Dueling�Network�Architectures�for�Deep�Reinforcement�Learning�

2016-06-28

Taehoon�Kim

Motivation

• Recent�advances• Design�improved�control�and�RL�algorithms

• Incorporate�existing�NN�into�RL�methods

• We,• focus�on�innovating�a�NN�that�is�better�suited�for�model-free�RL

• Separate

• the�representation�of�state�value

• (state-dependent)�action�advantages

2

Overview

3

statevaluefunction

advantagefunction

sharing convolutional featurelearningmodule

aggregating layer

state-actionvaluefunction

Dueling�network

• Single�Q�network�with�two�streams

• Produce�separate�estimations�of�state�value�func and�advantage�func

• without�any�extra�supervision

• which�states�are�valuable?

• without�having�to�learn�the�effect�of�each�action�for�each�state

4

Saliency�map�on�the�Atari�game�Enduro

5

1.Focusonhorizon,wherenewcarsappear

2.Focusonthescore

Notpaymuchattentionwhentherearenocarsinfront

Attentiononcarimmediatelyinfrontmakingitschoiceofactionveryrelevant

Definitions

• Value 𝑉(𝑠),�how�good�it�is�to�be�in�particular�state�𝑠

• Advantage 𝐴(𝑠, 𝑎)

• Policy 𝜋

• Return 𝑅* = ∑ 𝛾./*𝑟.∞.1* ,�where�𝛾 ∈ [0,1]

• Q function 𝑄8 𝑠, 𝑎 = 𝔼 𝑅* 𝑠* = 𝑠, 𝑎* = 𝑎, 𝜋

• State-value function 𝑉8 𝑠 = 𝔼:~8(<)[𝑄8 𝑠, 𝑎 ]

6

Bellman�equation

• Recursively�with�dynamic�programming

• 𝑄8 𝑠, 𝑎 = 𝔼<` 𝑟 + 𝛾𝔼:`~8(<`) 𝑄8 𝑠`, 𝑎` |𝑠, 𝑎, 𝜋

• Optimal�Q∗ 𝑠, 𝑎 = max8𝑄8 𝑠, 𝑎

• Deterministic�policy�a = argmax:`∈𝒜

Q∗ 𝑠, 𝑎`

• Optimal�V∗ 𝑠 = max:𝑄∗ 𝑠, 𝑎

• Bellman�equation Q∗ 𝑠, 𝑎 = 𝔼<` 𝑟 + 𝛾max:`𝑄∗ 𝑠`, 𝑎` |𝑠, 𝑎

7

Advantage�function

• Bellman�equation Q∗ 𝑠, 𝑎 = 𝔼<` 𝑟 + 𝛾 max:`𝑄∗ 𝑠`,𝑎` |𝑠, 𝑎

• Advantage�function A8 𝑠, 𝑎 = 𝑄8 𝑠, 𝑎 − 𝑉8(𝑠)

• 𝔼:~8(<`) 𝐴8 𝑠, 𝑎 = 0

8

Advantage�function

• Value�𝑉(𝑠),�how�good�it�is�to�be�in�particular�state�𝑠

• Q(𝑠, 𝑎),�the�value�of�choosing�a�particular�action�𝑎 when�in�state�𝑠

• A = 𝑉 − 𝑄 to�obtain�a�relative�measure of�importance�of�each�action

9

Deep�Q-network�(DQN)

• Model�Free

• states�and�rewards�are�produced�by�the�environment

• Off�policy

• states�and�rewards�are�obtained�with�a�behavior�policy�(epsilon�greedy)

• different�from�the�online�policy�that�is�being�learned

10

Deep�Q-network:�1)�Target�network

• Deep�Q-network�𝑄 𝑠, 𝑎; 𝜽

• Target�network�𝑄 𝑠, 𝑎; 𝜽/

• 𝐿O 𝜃O = 𝔼<,:,Q,<` 𝑦OSTU − 𝑄 𝑠, 𝑎; 𝜽𝒊

W

• 𝑦OSTU = 𝑟 + 𝛾max

:`𝑄 𝑠`, 𝑎`; 𝜽/

• Freeze�parameters�for�a�fixed�number�of�iterations

• 𝛻YZ𝐿O 𝜃O = 𝔼<,:,Q,<` 𝑦OSTU − 𝑄 𝑠, 𝑎; 𝜽𝒊 𝛻YZ𝑄 𝑠, 𝑎; 𝜽𝒊

11

𝑠`

𝑠

Deep�Q-network:�2)�Experience�memory

• Experience�𝑒* = (𝑠*, 𝑎*, 𝑟*, 𝑠*\])

• Accumulates�a�dataset�𝒟* = 𝑒], 𝑒W,… , 𝑒*

• 𝐿O 𝜃O = 𝔼 <,:,Q,<` ~𝒰(𝒟) 𝑦OSTU − 𝑄 𝑠, 𝑎; 𝜽𝒊

W

12

Double�Deep�Q-network�(DDQN)

• In�DQN

• the�max operator�uses�the�same�values�to�both�select and�evaluate an�action

• lead�to�overoptimistic�value�estimates

• 𝑦OSTU = 𝑟 + 𝛾max

:`𝑄 𝑠`, 𝑎`;𝜽/

• To�migrate�this�problem,�in�DDQN

• 𝑦OSSTU = 𝑟 + 𝛾𝑄 𝑠`, argmax

:`𝑄(𝑠`, 𝑎`; 𝜃O) ; 𝜽/

13

Prioritized�Replay��(Schaul et�al.,�2016)�

• To�increase�the�replay�probability�of�experience�tuples

• that�have�a�high�expected�learning�progress

• use�importance�sampling�weight�measured�via�the�proxy�of�absolute�TD-error

• sampling�transitions�with�high�absolute�TD-errors

• Led�to�faster�learning�and�to�better�final�policy�quality

14

Dueling�Network�Architecture�:�Key�insight

• For�many�states

• unnecessary to�estimate�the�value�of�each�action�choice

• For�example,�move�left�or�right�only�matters�when�a�collision�is�eminent

• In�most�of�states,�the�choice�of�action�has�no�affect�on�what�happens

• For�bootstrapping based�algorithm

• the�estimation�of�state�value is�of�great�importance for�every�state

• bootstrapping:�update�estimates�on�the�basis�of�other�estimates.�

15

Formulation

• A8 𝑠, 𝑎 = 𝑄8 𝑠, 𝑎 − 𝑉8(𝑠)

• 𝑉8 𝑠 = 𝔼:~8(<) 𝑄8 𝑠, 𝑎

• A8 𝑠, 𝑎 = 𝑄8 𝑠, 𝑎 − 𝔼:~8(<) 𝑄8 𝑠, 𝑎

• 𝔼:~8(<) 𝐴8 𝑠, 𝑎 = 0

• For�a�deterministic�policy,�𝑎∗ = argmaxa`∈𝒜

𝑄 𝑠, 𝑎`

• 𝑄 𝑠,𝑎∗ = 𝑉(𝑠) and�𝐴 𝑠, 𝑎∗ = 0

16

Formulation

• Dueling�network�=�CNN�+�fully-connected�layers�that�output

• a�scalar�𝑉 𝑠; 𝜃, 𝛽

• an� 𝒜 -dimensional�vector�𝐴(𝑠, 𝑎; 𝜃, 𝛼)

• Tempt�to�construct�the�aggregating�module

• 𝑄 𝑠,𝑎; 𝜃, 𝛼, 𝛽 = 𝑉 𝑠; 𝜃, 𝛽 + 𝐴(𝑠, 𝑎; 𝜃, 𝛼)

17

Aggregation�module�1:�simple�add

• But�𝑄 𝑠, 𝑎; 𝜃, 𝛼, 𝛽 is�only�a�parameterized�estimate�of�the�true�Q-

function

• Unidentifiable

• Given�𝑄,�𝑉 and�𝐴 can’t�uniquely�be�recovered

• We�force�the�𝐴 to�have�zero at�the�chosen�action

• 𝑄 𝑠,𝑎; 𝜃, 𝛼, 𝛽 = 𝑉 𝑠; 𝜃, 𝛽 + 𝐴 𝑠,𝑎; 𝜃, 𝛼 −max:`∈𝒜

𝐴(𝑠, 𝑎`; 𝜃,𝛼)

18

Aggregation�module�2:�subtract�max

• For�𝑎∗ = argmaxa`∈𝒜

𝑄(𝑠, 𝑎`;𝜃, 𝛼, 𝛽) = argmaxa`∈𝒜

𝐴 𝑠, 𝑎; 𝜃, 𝛼

• obtain�𝑄 𝑠, 𝑎∗;𝜃, 𝛼, 𝛽 = 𝑉 𝑠; 𝜃, 𝛽

• 𝑄 𝑠,𝑎∗ = 𝑉(𝑠)

• An�alternative�module�replace�max�operator�with�an�average

• 𝑄 𝑠,𝑎; 𝜃, 𝛼, 𝛽 = 𝑉 𝑠; 𝜃, 𝛽 + 𝐴 𝑠,𝑎; 𝜃, 𝛼 − ]𝒜∑ 𝐴(𝑠, 𝑎`; 𝜃, 𝛼):`

19

Aggregation�module�3:�subtract�average

• An�alternative�module�replace�max�operator�with�an�average

• 𝑄 𝑠,𝑎; 𝜃, 𝛼, 𝛽 = 𝑉 𝑠; 𝜃, 𝛽 + 𝐴 𝑠,𝑎; 𝜃, 𝛼 − ]𝒜∑ 𝐴(𝑠, 𝑎`; 𝜃, 𝛼):`

• Now�loses�the�original�semantics�of�𝑉 and�𝐴

• because�now�off-target�by�a�constant,� ]𝒜 ∑ 𝐴(𝑠, 𝑎`; 𝜃, 𝛼):`

• But�increases�the�stability�of�the�optimization• 𝐴 only�need�to�change�as�fast�as�the�mean

• Instead�of�having�to�compensate any�change�to�the�optimal�action’s�advantage

20

max:`∈𝒜

𝐴(𝑠, 𝑎`; 𝜃, 𝛼)

Aggregation�module�3:�subtract�average

• Subtracting�mean�is�the�best

• helps�identifiability

• not�change�the�relative�rank�of�𝐴 (and�hence�Q)

• Aggregation�module�is�a�part�of�the�network�not�a�algorithmic�step

• training�of�dueling�network�requires�only�back-propagation

21

Compatibility

• Because�the�output�of�dueling�network�is�Q�function

• DQN

• DDQN

• SARSA

• On-policy,�off-policy,�whatever

22

Definition:�Generalized�policy�iteration

23

Experiments:�Policy�evaluation

• Useful�for�evaluating�network�architecture

• devoid of�confounding�factors�such�as�choice�of�exploration strategy,�and�

interaction�between�policy�improvement and�policy�evaluation

• In�experiment,�employ�temporal�difference�learning

• optimizing�𝑦O = 𝑟 + 𝛾𝔼:`~8(<`) 𝑄 𝑠`, 𝑎`, 𝜃O

• Corridor�environment

• exact�𝑄8(𝑠,𝑎) can�be�computed�separately�for�all� 𝑠, 𝑎 ∈ 𝒮×𝒜

24

Experiments:�Policy�evaluation

• Test�for�5,�10,�and�20�actions�(first�tackled�by�DDQN)

• The�stream�𝑽 𝒔; 𝜽, 𝜷 learn�a�general�value�shared�across�many�

similar�actions�at�𝑠

• Hence�leading�to�faster�convergence

25

Performancegapincreasingwiththenumberofactions

Experiments:�General�Atari�Game-Playing

• Similar�to�DQN�(Mnih et al.,�2015)�and add fully-connected layers

• Rescale�the�combined�gradient�entering�the�last�convolutional�layer�

by�1/ 2,�which�mildly�increases�stability

• Clipped�gradients�to�their�norm�less�than�or�equal�to�10

• clipping�is�not�a�standard�practice�in�RL

26

Performance:�Up-to�30�no-op�random�start

• Duel�Clip�>�Single�Clip�>�Single

• Good�job�Dueling�network

27

Performance:�Human�start

28

• Not�necessarily�have�to�generalize�well�to�play�the�Atari�games

• Can�achieve�good�performance�by�simply�remembering�sequences�of�

actions

• To�obtain�a�more�robust�measure�that�use�100�starting�points�

sampled�from�a�human�expert’s�trajectory

• from�each�starting�points,�evaluate�up�to�108,000�frames

• again,�good�job�Dueling�network

Combining�with�Prioritized�Experience�Replay�

• Prioritization�and�the�dueling�architecture�address very�different�

aspects�of�the�learning�process

• Although�orthogonal�in�their�objectives,�these�extensions�

(prioritization,�dueling�and�gradient�clipping)� interact�in�subtle�ways�

• Prioritization�interacts�with�gradient�clipping

• Sampling�transitions�with�high�absolute�TD-errors�more�often�leads�to�gradients�with�higher�

norms�so re-tuned�the�hyper�parameters

29

References

1. [Wang,�2015]�Wang,�Z.,�de�Freitas,�N.,�&�Lanctot,�M.�(2015).�Dueling�network�architectures�for�

deep�reinforcement�learning. arXiv preprint�arXiv:1511.06581.

2. [Van,�2015]�Van�Hasselt,�H.,�Guez,�A.,�&�Silver,�D.�(2015).�Deep�reinforcement�learning�with�

double�Q-learning. CoRR,�abs/1509.06461.

3. [Schaul,�2015]�Schaul,�T.,�Quan,�J.,�Antonoglou,�I.,�&�Silver,�D.�(2015).�Prioritized�experience�

replay. arXiv preprint�arXiv:1511.05952.

4. [Sutton,�1998]�Sutton,�R.�S.,�&�Barto,�A.�G.�(1998). Reinforcement�learning:�An�introduction(Vol.�

1,�No.�1).�Cambridge:�MIT�press.

30