Reinforcement Learning for Active Model Selection

2 downloads 0 Views 148KB Size Report
problem, "active model selection" [10]. As this is a Markov decision problem, we consider applying reinforcement learn- ing (RL) techniques to learn an effective ...
Reinforcement Learning for Active Model Selection Aloak Kapoor [email protected]

Russell Greiner [email protected]

Department of Computing Science University of Alberta Edmonton, AB T6J 2E8 _A+S#V/Ÿ%'#,.*.+ KF6D0 #,"0 ; #S  +F"(€1

ABSTRACT

`*\=%'/*  G*D 0E; #7F6D0 #,G124*3%H  6IFJ *+8>8:;6  +=`%-# (D%'*  2%' JD0#= J+,""%"M

+,F ¥-"%GD%>/#"%-2` J07

+,FJ ¤

+8RS%' J"3[D0 +,F ¥-"* 6 %*S%- !ME,003+,N(%2F60#0

 +F"©""6 "3 6 "%c2' /#– D%'B+8!F60#0  #XL#," m(Gk12o8:;6 "%K+8\/FJ"/"0L' #

Categories and Subject Descriptors

M x M y9d z{;|}^~€V}^D‚Uƒ„U|"… ‚:‚:}s†…„A,….gˆ‡)!; #;‰OŠrD‹ŒŽmqhwrJ>rJw‘2’3“ u'‹”.u–•V—Œ qs˜qsŽmqhwr)™2•v'•š*uŽuv9’/u•vrq:r ”

%-""%9XFJ* 6 D%'0X  1.(A12 7%'/#,K+,@ *G 1b 0EjmmM ,M( G"/%'%'3[D;+,E""6 nCFJ"#9 ""  ++ J2 J '8:+ G+8N%-' "#/%c8:+

Keywords

#, AF60#07; # M

+,%-%

# jmlH)n"/5.6"%2+*9/0 "60 £Mcµ ">µ "+ x M

+Ex>  "%'".%cR1+ &UMbiHA

+.+8:%H

J+,*%?>  #

FJR8s+,6D07$>iH  J"D0 3`NM

%' J""3["%?G0`I+8U2"+/*+

F%'I++,%B+8C / 8:+ ""*".R; # ·N8C+(J1\ 8s B

+,%G+; 6 b+IB  +,F"



  i¬%' +8

%"‡

µ " "%'63?+8U"+,6+,*R+8€"+/

 7+=6 U0;

/  J0R"+/NM + )`* ,(;38"+

/  J0SD0E6 %B6 4,0 %"(N"S%R0"%'E8:6+,

%b6 U0"0I+

j

n· +8!"+,6 %',(9%'*/ b6 U0;

+."6 %>8:+ =%–+6"+*,MEj:®*0"%' /FJ–=`\8:+ ;



87

+8b*0 "%'3E8:6+,XD0 =6 U0%94+,6 9%'* 38s # %'%'6*

/+%!FJ"+;1>M n

c+,

',

 %!0>0"%'R6 U0%!"+./ 6

6./!9F6D0 #,B%R` D6%-0ajm£

:9

«

fVnM

"\9F60#,B%

`6%-0N(GS /#Q J /+ 0L%+;+%'"7‰ 6%'0/7U8:6 6 

%

%H–67/; #, HD

‚ Ž„

D  +,FF/3P.M « FJ+

XD0

j

+.+%'#*"+/

lG"# ;j

0  +FDFS+=@? A

8 =B? A ,DCE26F   

  O

 ,HGJI L=@? ANM  "  P 3ER K P(Q S UT  

:9 , WVX1T 

§

M?!+.+%'"3$%' J"& #( 12 j Zn § j €n(G*%bD;G="+,%G/&,"37+

+ ?`* ,(38NB"+,=%C3/3=>±cjhy

$+7FJ=I;` /–6

"%'R12 >£R%GM

+.""6 %IFJ""6%'7µ k"+,* 6"%9%'"+ K8s+ –"+,





%

/ %c=GF60#?+9>%'#/H"+,

7

M

  J0awrJu(U07 µ  " J"%2>  < +6%G"6^;@

7

W7

±cl\(?µ 

D%=%-' +,#4 J '8s+ ",(cF6 I%*%-G%'6FJ+,

/mM

P  '"6/; (Dµ E%'6TU %?/%'36D;+,%12 GI63 G"+,%



"+%'/0 #a+1

W7



M

$ #

12+,6 

eE+,%"+6/0Q.  ¤ +9*  +;0 ""%'+,$' ,M

(

)

/+%

5%* # ' 8^ ' ‚ „  ‚  „  a F

9(C 1b; 0a8:6+,

#

(9%'2+8

(CD0X$' %'3/+X8:6+,

\MCµ J"3[D/3.(.1cB/0".38s*\%-;

+8!H$p>³XF –

–

"/#\F60#,£ h( 0*F IG"+,"+,7+8A±c\. J  D@ * %+;+8?–8:+ ‡ ,£ M9_I"+*  § § %'9+8c DF%-"%\"+ ' "%' J+,0%\+E€* J+,%'%'F/ J+%-@

F,+

 + H±c=0%-' F6 /+%bD;b+.""6 2+/712$£



2W-* (  5 2 5' 5

/+%b+8!R$p>³X"+ ' "%' J+,0+\ "+/

2.3 Standard Policies *+ 0N%'/+,!M



%)c[D3@ˆ+ "+9 &V+;
+,

\/0>; F3'  37^; #,,‡ q:¡ur •r

u qs˜Žm˜B•

w˜q:Žmq:¡uBq:rŽuh”.uv

v'w "’/uš ‘CqsŽ j Exn ¤ jˆx ¤ nR˜Œ Ž •ŽŽ u •,Žmqhwr•Ž’/u'•˜ŽC©Žmq:š*u˜

”.uŽ)£

BuŽˆ•,j-e

Œ‹,“

w,’ qm *Žˆ• Vu˜9•˜Œ w DŽmq:š=•,’

_Rµ S#,+ 3

6"%97` J"0 "#  j

eZŽ uvu

enBwq:r˜Z•rJ‹

&

n

x,n

j

xn'nc%

6 #>6

y MŸiH%IK' %'3+, 8:6+,Ÿ%' J""3[D%

u

j

nc ^">+8 9j

nM

3.1 The Optimal Policy

   WN  % n ! y  n , "{– $# L , –  k n n %#&/  ; W ( ' c 7 Single Coin Lookahead (SCL). e,M

*j

j

"+=@

5 6+,Qxn–+8GK J+,XD;

6D0  G$+, N J+/N‡ j

n

`

j

nd

j

±c"##*bR"D0%-%GK127£

n

j

ng



j .n

fID0 J '8s+ *#

Z

$Z

Z‚s…!… | A{;… "… |  "… D{QA{

nZ

D‚ A… )„A,|} J„





µ 

;6 "%

e





&

!}4|"}:„A†]| A…

±cl

lH?jm%'en

±c60#,(J$%G0Eµ D0 07pB"< ^;+,%



±c6D0 #,(D$ µ ;%"( c+[J0 ">P.    1 0 %"( D0=# ,0 6D3*; #9R J+,=+





*/*"HR` J"0K "# M

0.25

nZ nZ Z

%'/+,!M?lG" J"0K" %'+ 0"%2+8!\$pR³Ÿ7FJB%'/–6/"0N(

0.50

0.75

1.0

Lambda

Z

D‚< A…

\‚s8=>A@?BADC¬D„FEGIH

+ =+6 IlBo 1%I""6/;0a8:+ =±cl\(€µ €(ZD0=[%'+;12$/_)F/9x.M

8^ "D0 9DB8:+ 9C

+,F/*%B"+%'/0 "0N(!"3@  R±cl

+ \µ  

+ 06"0$\%'/"%-R` J"0$ "# MHP

12/0 \ #9+8C J+,.%G$> D; * H%' D",(/"6D0 /#*+*FJ;>µ +, •r K+8Z\

+,F"*%"M?®a

7

Djm,M #MC"+[D0""B. +,F%' %'"G`3 12Ktu'•ŽmŒ vu˜–; –/600Et"u•ŽmŒ v'u\”v'wŒ DM n

+ R+,6

,



+ R`* ,( )#,6 7e9%'+12%B\ %'63%R+8?+ !+8CE%-;,M "+,6^0L"+* 6  4+ B J+,

%'+,*G%'""2%'+8A%-"%c/G%*B*+6.c+8!*G3

&,"%A " 8:+ "*".A; #H#".!+2"+* Z' /#MAP TU"( H+ ³

8s+ I6/+,O¯@

+.+%R*+,#=+,F ¥-"%"M

%"M ³) D %)/" "%-? #6*".G•”V•q:r˜Ž)6%'#BlHK8s+ ?/SI#" €F60#,0S; #7  +F@



'!rDuvŽˆ•q:rŽ

:

«

 

+;` + #,+ M

%=+46%'$" 8s6GF%-' +,%*+S' %-8:+ 

:Z

J„N|{}

nZ

 A|} D„ 7

l9M$+.+,.MJiG%'""+,NMDPGŠ(Jxf,f .M

1 &

7

d g>pIMi9M c+,!( ZM

D m(04MDM,+ 0!MDiH "< 1 %98:+ I –"+**".%"MlG6%'%'" #M

4w‹.uvr

"* J+ @0 TU ""B/" #=1278:6+,

#" +,6%%'6  J+ 'MQ®a$%'+aD&ap>oN/+' 0

%'+HD& %U?iHFJ 'bP#," 63

n

,x.M

d3e !;g1M 9MD_%'3%'& /%H0±HM –MlH+.M iG3 %'%H+8

c «

(Ae

d3e g\l9MµUMDµ 6 '+,$D07i9M

12/!  +;*+ 0+8

FJ' >8:;6 "%>8s+ \8:6+,   +` /;+,!(U1c= ""+,**D0



—i— — n

$•rJ‹+9•Žˆ•+ q:rq:r ” Cwrt"uvurJu(!e

d3e g–µUMlH6%'%'"AD0$³)M B+ < # M.GvŽmq )cqm•,’UŠrŽu’:’ q/”.urJu/0

+,F/]+8JF60#0I; # M?µ J""3[3.(VI2F%'"b+8

US y7

L*"".

` J *".%"M bŒ’s’/uŽmq:rSw-tRŽ u,Gš*uvqm•r-4•Ž uš*•Žmqm•’



_%' "%'63%

; ?*+%-)" 8:612"\"+,%'^0  #H   +V"%!8s+ !C#," 

±?+ 6 + %\12%'X+$&

;"%"M

d3e g\›–M lH+FF/%"MJµ +,*>%' J"%2+8ZR%'5.6 /A0 "%'#,+8

%'* /> J+//%B; \FR+*"+1 2 "# 21278m H"%'%

c

k

>q:˜w¡uv



6 –` @

J /* %"(A+,4

Acknowledgments

cMD³Z +; "@

0 3TU ".–8s6 $%'%98:+ I8:6+Q 

+;` +!M

&



/ #,$  +,F"*%

8s+ "*".H; #=+0 "%'/A R#6D; ."0DG±?l   #[ %-K"+/LkE%'M

12

_ 6%"(G±cl

12&V%'6FJ+,

/0 ""%'+,a12"