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Bernoulli, Caratheodory
(optimality principles)
Wald (sequential analysis)
Raiffa & Schlaifer
(Decision theory)
Robbins/Monro
Howard
Stochastic
Approximation
Fel’dbaum
(Dual control)
Bellman
Bar-Shalom, Tse
Bandit Problems
Gittins
Adaptive Control Processes
Dynamic Programming
Simulation
Function approximation
AI
Decision theoretic planning
Reinforcement learning
Exploration vs. exploitation
Kaelbling(IE); Meuleau et al (IQL+)
Dearden et al (Bayesian Q-learning)
Reinforcement Learning
Policy gradient techniques
Williams, Sutton et al, Konda & Tsitsiklis
POMDP’s
Meuleau et al (finite state controllers)
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O f CBFGDN<iHJI Hkp=DBjLA FGzJAIMD [ ><?IMDNAPCNF l DN<?HkI < DNj FkI f I l [email protected]<?I |HJC@n?O R K f C@> f <iILz FGI FkK
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Az R <?IMDBACB>@K APCB>@A> A mKLniHJ<iDNFGDN<iHJI >EDNAPKL> g<?I jL< l j\DNjLAxOLA l B<?HkI h F GACFk>@>B<izJIL>FKtFGDB<?AIMDDBH
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DNC@A FGD h APIMD { jL<?> h AFkIL> DBjtFGD KtF DN<?APIMDN> Fk>@>B<izJILAO DBCBAFGD h AIMDN> nuFGDBAC <iIDNjrAxFkn?niH l FGDN<iHJIKLCBH l AP>B>FkC@A\n?< GAn R DBH Q A DBCBA F DNAO Q APDBDBAC o>B<?I l A DNjrA RQ AILA1vLDpyCBH h DNjrA\CBA>@K HkIL>BAP>#Hkp A FGCBn?<iAC
KtFGDB<?API DB> θ1 =Pr{Cure | Drug 1}
θ2 =Pr{Cure | Drug 2}
..
Failure
Success
Failure
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p1
-1
p212
+1
11
p122
1
21
p
p211
+1
-1
p22 1
(a)
p22 2
(b)
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F Q Fkn?FkI l A Q APD AAPI DNjrA>BAD H l CB<iDBAC@<uF !#"%$
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{ jLA!KLCB<iHJCB> h F RZQ A <iIrpyHJC h APO QMR KLCBA1sm<iH f >@n R zMF DNjLAPCBAO A h KL<?C@< l FknxOtFGDNF C@A> f niDB>
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1.0
1.0
f(θ1)
f(θ2)
θ1
,
1.0
θ2
Drug 2
Drug 1
Success
1.0
Failure
Success
Failure
1.0
2.0
f(θ1)
f(θ2)
θ1
1.0
,
θ2
1.0
1.0
1.0
1.0
1.0
2.0
f(θ1)
f(θ2)
f(θ1)
f(θ2 )
f(θ1)
f(θ2)
1.0
θ1 1.0
Drug 1
θ2 1.0
θ1
1.0
,
θ2
1.0
θ1
1.0
,
θ2
1.0
Drug 2
Success
3.0
,
Failure
1.0
1.0
f(θ)
1
f(θ2)
θ 1.0
1
,
f(θ1)
f(θ2)
θ2 1.0
θ1 1.0
,
θ2 1.0
..
.
Failure
f(θ1)
c <iz f CBA
f(θ2)
,
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θ1
1.0
θ2
1.0
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(( α 1,β1) ,(α2 ,β2 ))
Drug 1
α1
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Drug 2
...
Success
β1
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((α1+1,β1),( α 2,β2))
Failure
((α1 ,β1+1),( α 2 ,β2))
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a1
2
2
[
][ ]
2
][ ]
[
α 12
α 1+ β2
t+1
1
2
β1
2
α 1+ β21
1
α1 β1
1
β12 α 12
2
α1 +1 β2
1
β 22 α22
[ ][
s =1:
2
a2
2
1 1
s =2: α 1 β1 +1 , α 1 β1
t+1
β21 α 12
β22 α 22
β11
1
α 1 + β1
1
1 1
st =1: α1 β1 , α1 β1
β 21 α21
β 22 α 22
][ ]
[
α11 + β1
1
a1
2 2
, α1 β1
2
2
β 2 α2
st+1=1: α11+1 β1
1
1
β 2 α 12
,
]
2
s t+1
=2: α11 β11 , α 1 β21+1
β12 α12 β22 α2
[ ][ ]
a2
a1
a2
a1
a
...
2
a1
2
a2
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Node structure:
state
n111 n112 n211 n212
n1 n1 n221 n 2
21 22
successor
next_in_layer next_in_hashlist
22
Depth
Other:
layer
0
1
hash_table
ptr to 1st node in layer 0
0
ptr to 1st node in layer 1
1
last_node
ptr to 1st node in list 0
ptr to 1st node in list 1
horizon
^
hashsize-1
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J-J
220
Number of keys per slot
210
200
190
180
170
160
0
2000
4000
6000
8000
10000
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Q
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:95 J9O 5 RrG"G Q G?D 5 8HI3L 5 `5 JG?8HWoMG 5 JUgBG?DeMG8=Jh[BMPI 8=I3R :95 MPJ98HI3VO :9Q O QB: G?8 Q I Q TG : MPI3J8HG?8 Q I7J9MPW<O3Y 5 J
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k
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(1)
1−ρ i / 0
1
1
ρ i/ 1
(2)
1−ρ i / 0
2
1−ρ / 0
3
ρ(2)/ 1
ρ(1)/ 1
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0
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Multi-armed bandit problems
Static
Stochastic
Scheduling
Gittins indices:
{g , g ,...
2
...gNn }
V1
V2
VNn
MDP1
MDP
1
(Gittins & Jones, 1974)
(Katahakis & Veinott, 1987)
MDP
Nn
2
Q-Learning
(Watkins, 1989)
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%
%
5
|
0.30
|
0.20
|
0.10
|
Relative Degradation
0.40
|
0.00 |
0
|
500
|
1000
|
1500
|
2000
|
2500
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3000
|
3500
|
4000
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|
4500
|
5000
Trial
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4
3
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1000
1500
2000
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1
2
1
3
4
2
8
3
6
4
4
12
2
5
6
1
7
24
12
6
3
12
24
12
6
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3
1
1
6
2
3
2
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4
5
6
4
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0.4
0.5
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0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
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2
0.5
0.6
0.7
0.8
0.9
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5
5
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4
4
4
4
4
4
3
3
3
3
3
3
3
2
2
2
2
2
2
2
1
1
1
1
1
1
0
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
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0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
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0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
6
6
6
6
6
6
6
5
5
5
5
5
5
5
4
4
4
4
4
4
4
3
3
3
3
3
3
3
2
2
2
2
2
2
2
1
1
1
1
1
1
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
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0.4
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0.7
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0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
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0.2
0.3
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0.025
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[ 1;M ( m111++ ),N (1n11++
1
r
2
[ 1;M ( m1++ ),N (2n111++ ) ]
r
[ 2 ;M (m112++ ),N (1n112++ ) ]
r
2
[ 2 ;M (m112++ ),N (2n11++
)]
2
r
[ 1;M ( m2++ ),N (1n211++ ) ]
r2
[ 1;M ( m112++ ),N (2n211++ ) ]
r
[ 2 ;M (m122++ ),N (1n212++ ) ]
r2
[ 2 ;M (m122++ ),N (2n212++ ) ]
1
1
1
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