebb-Rule


(”slow”)

supervised L.


=

=

=



STDP

LTP(LTD=anti)

ISO-Control


You are here !

Back to the question: To get the value of a given state, will we have to evaluate ALL states and actions onwards?There is no unique answer to this! Different methods exist which assign the value of a state by using differently many (weighted) values of subsequent states. We will discuss a few but concentrate on the most commonly used TD-algorithm(s).

Temporal Difference (TD) Learning

Towards TD-learning – Pictorial View

In the following slides we will treat “Policy evaluation”: We define some given policy and want to evaluate the state space. We are at the moment still not interested in evaluating actions or in improving policies.

Formalising RL: Policy Evaluation with goal to find the optimal value function of the state spaceWe consider a sequence st, rt+1, st+1, rt+2, . . . , rT , sT . Note, rewards occur downstream (in the future) from a visited state. Thus, rt+1 is the next future reward which can be reached starting from state st. The complete return Rt to be expected in the future from state st is, thus, given by:

where ≤1 is a discount factor. This accounts for the fact that rewards in the far future should be valued less.Reinforcement learning assumes that the value of a state V(s) is directly equivalent to the expected return E at this state, where denotes the (here unspecified) action policy to be followed.

Thus, the value of state st can be iteratively updated with:

We use as a step-size parameter, which is not of great importance here, though, and can be held constant.Note, if V(st) correctly predicts the expected complete return Rt, the update will be zero and we have found the final value. This method is called constant- Monte Carlo update. It requires to wait until a sequence has reached its terminal state (see some slides before!) before the update can commence. For long sequences this may be problematic. Thus, one should try to use an incremental procedure instead. We define a different update rule with:

The elegant trick is to assume that, if the process converges, the value of the next state V(st+1) should be an accurate estimate of the expected return downstream to this state (i.e., downstream to st+1). Thus, we would hope that the following holds:Indeed, proofs exist that under certain boundary conditions this procedure, known as TD(0), converges to the optimal value function for all states.

This is why it is called TD (temp. diff.) Learning

| {z }




d-Rule

Monte CarloControl

Q-Learning

TD( )often =0

ll

TD(1) TD(0)

Rescorla/Wagner


Neur.TD-formalism


(”fast”)




SARSA


(non-evaluative)

ISO-Learning

ISO-Modelof STDP


Eligibility Traces

Hebb-Rule


(”slow”)

supervised L.


=

=

=



STDP

LTP(LTD=anti)

ISO-Control

Reinforcement Learning – Relations to Brain Function I

You are here !

Trace

d

1

X

x1

r

vv’

E

u1

How to implement TD in a Neuronal Way

Now we have:


We had defined:(first lecture!)

X0

X1

Xn

v(t)

x

x

v’

reward

(n-i)

d

How to implement TD in a Neuronal Way

v(t+1)-v(t)

Note: v(t+1)-v(t) is acausal (future!). Make it “causal” by using delays.

x

w = 10X0

X1

reward

d

v(t)v(t- )

r

Serial-Compound representations X1,…Xn for defining an eligibility trace.




d-Rule

Monte CarloControl

Q-Learning

TD( )often =0

ll

TD(1) TD(0)

Rescorla/Wagner


Neur.TD-formalism


(”fast”)




SARSA


(non-evaluative)

ISO-Learning

ISO-Modelof STDP


Eligibility Traces

Hebb-Rule


(”slow”)

supervised L.


=

=

=



STDP

LTP(LTD=anti)

ISO-Control

Reinforcement Learning – Relations to Brain Function II

You are here !

TD-learning & Brain FunctionNovelty Response:no prediction,reward occurs

no CS r

After learning:predicted reward occurs

CS r

DA-responses in the basal ganglia pars compacta of thesubstantia nigra and the medially adjoining ventral tegmental area (VTA).

This neuron is supposed to represent the d-error of TD-learning, which has moved forward as expected.

After learning:predicted reward does notoccur

CS 1.0 s

Omission of reward leads to inhibition as also predicted by the TD-rule.

TD-learning & Brain Function

1.5 srTr

RewardExpectation

This neuron is supposed to represent the reward expectation signal v. It has extended forward (almost) to the CS (here called Tr) as expected from the TD-rule. Such neurons are found in the striatum, orbitofrontal cortex and amygdala.

1.0 s

Reward Expectation(Population Response)

Tr r

This is even better visible from the population response of 68 striatal neurons

Reinforcement Learning – The Control ProblemSo far we have concentrated on evaluating and

unchanging policy. Now comes the question of how to actually improve a policy trying to find the optimal policy.

We will discuss:

1) Actor-Critic Architectures

But not:

2) SARSA Learning

3) Q-Learning

Abbreviation for policy:




d-Rule

Monte CarloControl

Q-Learning

TD( )often =0

ll

TD(1) TD(0)

Rescorla/Wagner


Neur.TD-formalism


(”fast”)




SARSA


(non-evaluative)

ISO-Learning

ISO-Modelof STDP


Eligibility Traces

Hebb-Rule


(”slow”)

supervised L.


=

=

=



STDP

LTP(LTD=anti)

ISO-Control

Reinforcement Learning – Control Problem I

You are here !

Control Loops

ControllerControlled

SystemControlSignals

Feedback

DisturbancesSet-Point

X0

A basic feedback–loop controller (Reflex) as in the slide before.

Actor(Controller)

Environment(Controlled System)

Feedback

Disturbances

Context

Critic

Actions(Control Signals)

ReinforcementSignal

X0

Control Loops

An Actor-Critic Architecture: The Critic produces evaluative, reinforcement feedback for the Actor by observing the consequences of its actions. The Critic takes the form of a TD-error which gives an indication if things have gone better or worse than expected with the preceding action. Thus, this TD-error can be used to evaluate the preceding action: If the error is positive the tendency to select this action should be strengthened or else, lessened.

ù(s;a) = Pbep(s;b)

ep(s;a)

Example of an Actor-Critic Procedure

Action selection here follows the Gibb’s Softmax method:

where p(s,a) are the values of the modifiable (by the Critcic!) policy parameters of the actor, indicting the tendency to select action a when being in state s.

p(st;at) p(st;at) + ì î t

We can now modify p for a given state action pair at time t with:

where dt is the d-error of the TD-Critic.




d-Rule

Monte CarloControl

Q-Learning

TD( )often =0

ll

TD(1) TD(0)

Rescorla/Wagner


Neur.TD-formalism


(”fast”)




SARSA


(non-evaluative)

ISO-Learning

ISO-Modelof STDP


Eligibility Traces

Hebb-Rule


(”slow”)

supervised L.


=

=

=



STDP

LTP(LTD=anti)

ISO-Control

Reinforcement Learning – Control I & Brain Function III

You are here !

Cortex (C)FrontalCortex

VP SNr GPi

DA-System(SNc,VTA,RRA)

Thalamus

Striatum (S)GPe

STN

Actor-Critics and the Basal Ganglia

VP=ventral pallidum,SNr=substantia nigra pars reticulata,SNc=substantia nigra pars compacta,GPi=globus pallidus pars interna,GPe=globus pallidus pars externa,VTA=ventral tegmental area,RRA=retrorubral area, STN=subthalamic nucleus.

The basal ganglia are a brain structure involved in motor control. It has been suggested that they learn by ways of an Actor-Critic mechanism.

So called striosomal modules fulfill the functions of the adaptive Critic. The prediction-error (d) characteristics of the DA-neurons of the Critic are generated by: 1) Equating the reward r with excitatory input from the lateral hypothalamus. 2) Equating the term v(t) with indirect excitation at the DA-neurons which is initiated from striatal striosomes and channelled through the subthalamic nucleus onto the DA neurons. 3) Equating the term v(t−1) with direct, long-lasting inhibition from striatal striosomes onto the DA-neurons. There are many problems with this simplistic view though: timing, mismatch to anatomy, etc.

C

S

STN

DA r+

-Cortex=C, striatum=S, STN=subthalamic Nucleus, DA=dopamine system, r=reward.

Actor-Critics and the Basal Ganglia: The Critic

DAGlu

Cortico-striatal(”pre”)

Nigro-striatal(”DA”)

Medium-sized Spiny ProjectionNeuron in the Striatum (”post”)

CDA

overview over different methods. different types/classes of learning unsupervised learning...

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