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cleartomark
%%EndFont 
%%BeginFont: CMMI10
%!PS-AdobeFont-1.1: CMMI10 1.100
%%CreationDate: 1996 Jul 23 07:53:57
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.100) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMMI10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-32 -250 1048 750}readonly def
/UniqueID 5087385 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMSY6
%!PS-AdobeFont-1.1: CMSY6 1.0
%%CreationDate: 1991 Aug 15 07:21:34
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMSY6) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.035 def
/isFixedPitch false def
end readonly def
/FontName /CMSY6 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-4 -948 1329 786}readonly def
/UniqueID 5000816 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: TeX-cmex8
%!PS-AdobeFont-1.0: TeX-cmex8 001.001
% Filtered by type1fix.pl 0.05
%%EndComments
13 dict dup begin
/FontInfo 16 dict dup begin
/Copyright (see\040copyright\040of\040original\040TeX\040font) def
/FamilyName (TeX\040cmex8) def
/FullName (TeX\040cmex8\040Regular) def
/ItalicAngle 0 def
/Notice (converted\040after\040April\0402001) def
/UnderlinePosition -100 def
/UnderlineThickness 50 def
/Weight (Regular) def
/isFixedPitch false def
/version (001.001) def
end readonly def
/FontName /TeX-cmex8 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 80 /P put
readonly def
/FontBBox {-29 -2957 1554 772} readonly def
/UniqueID 4314416 def
currentdict end
currentfile eexec
D9D66F633B846A97B686A97E45A3D0AA06DA87FC7163A5A2A756A598FAB07633
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cleartomark
%%EndFont 
%%BeginFont: CMSY8
%!PS-AdobeFont-1.1: CMSY8 1.0
%%CreationDate: 1991 Aug 15 07:22:10
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMSY8) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.035 def
/isFixedPitch false def
end readonly def
/FontName /CMSY8 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-30 -955 1185 779}readonly def
/UniqueID 5000818 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMEX10
%!PS-AdobeFont-1.1: CMEX10 1.00
%%CreationDate: 1992 Jul 23 21:22:48
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.00) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMEX10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMEX10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /parenleftbig put
dup 1 /parenrightbig put
dup 2 /bracketleftbig put
dup 3 /bracketrightbig put
dup 12 /vextendsingle put
dup 14 /slashbig put
dup 20 /bracketleftbigg put
dup 21 /bracketrightbigg put
dup 40 /braceleftBigg put
dup 56 /bracelefttp put
dup 58 /braceleftbt put
dup 60 /braceleftmid put
dup 62 /braceex put
dup 80 /summationtext put
dup 88 /summationdisplay put
dup 104 /bracketleftBig put
dup 105 /bracketrightBig put
dup 112 /radicalbig put
dup 115 /radicalBigg put
readonly def
/FontBBox{-24 -2960 1454 772}readonly def
/UniqueID 5000774 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: MSBM10
%!PS-AdobeFont-1.1: MSBM10 2.1
%%CreationDate: 1993 Sep 17 11:10:37
% Math Symbol fonts were designed by the American Mathematical Society.
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (2.1) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (MSBM10) readonly def
/FamilyName (Euler) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /MSBM10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 67 /C put
dup 70 /F put
dup 82 /R put
dup 91 /hatwide put
readonly def
/FontBBox{-55 -420 2343 920}readonly def
/UniqueID 5031982 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMSY10
%!PS-AdobeFont-1.1: CMSY10 1.0
%%CreationDate: 1991 Aug 15 07:20:57
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMSY10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.035 def
/isFixedPitch false def
end readonly def
/FontName /CMSY10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-29 -960 1116 775}readonly def
/UniqueID 5000820 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMMI12
%!PS-AdobeFont-1.1: CMMI12 1.100
%%CreationDate: 1996 Jul 27 08:57:55
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.100) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMMI12) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI12 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-30 -250 1026 750}readonly def
/UniqueID 5087386 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMMI6
%!PS-AdobeFont-1.1: CMMI6 1.100
%%CreationDate: 1996 Jul 23 07:53:52
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.100) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMMI6) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI6 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{11 -250 1241 750}readonly def
/UniqueID 5087381 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMR6
%!PS-AdobeFont-1.1: CMR6 1.0
%%CreationDate: 1991 Aug 20 16:39:02
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMR6) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR6 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-20 -250 1193 750}readonly def
/UniqueID 5000789 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMMI8
%!PS-AdobeFont-1.1: CMMI8 1.100
%%CreationDate: 1996 Jul 23 07:53:54
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.100) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMMI8) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI8 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-24 -250 1110 750}readonly def
/UniqueID 5087383 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMR8
%!PS-AdobeFont-1.1: CMR8 1.0
%%CreationDate: 1991 Aug 20 16:39:40
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMR8) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR8 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-36 -250 1070 750}readonly def
/UniqueID 5000791 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMR12
%!PS-AdobeFont-1.1: CMR12 1.0
%%CreationDate: 1991 Aug 20 16:38:05
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMR12) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR12 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-34 -251 988 750}readonly def
/UniqueID 5000794 def
currentdict end
currentfile eexec
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/FontInfo 7 dict dup begin
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/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
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/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
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/FontName /CMSY7 def
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/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
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b(is)h Fy([)p Fr(\000k)p Ft(f)11 b Fr(k)2821 2824 y Fx(2)2821
2885 y Fn(1)2896 2860 y Ft(;)17 b Fr(k)p Ft(f)11 b Fr(k)3099
2824 y Fx(2)3099 2885 y Fn(1)3173 2860 y Fy(])35 b Fz(rather)257
2980 y(than)j Fy([)p Fr(\000)p Fy(1)p Ft(;)17 b Fy(1])p
Fz(,)40 b(implying)35 b(that)j(the)g(n)m(um)m(b)s(er)g(of)g(lab)s(eled)
e(examples)i(from)f Ft(f)49 b Fz(needed)257 3101 y(to)f(estimate)g
Ft(C)7 b Fy(\()p Ft(S;)17 b(k)s Fy(\))48 b Fz(additionally)d(dep)s
(ends)50 b(on)e(the)h(parameter)e Fr(k)p Ft(f)11 b Fr(k)3081
3116 y Fn(1)3155 3101 y Fz(.)91 b(Since)257 3221 y Fr(k)p
Ft(f)11 b Fr(k)32 b(\024)h(k)p Ft(f)11 b Fr(k)717 3236
y Fn(1)791 3221 y Fz(,)36 b(the)g(running-time)d(b)s(ecomes)i(p)s
(olynomial)d(in)i Ft(n)p Fz(,)j Fy(1)p Ft(=\022)s Fz(,)e
Fy(log)q(\(1)p Ft(=\016)t Fy(\))g Fz(and)257 3342 y Fr(k)p
Ft(f)11 b Fr(k)416 3357 y Fn(1)490 3342 y Fz(.)44 b(Let)33
b(us)g(state)g(this)f(extension)h(to)f(real-v)-5 b(alued)31
b(functions)i(for)f(later)f(use.)257 3367 y
SDict begin H.S end
 257 3367
a 257 3367 a
SDict begin 14.5 H.A end
 257 3367 a 257 3367 a
SDict begin [ /View [/XYZ H.V] /Dest (theorem.4) cvn H.B /DEST pdfmark
end
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b(4.)41 b Fz([)808 3522 y
SDict begin H.S end
 808 3522 a Fz(19)905 3458 y
SDict begin H.R end

905 3458 a 905 3522 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.J97) cvn H.B /ANN pdfmark end
 905 3522 a Fz(])34 b Fm(Ther)-5
b(e)33 b(is)g(a)h(r)-5 b(andomize)g(d)32 b(algorithm)h
Ft(A)h Fm(such)f(that)h(for)g(e)-5 b(ach)33 b(func-)257
3643 y(tion)40 b Ft(f)22 b Fy(:)35 b Fr(f)p Fy(0)p Ft(;)17
b Fy(1)p Fr(g)838 3606 y Fw(n)920 3643 y Fr(!)37 b Fp(R)i
Fm(satisfying)h Fr(k)p Ft(f)11 b Fr(k)1766 3658 y Fn(1)1876
3643 y Fr(\024)38 b Ft(k)k Fm(and)e(thr)-5 b(eshold)39
b Ft(\022)g(>)e Fy(0)p Fm(,)k Ft(A)p Fy(\()p Ft(\022)s(;)17
b(\016)n(;)g(k)s(;)g(f)11 b Fy(\))257 3763 y Fm(outputs)35
b(with)f(pr)-5 b(ob)g(ability)34 b Fy(1)20 b Fr(\000)g
Ft(\016)38 b Fm(a)c(c)-5 b(ol)5 b(le)-5 b(ction)33 b
Ft(G)28 b Fr(\022)g Fy(2)2300 3727 y Fx([)p Fw(n)p Fx(])2420
3763 y Fm(c)-5 b(ontaining)33 b(al)5 b(l)33 b(fr)-5 b(e)g(quencies)257
3883 y Ft(S)41 b Fm(with)35 b Fr(j)619 3857 y Fy(^)598
3883 y Ft(f)10 b Fy(\()p Ft(S)c Fy(\))p Fr(j)27 b(\025)h
Ft(\022)s Fm(.)45 b Ft(A)35 b Fm(runs)g(in)f(time)h Fy(p)s(oly)r(\()p
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(olean)g(functions)h Ft(f)38 b Fr(2)28 b Ft(B)2120 4079
y Fw(n)2191 4064 y Fz(satisfying)23 b Fr(k)2688 4038
y Fy(^)2667 4064 y Ft(f)10 b Fr(k)2775 4079 y Fx(1)2842
4064 y Fr(\024)29 b Ft(k)s Fz(,)c(w)m(e)g(can)f(use)257
4184 y(the)k(algorithm)d(of)i(Lemma)1309 4184 y
SDict begin H.S end
 1309
4184 a Fz(3)1358 4120 y
SDict begin H.R end
 1358 4120 a 1358 4184 a
SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (theorem.3) cvn H.B /ANN pdfmark end
 1358
4184 a 27 w Fz(to)g(\034nd)h(a)g(small)d(collection)g
Ft(G)j Fz(con)m(taining)e(all)f(frequen-)257 4305 y(cies)35
b Ft(S)k Fz(for)34 b(whic)m(h)g Fr(j)1023 4278 y Fy(^)1002
4305 y Ft(f)11 b Fy(\()p Ft(S)6 b Fy(\))p Fr(j)29 b(\025)i
Ft("=k)36 b Fz(in)d(time)g Fy(p)s(oly)q(\()p Ft(n;)17
b(k)s(;)g Fy(1)p Ft(=";)g Fy(log\(1)p Ft(=\016)t Fy(\)\))p
Fz(,)34 b(and)g(then)g(use)257 4425 y(the)c(generalized)f(lo)m
(w-degree)h(algorithm)c(on)k Ft(G)g Fz(to)f(output)h(an)f
Ft(O)s Fy(\()p Ft(")p Fy(\))p Fz(-appro)m(ximation)257
4545 y(of)47 b Ft(f)11 b Fz(.)88 b(This)47 b(algorithm)d(is)j(kno)m(wn)
i(as)e(the)h Fm(Kushilevitz-Mansour)f(algorithm)p Fz(,)k(or)257
4666 y(simply)31 b Fm(KM-algorithm)p Fz(,)h(and)h(w)m(e)g(state)g(its)f
(prop)s(erties)h(in)f(the)h(follo)m(wing)c(theorem.)257
4685 y
SDict begin H.S end
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SDict begin 14.5 H.A end
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SDict begin [ /View [/XYZ H.V] /Dest (theorem.5) cvn H.B /DEST pdfmark
end
 257
4685 a 161 x Fs(Theorem)k(5.)38 b Fz([)880 4846 y
SDict begin H.S end
 880
4846 a Fz(30)978 4782 y
SDict begin H.R end
 978 4782 a 978 4846 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.KM93) cvn H.B /ANN pdfmark end
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a Fz(])31 b Fm(F)-7 b(or)31 b(e)-5 b(ach)30 b(Bo)-5 b(ole)g(an)30
b(function)h Ft(f)42 b Fm(satisfying)30 b Fr(k)2764 4820
y Fy(^)2743 4846 y Ft(f)11 b Fr(k)2852 4861 y Fx(1)2919
4846 y Fr(\024)28 b Ft(k)s Fm(,)k(the)f(KM-)257 4967
y(algorithm,)e(given)e(inputs)h Ft(k)s Fm(,)i Ft(")p
Fm(,)f Ft(\016)j Fm(and)c(or)-5 b(acle)27 b(ac)-5 b(c)g(ess)27
b(to)i Ft(f)11 b Fm(,)29 b(outputs)g(with)f(pr)-5 b(ob)g(ability)257
5087 y Fy(1)22 b Fr(\000)h Ft(\016)39 b Fm(an)34 b Ft(O)s
Fy(\()p Ft(")p Fy(\))p Fm(-appr)-5 b(oximation)32 b(of)j
Ft(f)45 b Fm(in)35 b(time)f Fy(p)s(oly)r(\()p Ft(n;)17
b(k)s(;)g Fy(1)p Ft(=";)g Fy(log)o(\(1)p Ft(=\016)t Fy(\)\))p
Fm(.)404 5268 y Fz(As)30 b(already)f(men)m(tioned)g(at)g(the)h(b)s
(eginning)e(of)h(this)g(section,)i(an)m(y)f(Bo)s(olean)e(func-)257
5388 y(tion)e Ft(f)38 b Fz(computable)26 b(b)m(y)i(a)f(decision)g(tree)
g(with)g Ft(m)g Fz(no)s(des)h(satis\034es)g Fr(k)2840
5362 y Fy(^)2819 5388 y Ft(f)11 b Fr(k)2928 5403 y Fx(1)2994
5388 y Fr(\024)29 b Ft(m)p Fz(.)42 b(Th)m(us,)p eop end
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SDict begin H.S end
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SDict begin H.R end
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SDict begin [ /View [/XYZ H.V] /Dest (page.13) cvn H.B /DEST pdfmark
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(needed.)257 713 y
SDict begin H.S end
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SDict begin 14.5 H.A end
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713 a
SDict begin [ /View [/XYZ H.V] /Dest (theorem.6) cvn H.B /DEST pdfmark
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905 y
SDict begin H.S end
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SDict begin H.R end
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a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.KM93) cvn H.B /ANN pdfmark end
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b(P)m(arsev)-5 b(al's)32 b(iden)m(tit)m(y)-8 b(,)31 b(the)h(n)m(um)m(b)
s(er)g(of)f(frequencies)h Ft(S)37 b Fz(with)31 b Fr(j)2733
1842 y Fy(^)2712 1868 y Ft(f)11 b Fy(\()p Ft(S)6 b Fy(\))p
Fr(j)27 b Ft(>)g("=k)34 b Fz(is)d(less)257 1989 y(than)44
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b Fz(if)32 b(it)h(is)g Ft(")p Fz(-close)f(to)h(a)g Ft(k)s
Fz(-sparse)h(function.)46 b(It)33 b(is)g(not)g(hard)h(to)257
2229 y(sho)m(w)g(\(see)g([)722 2229 y
SDict begin H.S end
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2165 y
SDict begin H.R end
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SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.KM93) cvn H.B /ANN pdfmark end
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Fy(\()p Ft(";)17 b(k)s Fy(\))p Fz(-sparse)33 b(function)f
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2589 y Fx(^)1340 2607 y Fw(f)6 b Fx(\()p Fw(S)t Fx(\))p
Fn(j\024)p Fw("=k)1701 2446 y Fy(^)1679 2472 y Ft(f)11
b Fy(\()p Ft(S)6 b Fy(\))1880 2431 y Fx(2)1947 2472 y
Fr(\024)28 b Ft(")22 b Fy(+)g Ft(")2264 2431 y Fx(2)2303
2472 y Ft(=k)s(:)257 2821 y Fz(Th)m(us,)35 b(the)d(KM-algorithm)d(is)k
(applicable)d(to)i Fy(\()p Ft(";)17 b(k)s Fy(\))p Fz(-sparse)33
b(Bo)s(olean)e(functions.)257 2846 y
SDict begin H.S end
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a
SDict begin 14.5 H.A end
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SDict begin [ /View [/XYZ H.V] /Dest (theorem.7) cvn H.B /DEST pdfmark
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b Fz([)902 3033 y
SDict begin H.S end
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SDict begin H.R end
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a 1000 3033 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.KM93) cvn H.B /ANN pdfmark end
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b(ach)29 b Fy(\()p Ft(";)17 b(k)s Fy(\))p Fm(-sp)-5 b(arse)28
b(function)i Ft(f)11 b Fm(,)30 b(the)g(KM-algorithm)f(given)257
3153 y(inputs)36 b Ft(k)s Fm(,)f Ft(")p Fm(,)g Ft(\016)40
b Fm(and)35 b(or)-5 b(acle)34 b(ac)-5 b(c)g(ess)35 b(to)g
Ft(f)47 b Fm(outputs)36 b(an)f Ft(O)s Fy(\()p Ft(")p
Fy(\))p Fm(-appr)-5 b(oximation)32 b(of)j Ft(f)47 b Fm(in)257
3274 y(time)35 b Fy(p)s(oly)r(\()p Ft(n;)17 b(k)s(;)g
Fy(1)p Ft(=";)g Fy(log)o(\(1)p Ft(=\016)t Fy(\)\))p Fm(.)404
3485 y Fz(Mansour)48 b([)850 3485 y
SDict begin H.S end
 850 3485 a Fz(36)948
3421 y
SDict begin H.R end
 948 3421 a 948 3485 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.M95) cvn H.B /ANN pdfmark end
 948 3485 a Fz(])f(sho)m(w)m(ed)j(that)d
(eac)m(h)i(DNF)e(with)h(terms)f(of)g(size)h(at)f(most)g
Ft(d)h Fz(is)257 3606 y Fy(\()p Ft(";)17 b(k)s Fy(\))p
Fz(-sparse)56 b(for)e Ft(k)70 b Fy(=)c Ft(d)1311 3570
y Fw(O)r Fx(\()p Fw(d)11 b Fx(log)r(\(1)p Fw(=")p Fx(\)\))1722
3606 y Fz(.)112 b(By)55 b(an)h(argumen)m(t)e(attributed)h(to)g(W)-8
b(ar-)257 3726 y(m)m(uth)46 b(in)g([)683 3726 y
SDict begin H.S end
 683 3726
a Fz(44)780 3662 y
SDict begin H.R end
 780 3662 a 780 3726 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.V90) cvn H.B /ANN pdfmark end
 780 3726 a Fz(],)k(ev)m(ery)e
Ft(m)p Fz(-term)d(DNF)h(is)g Ft(")p Fz(-close)f(to)h(a)g(DNF)g(with)f
(terms)h(of)g(size)257 3847 y(at)28 b(most)g Fy(log)q(\()p
Ft(m=")p Fy(\))g Fz(\(b)m(y)h(simply)d(ignoring)g(all)g(terms)i(of)g
(size)g(larger)f(than)i Fy(log)q(\()p Ft(m=")p Fy(\))p
Fz(\).)257 3967 y(Hence,)57 b(an)51 b Ft(m)p Fz(-term)f(DNF)h
Ft(f)62 b Fz(is)50 b Ft(")p Fz(-close)g(to)g(a)h Fy(\()p
Ft(";)17 b(k)s Fy(\))p Fz(-sparse)51 b(function)f(for)g
Ft(k)62 b Fy(=)257 4087 y(\(log)q(\()p Ft(m=")p Fy(\)\))715
4051 y Fw(O)r Fx(\(log)q(\()p Fw(m=")p Fx(\))13 b(log)q(\(1)p
Fw(=")p Fx(\)\))1399 4087 y Fy(=)32 b(\()p Ft(m=")p Fy(\))1763
4051 y Fw(O)r Fx(\(log)12 b(log)q(\()p Fw(m=")p Fx(\))h(log)q(\(1)p
Fw(=")p Fx(\)\))2517 4087 y Fz(,)36 b(whic)m(h)g(b)m(y)g(the)f
(triangle)257 4208 y(inequalit)m(y)d(implies)e(that)i
Ft(f)43 b Fz(itself)32 b(is)g Fy(\()p Ft(O)s Fy(\()p
Ft(")p Fy(\))p Ft(;)17 b(k)s Fy(\))p Fz(-sparse.)257
4233 y
SDict begin H.S end
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SDict begin 14.5 H.A end
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SDict begin [ /View [/XYZ H.V] /Dest (theorem.8) cvn H.B /DEST pdfmark
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4233 a 186 x Fs(Corollary)28 b(8.)36 b Fz([)897 4419
y
SDict begin H.S end
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SDict begin H.R end
 995 4355 a 995 4419 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.M95) cvn H.B /ANN pdfmark end
 995
4419 a Fz(])27 b Fm(The)g(class)g(of)g Ft(m)p Fm(-term)h(DNF)f
(formulas)f(is)i(le)-5 b(arnable)26 b(with)h(mem-)257
4540 y(b)-5 b(ership)34 b(queries)h(in)f(time)h Fy(p)s(oly)q(\()p
Ft(n;)17 b Fy(\()p Ft(m=")p Fy(\))1842 4504 y Fx(log)c(log)q(\()p
Fw(m=")p Fx(\))g(log)q(\(1)p Fw(=")p Fx(\))2486 4540
y Ft(;)k Fy(1)p Ft(=";)g Fy(log\(1)p Ft(=\016)t Fy(\)\))p
Fm(.)257 4713 y
SDict begin H.S end
 257 4713 a 257 4713 a
SDict begin 14.5 H.A end
 257 4713 a 257
4713 a
SDict begin [ /View [/XYZ H.V] /Dest (subsection.3.3) cvn H.B /DEST
pdfmark end
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(quasip)s(olynomial-time)17 b(learning)k(algorithm)f(for)257
5147 y(DNF)32 b(form)m(ulas)e(based)i(on)f(the)h(sparseness)i(of)c
(these)j(functions.)43 b(Another)32 b(prop)s(ert)m(y)257
5268 y(of)38 b(the)g(F)-8 b(ourier)37 b(transform)g(of)g
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(e)g(lo)m(w)m(er)257 5388 y(b)s(ounded)i(in)e(terms)h(of)g
Ft(m)g Fz([)1329 5388 y
SDict begin H.S end
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2907 y Ft(M)2544 2922 y Fw(i)2573 2907 y Fy(\()p Ft(x)p
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Fz(together)g(with)g(the)h(v)-5 b(alue)30 b Ft(M)3063
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b(can)h(b)s(e)g(exactly)g(computed)g(from)e Ft(f)11 b
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SDict begin H.S end
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SDict begin H.R end
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3509 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
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SDict begin H.S end
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SDict begin 14.5 H.A end
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SDict begin [ /View [/XYZ H.V] /Dest (theorem.10) cvn H.B /DEST pdfmark
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SDict begin H.S end
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SDict begin H.R end
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SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
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y
SDict begin H.S end
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SDict begin H.R end
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SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
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SDict begin H.S end
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SDict begin 14.5 H.A end
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SDict begin [ /View [/XYZ H.V] /Dest (theorem.11) cvn H.B /DEST pdfmark
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SDict begin H.S end
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SDict begin H.R end
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SDict begin [ /View [/XYZ H.V] /Dest (Hfootnote.3) cvn H.B /DEST pdfmark
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SDict begin H.S end
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SDict begin H.S end
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SDict begin [ /View [/XYZ H.V] /Dest (subsection.3.4) cvn H.B /DEST
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SDict begin 14.5 H.A end

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SDict begin H.S end
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/Dest (cite.JKS02) cvn H.B /ANN pdfmark end
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SDict begin H.S end
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SDict begin H.R end
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SDict begin 14.5 H.A end
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SDict begin [ /View [/XYZ H.V] /Dest (theorem.13) cvn H.B /DEST pdfmark
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SDict begin H.S end
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SDict begin H.R end

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SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
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y Fw(n)2463 3526 y Fr(g)p Fz(.)404 3646 y(In)37 b(this)h(section)f(w)m
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3887 y(and)37 b(T)-8 b(amon's)37 b([)876 3887 y
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a Fz(9)925 3823 y
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 925 3823 a 925 3887 a
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/Dest (cite.BT96) cvn H.B /ANN pdfmark end
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f(monotone)f(functions,)h(Serv)m(edio's)h(learnabilit)m(y)c(result)257
4008 y(for)29 b(monotone)f(DNF)h([)1116 4008 y
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a Fz(40)1214 3943 y
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/Dest (cite.S04) cvn H.B /ANN pdfmark end
 1214 4008
a Fz(],)h(and)f(the)g(v)m(ery)h(recen)m(t)h(learning)c(algorithm)f(of)i
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g(trees)i([)2062 4128 y
SDict begin H.S end
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2160 4064 a 2160 4128 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.ODS06) cvn H.B /ANN pdfmark end
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b(start)f(with)g(a)f(discussion)i(of)257 4248 y(the)g(in\035uence)h
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Ft(f)11 b Fz(.)257 4507 y Fs(In\035uence)39 b(and)f(sensitivit)m(y)257
4591 y
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4591 a 101 x Fz(The)51 b(concept)g(of)e(in\035uence)i(of)e(a)h(v)-5
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4812 y(duced)45 b(b)m(y)g(Ben-Or)f(and)f(Linial)e([)1570
4812 y
SDict begin H.S end
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4812 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
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b(a)g(Bo)s(olean)e(function)h(on)h Ft(n)g Fz(v)-5 b(ari-)257
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b(:)g(:)g(:)f(;)h(x)851 5188 y Fw(n)898 5173 y Fy(\))32
b Fz(c)m(hanges)i(the)f(v)-5 b(alue)32 b(of)g Ft(f)11
b Fz(.)43 b(More)33 b(formally)-8 b(,)1253 5388 y Ft(I)1296
5403 y Fw(j)1333 5388 y Fy(\()p Ft(f)11 b Fy(\))27 b(=)h(Pr[)p
Ft(f)11 b Fy(\()p Ft(x)p Fy(\))28 b Fr(6)p Fy(=)f Ft(f)11
b Fy(\()p Ft(x)23 b Fr(\010)f Ft(e)2370 5403 y Fw(j)2407
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2683 y Ft(I)1469 2698 y Fw(j)1506 2683 y Fy(\(MAJ)1756
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2963 y Fw(n)1480 2948 y Fy(\))41 b Ft(<)1676 2863 y Fo(p)p
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y
SDict begin H.S end
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a
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/Dest (cite.FK96) cvn H.B /ANN pdfmark end
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3309 y
SDict begin H.S end
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SDict begin H.R end
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3309 a
SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link
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3775 y(since)33 b(all)e(v)-5 b(ariables)31 b Ft(x)1335
3790 y Fw(j)1404 3775 y Fz(with)h Ft(j)i Fr(62)28 b Ft(R)34
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3999 y
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SDict begin H.S end
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2687 4240 a
SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link
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y
SDict begin H.S end
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y Fy(^)1038 4616 y Ft(f)1086 4631 y Fn(\010)p Fw(e)1174
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y Fy(^)2516 4622 y Ft(f)11 b Fy(\()p Ft(S)6 b Fy(\))p
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SDict begin H.S end
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SDict begin 17.5 H.A end
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SDict begin [ /View [/XYZ H.V] /Dest (equation.5) cvn H.B /DEST pdfmark
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b(or)48 b(a)h(bit)f Ft(b)56 b Fr(2)f(f)p Fy(0)p Ft(;)17
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Fw(j;b)2104 1960 y Fz(to)e(denote)i(the)f(Bo)s(olean)f(function)257
2080 y Ft(f)305 2095 y Fw(j;b)388 2080 y Fy(\()p Ft(x)p
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2080 y Ft(;)17 b(:)g(:)g(:)f(;)h(x)1116 2095 y Fw(j)t
Fn(\000)p Fx(1)1243 2080 y Ft(;)g(b;)g(x)1427 2095 y
Fw(j)t Fx(+1)1554 2080 y Ft(;)g(:)g(:)g(:)e(;)i(n)p Fy(\))p
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2861 y Fw(j)2030 2846 y Fy(])p 1376 2897 681 4 v 1692
2988 a(2)2094 2920 y(=)2229 2820 y(^)2208 2846 y Ft(f)11
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2846 y Ft(f)2558 2861 y Fn(\010)p Fw(e)2646 2871 y Fu(j)2682
2846 y Fy(\()p Ft(j)6 b Fy(\))p 2208 2897 597 4 v 2482
2988 a(2)2842 2920 y(=)2967 2893 y(^)2946 2920 y Ft(f)2994
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SDict begin [ /View [/XYZ H.V] /Dest (theorem.20) cvn H.B /DEST pdfmark
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SDict begin H.S end
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SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link
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SDict begin 14.5 H.A end
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SDict begin H.S end
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SDict begin H.R end
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SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link
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SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
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257 1269 y
SDict begin H.S end
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SDict begin 14.5 H.A end
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SDict begin [ /View [/XYZ H.V] /Dest (theorem.24) cvn H.B /DEST pdfmark
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SDict begin 14.5 H.A end
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SDict begin [ /View [/XYZ H.V] /Dest (theorem.25) cvn H.B /DEST pdfmark
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SDict begin H.S end
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SDict begin H.R end
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/Dest (theorem.24) cvn H.B /ANN pdfmark end
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SDict begin [ /View [/XYZ H.V] /Dest (theorem.27) cvn H.B /DEST pdfmark
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3445 a 257 3445 a
SDict begin 14.5 H.A end
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SDict begin [ /View [/XYZ H.V] /Dest (theorem.28) cvn H.B /DEST pdfmark
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x Fs(Corollary)38 b(28.)j Fz([)968 3545 y
SDict begin H.S end
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Fz(9)1017 3481 y
SDict begin H.R end
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SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.BT96) cvn H.B /ANN pdfmark end
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Fz(])35 b Fm(The)f(class)g(of)h(monotone)f Ft(k)s Fm(-juntas)g(is)h(le)
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SDict begin H.S end
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SDict begin 14.5 H.A end
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a 257 3987 a
SDict begin [ /View [/XYZ H.V] /Dest (subsection.4.3) cvn H.B /DEST
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SDict begin H.S end
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SDict begin H.R end
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Fy(\))27 b(=)997 826 y Fo(X)957 1037 y Fw(S)11 b Fx(:)23
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b Fy(\()p Ft(S)6 b Fy(\))1398 879 y Fx(2)1465 920 y Fy(=)1608
826 y Fo(X)1568 1037 y Fw(S)12 b Fx(:)23 b Fw(j)t Fn(2)p
Fw(S)1792 920 y Fy(\()1851 894 y(^)1830 920 y Ft(f)10
b Fy(\()p Ft(S)c Fy(\))22 b Fr(\000)2153 894 y Fy(^)2152
920 y Ft(h)p Fy(\()p Ft(S)6 b Fy(\)\))2388 879 y Fx(2)2455
920 y Fr(\024)28 b(k)p Ft(f)33 b Fr(\000)22 b Ft(h)p
Fr(k)2896 879 y Fx(2)2963 920 y Fr(\024)28 b Ft(":)257
1229 y Fz(This)33 b(sho)m(ws)h(that)e(an)m(y)i(v)-5 b(ariable)30
b(in)i Ft(R)q Fy(\()p Ft(")p Fy(\))g Fz(m)m(ust)h(b)s(e)g(relev)-5
b(an)m(t)32 b(for)g Ft(h)p Fz(.)p 3446 1229 49 49 v 404
1441 a(In)40 b(the)h(next)g(prop)s(osition)e(w)m(e)i(b)s(ound)g(the)f
(w)m(eigh)m(t)h(of)f(the)h(high)e(order)h(F)-8 b(ourier)257
1562 y(sp)s(ectrum)33 b(of)f Ft(f)43 b Fz(under)33 b(the)g(assumption)f
(that)g Ft(f)43 b Fz(is)32 b(close)g(to)g(some)g(function)g
Ft(g)k Fz(with)257 1682 y(a)d(small)d(w)m(eigh)m(t)j(on)f(this)g
(region.)257 1702 y
SDict begin H.S end
 257 1702 a 257 1702 a
SDict begin 14.5 H.A end
 257 1702 a
257 1702 a
SDict begin [ /View [/XYZ H.V] /Dest (theorem.30) cvn H.B /DEST pdfmark
end
 257 1702 a 172 x Fs(Prop)s(osition)25 b(30.)35
b Fm(F)-7 b(or)25 b(any)h(function)g Ft(f)37 b Fm(that)27
b(is)f Ft(")p Fm(-close)f(to)h(some)g(Bo)-5 b(ole)g(an)25
b(function)257 1994 y Ft(g)39 b Fm(with)554 1920 y Fo(P)659
2023 y Fn(j)p Fw(S)t Fn(j\025)p Fw(t)850 1994 y Fy(^)-53
b Ft(g)t Fy(\()p Ft(S)6 b Fy(\))1039 1958 y Fx(2)1105
1994 y Fr(\024)29 b Ft(")p Fm(,)1549 2058 y Fo(X)1538
2274 y Fn(j)p Fw(S)t Fn(j\025)p Fw(t)1742 2127 y Fy(^)1721
2153 y Ft(f)10 b Fy(\()p Ft(S)c Fy(\))1921 2112 y Fx(2)1988
2153 y Fr(\024)28 b Fy(4)p Ft(":)257 2476 y Fs(Pro)s(of.)51
b Fz(Using)34 b(the)h(inequalit)m(y)1533 2450 y Fy(^)1512
2476 y Ft(f)1571 2440 y Fx(2)1642 2476 y Fy(=)c(\()1809
2450 y(^)1787 2476 y Ft(f)k Fr(\000)27 b Fy(^)-52 b Ft(g)27
b Fy(+)g(^)-52 b Ft(g)s Fy(\))2233 2440 y Fx(2)2304 2476
y Fr(\024)32 b Fy(2\()2521 2450 y(^)2500 2476 y Ft(f)i
Fr(\000)28 b Fy(^)-53 b Ft(g)t Fy(\))2772 2440 y Fx(2)2835
2476 y Fy(+)23 b(2)s(^)-52 b Ft(g)3034 2440 y Fx(2)3108
2476 y Fz(\(Cauc)m(h)m(y-)257 2596 y(Sc)m(h)m(w)m(arz\))35
b(it)c(follo)m(ws)h(that)752 2726 y Fo(X)741 2942 y Fn(j)p
Fw(S)t Fn(j\025)p Fw(t)945 2794 y Fy(^)924 2820 y Ft(f)11
b Fy(\()p Ft(S)6 b Fy(\))1125 2779 y Fx(2)1191 2820 y
Fr(\024)28 b Fy(2)1373 2726 y Fo(X)1362 2942 y Fn(j)p
Fw(S)t Fn(j\025)p Fw(t)1528 2820 y Fy(\()1587 2794 y(^)1566
2820 y Ft(f)11 b Fy(\()p Ft(S)6 b Fy(\))22 b Fr(\000)k
Fy(^)-53 b Ft(g)s Fy(\()p Ft(S)6 b Fy(\)\))2118 2779
y Fx(2)2180 2820 y Fy(+)22 b(2)2354 2726 y Fo(X)2344
2942 y Fn(j)p Fw(S)t Fn(j\025)p Fw(t)2529 2820 y Fy(^)-52
b Ft(g)s Fy(\()p Ft(S)6 b Fy(\))2718 2779 y Fx(2)2785
2820 y Fr(\024)28 b Fy(4)p Ft(":)p 3446 2993 49 49 v
404 3206 a Fz(No)m(w)h(assume)g(that)f Ft(f)39 b Fz(is)28
b(an)g Fy(\()p Ft("=n;)17 b(k)s Fy(\))p Fz(-jun)m(ta)28
b(with)g(the)h(additional)c(prop)s(ert)m(y)k(that)257
3326 y(it)f(is)f Ft(")p Fz(-close)h(to)f(some)h(Bo)s(olean)f(function)h
Ft(g)j Fz(whose)e(F)-8 b(ourier)27 b(w)m(eigh)m(t)i(on)f(the)g
(frequen-)257 3446 y(cies)d Ft(S)31 b Fz(with)24 b Fr(j)p
Ft(S)6 b Fr(j)27 b(\025)h Ft(t)d Fz(is)g(b)s(ounded)g(b)m(y)h
Ft(")p Fz(.)41 b(Then)26 b(b)m(y)f(using)g(Prop)s(osition)2915
3446 y
SDict begin H.S end
 2915 3446 a Fz(24)3012 3382 y
SDict begin H.R end
 3012 3382 a 3012
3446 a
SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (theorem.24) cvn H.B /ANN pdfmark end
 3012 3446 a 25 w Fz(with)g Ft(k)30 b Fy(=)e Ft(n)257
3567 y Fz(as)39 b(w)m(ell)e(as)h(Prop)s(osition)1241
3567 y
SDict begin H.S end
 1241 3567 a Fz(30)1339 3503 y
SDict begin H.R end
 1339 3503 a 1339
3567 a
SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (theorem.30) cvn H.B /ANN pdfmark end
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(collection)c Ft(G)p Fy(\()p Ft("=n;)17 b(t)p Fy(\))257
3687 y Fz(has)33 b(the)g(prop)s(ert)m(y)871 3811 y Fo(X)764
4027 y Fw(S)t Fn(62)p Fw(G)p Fx(\()p Fw("=n;t)p Fx(\))1161
3879 y Fy(^)1139 3906 y Ft(f)11 b Fy(\()p Ft(S)6 b Fy(\))1340
3864 y Fx(2)1407 3906 y Fr(\024)1600 3811 y Fo(X)1512
4027 y Fw(S)t Fn(6\022)p Fw(R)p Fx(\()p Fw("=n)p Fx(\))1871
3879 y Fy(^)1849 3906 y Ft(f)11 b Fy(\()p Ft(S)6 b Fy(\))2050
3864 y Fx(2)2111 3906 y Fy(+)2220 3811 y Fo(X)2209 4027
y Fn(j)p Fw(S)t Fn(j\025)p Fw(t)2413 3879 y Fy(^)2392
3906 y Ft(f)11 b Fy(\()p Ft(S)6 b Fy(\))2593 3864 y Fx(2)2660
3906 y Fy(=)27 b Ft(O)s Fy(\()p Ft(")p Fy(\))p Ft(:)257
4223 y Fz(This)38 b(means)g(that)g Ft(f)49 b Fz(can)38
b(b)s(e)g Ft(O)s Fy(\()p Ft(")p Fy(\))p Fz(-appro)m(ximated)e(b)m(y)j
(using)e(only)h(F)-8 b(ourier)36 b(co)s(e\036-)257 4343
y(cien)m(ts)30 b(corresp)s(onding)f(to)g(frequencies)i(in)e
Ft(G)p Fy(\()p Ft("=n;)17 b(t)p Fy(\))p Fz(.)42 b(Since)29
b(b)m(y)h(Prop)s(osition)3233 4343 y
SDict begin H.S end
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4279 y
SDict begin H.R end
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SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (theorem.29) cvn H.B /ANN pdfmark end
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4463 y(size)g(of)f Ft(R)q Fy(\()p Ft("=n)p Fy(\))h Fz(is)f(b)s(ounded)h
(b)m(y)h Ft(k)s Fz(,)f(it)f(further)h(follo)m(ws)e(that)h(the)i(size)e
(of)h Ft(G)p Fy(\()p Ft("=n;)17 b(t)p Fy(\))29 b Fz(is)257
4584 y(b)s(ounded)h(b)m(y)h Ft(k)839 4548 y Fw(t)868
4584 y Fz(.)43 b(Th)m(us)31 b(w)m(e)f(can)g(use)g(the)g(algorithm)c
Fg(Find-Variables)33 b Fz(to)c(compute)257 4704 y(a)h(sup)s(erset)i
Ft(R)792 4668 y Fn(\003)862 4704 y Fz(of)e Ft(R)q Fy(\()p
Ft("=)p Fy(2)p Ft(n)p Fy(\))f Fz(con)m(taining)g(only)h(v)-5
b(ariables)29 b(in)g Ft(R)q Fy(\()p Ft("=n)p Fy(\))h
Fz(in)g(order)g(to)g(get)257 4825 y(the)j(follo)m(wing)d(learning)h
(result.)257 4844 y
SDict begin H.S end
 257 4844 a 257 4844 a
SDict begin 14.5 H.A end
 257 4844 a
257 4844 a
SDict begin [ /View [/XYZ H.V] /Dest (theorem.31) cvn H.B /DEST pdfmark
end
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Fm(Ther)-5 b(e)61 b(is)g(a)h(r)-5 b(andomize)g(d)60 b(algorithm)h
Ft(A)g Fm(such)h(that)g(for)f(e)-5 b(ach)257 5137 y(monotone)41
b Fy(\()p Ft("=n;)17 b(k)s Fy(\))p Fm(-junta)41 b Ft(f)53
b Fm(that)42 b(is)f Ft(")p Fm(-close)g(to)h(some)f(Bo)-5
b(ole)g(an)40 b(function)h Ft(g)k Fm(with)257 5182 y
Fo(P)363 5286 y Fn(j)p Fw(S)t Fn(j\025)p Fw(t)553 5257
y Fy(^)-53 b Ft(g)t Fy(\()p Ft(S)6 b Fy(\))742 5221 y
Fx(2)809 5257 y Fr(\024)28 b Ft(")p Fm(,)33 b Ft(A)p
Fy(\()p Ft(k)s(;)17 b(t;)g(";)g(\016)n(;)g(E)6 b(X)i
Fy(\()p Ft(f)j Fy(\)\))32 b Fm(outputs)i(with)g(pr)-5
b(ob)g(ability)33 b Fy(1)19 b Fr(\000)g Ft(\016)38 b
Fm(an)33 b Ft(O)s Fy(\()p Ft(")p Fy(\))p Fm(-)257 5388
y(appr)-5 b(oximation)34 b(for)g Ft(f)46 b Fm(in)34 b(time)h
Fy(p)s(oly)r(\()p Ft(n;)17 b(k)1869 5352 y Fw(t)1898
5388 y Ft(;)g Fy(log)q(\(1)p Ft(=\016)t Fy(\)\))p Fm(.)p
eop end
%%Page: 27 27
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SDict begin H.S end
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a 257 266 a
SDict begin H.R end
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SDict begin [ /View [/XYZ H.V] /Dest (page.27) cvn H.B /DEST pdfmark
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b(b)m(y)h(Prop)s(osition)1317 573 y
SDict begin H.S end
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509 y
SDict begin H.R end
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SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (theorem.21) cvn H.B /ANN pdfmark end
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b Fy(\()p Ft(g)t Fy(\))27 b Fr(\024)h Ft(l)34 b Fz(implies)c(that)h
(the)i(F)-8 b(ourier)30 b(w)m(eigh)m(t)j(of)e Ft(g)k
Fz(on)257 693 y(the)27 b(frequencies)g Ft(S)k Fz(with)26
b Fr(j)p Ft(S)6 b Fr(j)26 b Ft(>)i(l)r(=")d Fz(is)h(b)s(ounded)g(b)m(y)
h Ft(")p Fz(,)g(w)m(e)f(also)f(obtain)g(the)h(follo)m(wing)257
814 y(corollary)-8 b(.)257 833 y
SDict begin H.S end
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SDict begin 14.5 H.A end

257 833 a 257 833 a
SDict begin [ /View [/XYZ H.V] /Dest (theorem.32) cvn H.B /DEST pdfmark
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b Fm(Ther)-5 b(e)30 b(is)i(a)f(r)-5 b(andomize)g(d)30
b(algorithm)h Ft(A)h Fm(such)f(that)h(for)f(e)-5 b(ach)31
b(mono-)257 1131 y(tone)39 b Fy(\()p Ft("=n;)17 b(k)s
Fy(\))p Fm(-junta)38 b Ft(f)49 b Fm(that)39 b(is)g Ft(")p
Fm(-close)e(to)i(some)f(Bo)-5 b(ole)g(an)37 b(function)i
Ft(g)j Fm(with)c(in\035u-)257 1251 y(enc)-5 b(e)44 b
Ft(I)8 b Fy(\()p Ft(g)t Fy(\))45 b Fr(\024)h Ft(l)r Fm(,)h
Ft(A)p Fy(\()p Ft(k)s(;)17 b(l)r(;)g(";)g(\016)n(;)g(E)6
b(X)i Fy(\()p Ft(f)j Fy(\)\))43 b Fm(outputs)j(with)e(pr)-5
b(ob)g(ability)44 b Fy(1)29 b Fr(\000)h Ft(\016)48 b
Fm(an)d Ft(O)s Fy(\()p Ft(")p Fy(\))p Fm(-)257 1371 y(appr)-5
b(oximation)34 b(for)g Ft(f)46 b Fm(in)34 b(time)h Fy(p)s(oly)r(\()p
Ft(n;)17 b(k)1869 1335 y Fw(l)q(=")1962 1371 y Ft(;)g
Fy(log)q(\(1)p Ft(=\016)t Fy(\)\))p Fm(.)404 1568 y Fz(Since)34
b(for)g(an)m(y)h Ft(")c(>)g Fy(0)p Fz(,)j(an)h Ft(m)p
Fz(-term)e(DNF)i(is)f(an)g Fy(\()p Ft(";)17 b(m)g Fy(log)q(\()p
Ft(m=")p Fy(\)\))p Fz(-jun)m(ta,)34 b(Corol-)257 1688
y(lary)445 1688 y
SDict begin H.S end
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SDict begin H.R end
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a 542 1688 a
SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (theorem.32) cvn H.B /ANN pdfmark end
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(learning)g(result)i(for)g(monotone)g Ft(m)p Fz(-term)f(DNF,)i(whic)m
(h)257 1809 y(is)32 b(in)g(fact)h(a)f(consequence)j(of)d(a)h(stronger)f
(result)h(from)e([)2394 1809 y
SDict begin H.S end
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1745 y
SDict begin H.R end
 2443 1745 a 2443 1809 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.BT96) cvn H.B /ANN pdfmark end
 2443 1809 a Fz(].)257
1834 y
SDict begin H.S end
 257 1834 a 257 1834 a
SDict begin 14.5 H.A end
 257 1834 a 257 1834 a
SDict begin [ /View [/XYZ H.V] /Dest (theorem.33) cvn H.B /DEST pdfmark
end
 257
1834 a 171 x Fs(Corollary)g(33.)37 b Fz([)957 2005 y
SDict begin H.S end

957 2005 a Fz(9)1006 1941 y
SDict begin H.R end
 1006 1941 a 1006 2005 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.BT96) cvn H.B /ANN pdfmark end
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2005 a Fz(])29 b Fm(The)g(class)f(of)h(monotone)f Ft(m)p
Fm(-term)i(DNF)f(formulas)f(is)h(le)-5 b(arnable)257
2126 y(in)35 b(time)967 2246 y Fy(p)s(oly)r(\()p Ft(n;)17
b Fy(\()p Ft(m)g Fy(log\()p Ft(n=")p Fy(\)\))1825 2139
y Fr(p)p 1908 2139 351 4 v 66 x Fw(m)12 b Fx(log)q(\()p
Fw(m=")p Fx(\))q Fw(=")2330 2246 y Ft(;)17 b Fy(log)q(\(1)p
Ft(=\016)t Fy(\)\))p Ft(:)257 2417 y Fm(Henc)-5 b(e,)43
b(for)f Ft(m)e Fy(=)g Ft(O)s Fy(\(\(log)17 b Ft(n)p Fy(\))1383
2381 y Fx(2)1423 2417 y Ft(=)p Fy(\(log)g(log)g Ft(n)p
Fy(\))1890 2381 y Fx(3)1930 2417 y Fy(\))42 b Fm(and)f(c)-5
b(onstant)41 b Ft(")p Fm(,)i(monotone)d Ft(m)p Fm(-term)257
2538 y(DNF)35 b(formulas)f(ar)-5 b(e)34 b(le)-5 b(arnable)34
b(in)h(p)-5 b(olynomial)33 b(time.)404 2734 y Fz(Serv)m(edio)44
b([)832 2734 y
SDict begin H.S end
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SDict begin H.R end
 930 2670 a
930 2734 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.S04) cvn H.B /ANN pdfmark end
 930 2734 a Fz(])g(used)h(a)f(b)s(ound)g(of)g(Mansour)h([)
2206 2734 y
SDict begin H.S end
 2206 2734 a Fz(36)2303 2670 y
SDict begin H.R end
 2303 2670 a
2303 2734 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.M95) cvn H.B /ANN pdfmark end
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Ft(m)p Fz(-term)257 2855 y(DNF)33 b Ft(f)43 b Fz(is)31
b Ft(")p Fz(-close)h(to)g(a)g(Bo)s(olean)e(function)i
Ft(g)j Fz(satisfying)2474 2780 y Fo(P)2579 2884 y Fn(j)p
Fw(S)t Fn(j)p Fw(>t)2769 2855 y Fy(^)-52 b Ft(g)s Fy(\()p
Ft(S)6 b Fy(\))2958 2818 y Fx(2)3025 2855 y Fr(\024)28
b Ft(")k Fz(for)f Ft(t)d Fy(=)257 2983 y Ft(O)s Fy(\(log)q(\()p
Ft(m=")p Fy(\))17 b(log\(1)p Ft(=")p Fy(\)\))p Fz(.)40
b(By)27 b(Theorem)1775 2983 y
SDict begin H.S end
 1775 2983 a Fz(31)1873
2919 y
SDict begin H.R end
 1873 2919 a 1873 2983 a
SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (theorem.31) cvn H.B /ANN pdfmark end
 1873 2983 a Fz(,)h(this)e(implies)e(the)
j(follo)m(wing)d(exp)s(onen)m(tial)257 3104 y(impro)m(v)m(emen)m(t)33
b(of)f(Corollary)1384 3104 y
SDict begin H.S end
 1384 3104 a Fz(33)1482 3039
y
SDict begin H.R end
 1482 3039 a 1482 3104 a
SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (theorem.33) cvn H.B /ANN pdfmark end
 1482 3104 a Fz(.)257 3123 y
SDict begin H.S end

257 3123 a 257 3123 a
SDict begin 14.5 H.A end
 257 3123 a 257 3123 a
SDict begin [ /View [/XYZ H.V] /Dest (theorem.34) cvn H.B /DEST pdfmark
end
 257 3123
a 177 x Fs(Corollary)25 b(34.)34 b Fz([)948 3300 y
SDict begin H.S end
 948
3300 a Fz(40)1045 3236 y
SDict begin H.R end
 1045 3236 a 1045 3300 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.S04) cvn H.B /ANN pdfmark end
 1045
3300 a Fz(])25 b Fm(The)g(class)f(of)g(monotone)g Ft(m)p
Fm(-term)h(DNF)g(formulas)f(is)g(le)-5 b(arnable)257
3421 y(in)35 b(time)949 3541 y Fy(p)s(oly)r(\()p Ft(n;)17
b Fy(\()p Ft(m)g Fy(log\()p Ft(n=")p Fy(\)\))1807 3500
y Fx(log)q(\()p Fw(m=")p Fx(\))c(log)q(\(1)p Fw(=")p
Fx(\))2348 3541 y Ft(;)k Fy(log)q(\(1)p Ft(=\016)t Fy(\)\))p
Ft(:)257 3712 y Fm(Henc)-5 b(e,)34 b(monotone)e Ft(O)s
Fy(\(2)1192 3626 y Fn(p)p 1250 3626 146 3 v 1250 3676
a Fx(log)13 b Fw(n)1400 3712 y Fy(\))p Fm(-term)33 b(DNF)h(formulas)e
(ar)-5 b(e)34 b(le)-5 b(arnable)32 b(in)i(p)-5 b(olynomial)257
3832 y(time.)257 3971 y
SDict begin H.S end
 257 3971 a 257 3971 a
SDict begin 14.5 H.A end
 257 3971
a 257 3971 a
SDict begin [ /View [/XYZ H.V] /Dest (subsection.4.4) cvn H.B /DEST
pdfmark end
 257 3971 a 149 x Fj(4.4)131 b(Monotone)43
b(functions)h(with)h(b)t(ounded)e(in\035uence)257 4305
y Fz(Since)36 b Ft(k)s Fz(-jun)m(tas)g(ha)m(v)m(e)i(lo)m(w)d
(in\035uence,)j(it)d(is)g(clear)g(that)h(all)e(functions)i(that)f(are)h
(suf-)257 4425 y(\034cien)m(tly)k(close)g(to)g(some)g
Ft(k)s Fz(-jun)m(ta)g(also)f(m)m(ust)h(ha)m(v)m(e)i(lo)m(w)d
(in\035uence.)67 b(As)41 b(sho)m(wn)g(b)m(y)257 4545
y(F)-8 b(riedgut)24 b([)668 4545 y
SDict begin H.S end
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4481 y
SDict begin H.R end
 766 4481 a 766 4545 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.F98) cvn H.B /ANN pdfmark end
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(erse)j(is)c(true:)40 b(an)m(y)25 b(Bo)s(olean)f(function)g
Ft(f)36 b Fz(ha)m(ving)24 b(lo)m(w)g(in-)257 4666 y(\035uence)30
b(m)m(ust)e(b)s(e)g(close)g(to)g(a)g Ft(k)s Fz(-jun)m(ta)f(where)j
Ft(k)h Fz(only)c(dep)s(ends)j(on)e(the)g(in\035uence)h(and)257
4786 y(not)g(on)g(the)h(arit)m(y)e(of)h Ft(f)11 b Fz(.)42
b(More)29 b(precisely)-8 b(,)30 b(if)e Ft(I)8 b Fy(\()p
Ft(f)j Fy(\))27 b Fr(\024)h Ft(l)k Fz(then)d Ft(f)40
b Fz(is)28 b(a)h Fy(\()p Ft(";)17 b Fy(2)2972 4750 y
Fw(O)r Fx(\()p Fw("=l)q Fx(\))3176 4786 y Fy(\))p Fz(-jun)m(ta.)257
4907 y(V)-8 b(ery)28 b(recen)m(tly)-8 b(,)29 b(this)e(relationship)e
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4907 y
SDict begin H.S end
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SDict begin H.R end
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4907 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.ODS06) cvn H.B /ANN pdfmark end
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(applicabilit)m(y)33 b(of)k(the)g(generalized)g(lo)m(w-degree)g
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b(result)g(yields)g(the)h(follo)m(wing)d(appro)m(ximabilit)m(y)f
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(in\035uence.)257 5388 y
SDict begin H.S end
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SDict begin 14.5 H.A end
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a 257 5388 a
SDict begin [ /View [/XYZ H.V] /Dest (theorem.35) cvn H.B /DEST pdfmark
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SDict begin H.S end
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SDict begin H.R end
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SDict begin [ /View [/XYZ H.V] /Dest (page.28) cvn H.B /DEST pdfmark
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SDict begin H.S end
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SDict begin H.R end
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/Dest (cite.F98) cvn H.B /ANN pdfmark end
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SDict begin H.S end
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1920 y Fy(^)1548 1946 y Ft(f)11 b Fy(\()p Ft(S)6 b Fy(\))1749
1905 y Fx(2)1810 1946 y Fy(+)1914 1851 y Fo(X)1908 2063
y Fw(S)t Fn(2)p Fw(H)2103 1920 y Fy(^)2081 1946 y Ft(f)11
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Fr(\024)28 b Ft(")22 b Fy(+)2626 1851 y Fo(X)2620 2063
y Fw(S)t Fn(2)p Fw(H)2814 1920 y Fy(^)2793 1946 y Ft(f)11
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SDict begin H.S end
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/Dest (equation.4) cvn H.B /ANN pdfmark end
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a Fz(\))h(in)257 2510 y(Section)603 2510 y
SDict begin H.S end
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Fz(2)652 2446 y
SDict begin H.R end
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SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (section.2) cvn H.B /ANN pdfmark end
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2630 y(p)s(osition)31 b Ft(j)39 b Fz(that)403 2765 y
Fo(X)383 2977 y Fw(S)t Fn(2)p Fw(H)535 2987 y Fu(j)584
2860 y Fy(2)633 2819 y Fn(\000)p Fw(t)739 2834 y Fy(^)717
2860 y Ft(f)11 b Fy(\()p Ft(S)6 b Fy(\))918 2819 y Fx(2)985
2860 y Fr(\024)1130 2765 y Fo(X)1090 2977 y Fw(S)11 b
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y Fn(\000j)p Fw(S)t Fn(j)1545 2834 y Fy(^)1524 2860 y
Ft(f)11 b Fy(\()p Ft(S)6 b Fy(\))1725 2819 y Fx(2)1791
2860 y Fy(=)28 b Fr(k)p Ft(T)2002 2887 y Fx(1)p Fw(=)2072
2832 y Fn(p)p 2131 2832 36 3 v 55 x Fx(2)2171 2860 y
Ft(f)2219 2875 y Fw(j)2255 2860 y Fr(k)2305 2819 y Fx(2)2372
2860 y Fr(\024)g Fy(\()p Fr(k)p Ft(f)2613 2875 y Fw(j)2650
2860 y Fr(k)2700 2819 y Fx(2)2739 2860 y Fy(\))2777 2819
y Fx(4)p Fw(=)p Fx(3)2915 2860 y Fy(=)f Ft(I)3061 2875
y Fw(j)3098 2860 y Fy(\()p Ft(f)11 b Fy(\))3233 2819
y Fx(4)p Fw(=)p Fx(3)3342 2860 y Ft(;)257 3208 y Fz(implying)32
b(that)876 3133 y Fo(P)981 3237 y Fw(S)t Fn(2)p Fw(H)1133
3247 y Fu(j)1207 3182 y Fy(^)1186 3208 y Ft(f)10 b Fy(\()p
Ft(S)c Fy(\))1386 3172 y Fx(2)1456 3208 y Fr(\024)31
b Fy(2)1613 3172 y Fw(t)1642 3208 y Ft(I)1685 3223 y
Fw(j)1722 3208 y Fy(\()p Ft(f)11 b Fy(\))1857 3172 y
Fx(4)p Fw(=)p Fx(3)1966 3208 y Fz(.)48 b(Note)35 b(that)f(for)f(eac)m
(h)i Ft(S)h Fr(2)31 b Ft(H)42 b Fz(there)35 b(is)257
3340 y(some)d Ft(j)i Fr(62)28 b Ft(R)q Fy(\()p Ft(\022)s
Fy(\))k Fz(suc)m(h)h(that)e Ft(S)j Fr(2)28 b Ft(H)1598
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(using)f(the)i(b)s(ounds)257 3460 y Ft(I)8 b Fy(\()p
Ft(f)j Fy(\))27 b Fr(\024)i Ft(l)34 b Fz(and)f Ft(I)872
3475 y Fw(j)909 3460 y Fy(\()p Ft(f)11 b Fy(\))27 b Fr(\024)h
Ft(\022)36 b Fz(for)c(all)e Ft(j)k Fr(62)28 b Ft(R)q
Fy(\()p Ft(\022)s Fy(\))k Fz(w)m(e)i(get)321 3604 y Fo(X)315
3815 y Fw(S)t Fn(2)p Fw(H)510 3672 y Fy(^)488 3699 y
Ft(f)11 b Fy(\()p Ft(S)6 b Fy(\))689 3657 y Fx(2)756
3699 y Fr(\024)900 3604 y Fo(X)861 3820 y Fw(j)t Fn(62)p
Fw(R)p Fx(\()p Fw(\022)r Fx(\))1120 3604 y Fo(X)1100
3815 y Fw(S)t Fn(2)p Fw(H)1252 3825 y Fu(j)1322 3672
y Fy(^)1301 3699 y Ft(f)11 b Fy(\()p Ft(S)6 b Fy(\))1502
3657 y Fx(2)1569 3699 y Fr(\024)1680 3604 y Fo(X)1640
3820 y Fw(j)t Fn(62)p Fw(R)p Fx(\()p Fw(\022)r Fx(\))1847
3699 y Fy(2)1896 3657 y Fw(t)1925 3699 y Ft(I)1968 3714
y Fw(j)2005 3699 y Fy(\()p Ft(f)11 b Fy(\))2140 3657
y Fx(4)p Fw(=)p Fx(3)2277 3699 y Fr(\024)28 b Fy(2)2431
3657 y Fw(t)2460 3699 y Ft(\022)2508 3657 y Fx(1)p Fw(=)p
Fx(3)2635 3604 y Fo(X)2691 3814 y Fw(j)2796 3699 y Ft(I)2839
3714 y Fw(j)2875 3699 y Fy(\()p Ft(f)11 b Fy(\))28 b
Fr(\024)g Fy(2)3192 3657 y Fw(t)3221 3699 y Ft(\022)3269
3657 y Fx(1)p Fw(=)p Fx(3)3379 3699 y Ft(l)r(:)257 4031
y Fz(Th)m(us,)43 b(c)m(ho)s(osing)c Ft(\022)k Fy(=)c(\()p
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4152 y Fy(2)1830 4115 y Fx(3)p Fw(l)q(=")1959 4152 y
Ft(=")2054 4115 y Fx(3)2127 4152 y Fy(=)g(2)2286 4115
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(y)g Fy(2)1343 4236 y Fw(O)r Fx(\()p Fw(l)q(=")p Fx(\))1543
4212 y Fv(2)1582 4272 y Fz(.)p 3446 4272 49 49 v 404
4491 a(As)g(an)f(immediate)e(consequence)36 b(w)m(e)d(can)g(state)g
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b(ach)33 b(mono-)257 4813 y(tone)i(Bo)-5 b(ole)g(an)33
b(function)i Ft(f)46 b Fm(with)34 b Ft(I)8 b Fy(\()p
Ft(f)j Fy(\))27 b Fr(\024)i Ft(l)r Fm(,)35 b Ft(A)p Fy(\()p
Ft(l)r(;)17 b(";)g(\016)n(;)g(E)6 b(X)i Fy(\()p Ft(f)j
Fy(\)\))34 b Fm(outputs)h(with)g(pr)-5 b(ob)g(a-)257
4933 y(bility)35 b Fy(1)22 b Fr(\000)h Ft(\016)39 b Fm(an)34
b Ft(O)s Fy(\()p Ft(")p Fy(\))p Fm(-appr)-5 b(oximation)32
b(for)j Ft(f)45 b Fm(in)35 b(time)f Fy(p)s(oly)r(\()p
Ft(n;)17 b Fy(2)2733 4897 y Fx(\()p Fw(l)q(=")p Fx(\))2877
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b Fr(\024)3157 5180 y(p)p 3240 5180 228 4 v 75 x Fy(log)18
b Ft(m)p Fz(.)257 5388 y(If)33 b Ft(f)43 b Fz(is)32 b(monotone,)g(this)
g(implies)e(that)i Ft(I)8 b Fy(\()p Ft(f)j Fy(\))28 b(=)2070
5313 y Fo(P)2175 5417 y Fw(i)2241 5362 y Fy(^)2220 5388
y Ft(f)11 b Fy(\()p Ft(i)p Fy(\))28 b Fr(\024)2521 5313
y(p)p 2604 5313 V 75 x Fy(log)17 b Ft(m)q Fz(.)257 5417
y
SDict begin H.S end
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SDict begin H.S end
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y(parameters)33 b Ft(")f Fz(and)h Ft(\016)j Fz(for)c(clarit)m(y)f
(reasons.)257 2070 y Fs(Monotone)38 b(DNF)257 2134 y
SDict begin H.S end

257 2134 a 257 2134 a
SDict begin 14.5 H.A end
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SDict begin [ /View [/XYZ H.V] /Dest (section*.6) cvn H.B /DEST pdfmark
end
 257 2134
a 121 x Fz(In)25 b(the)g(ligh)m(t)d(of)i(O'Donnell)f(and)i(Serv)m
(edio's)g(e\036cien)m(t)g(learning)e(algorithm)e(for)j(mono-)257
2375 y(tone)36 b(decision)g(trees)g([)1113 2375 y
SDict begin H.S end
 1113
2375 a Fz(39)1211 2311 y
SDict begin H.R end
 1211 2311 a 1211 2375 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.ODS06) cvn H.B /ANN pdfmark end
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2375 a Fz(])g(an)g(in)m(teresting)f(question)h(to)g(ask)g(is)g(whether)
h(this)e(result)257 2495 y(can)e(b)s(e)g(extended)h(to)f(monotone)e
(DNF)i(form)m(ulas.)257 2495 y
SDict begin H.S end
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SDict begin 14.5 H.A end

257 2495 a 257 2495 a
SDict begin [ /View [/XYZ H.V] /Dest (theorem.38) cvn H.B /DEST pdfmark
end
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b(38.)57 b Fm(A)n(r)-5 b(e)63 b(monotone)f Ft(m)p Fm(-term)h(DNF)g
(formulas)f(le)-5 b(arnable)62 b(in)g(time)257 2804 y
Fy(p)s(oly)r(\()p Ft(n;)17 b(m)p Fy(\))p Fm(?)404 2993
y Fz(It)26 b(is)g(further)g(sho)m(wn)i(in)d([)1340 2993
y
SDict begin H.S end
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SDict begin H.R end
 1438 2928 a 1438 2993
a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.ODS06) cvn H.B /ANN pdfmark end
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Ft(I)8 b Fy(\()p Ft(f)j Fy(\))27 b Fr(\024)2871 2917
y(p)p 2954 2917 228 4 v 76 x Fy(log)18 b Ft(m)26 b Fz(cannot)257
3113 y(b)s(e)31 b(extended)i(to)e(monotone)f Ft(m)p Fz(-term)g(DNF)h
(form)m(ulas)f(or)g(ev)m(en)j(to)d(functions)h Ft(f)41
b Fz(that)257 3233 y(are)h(computable)g(b)m(y)h Fm(b)-5
b(oth)49 b Fz(an)42 b Ft(m)p Fz(-term)f(DNF)i(as)f(w)m(ell)f(as)i(b)m
(y)g(an)f Ft(m)p Fz(-clause)g(CNF.)257 3354 y(Th)m(us,)35
b(Problem)923 3354 y
SDict begin H.S end
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SDict begin H.R end
 1020
3290 a 1020 3354 a
SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (theorem.38) cvn H.B /ANN pdfmark end
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(y)f(solely)f(relying)f(on)i(Theorem)3072 3354 y
SDict begin H.S end
 3072
3354 a Fz(36)3170 3290 y
SDict begin H.R end
 3170 3290 a 3170 3354 a
SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (theorem.36) cvn H.B /ANN pdfmark end
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3354 a Fz(.)404 3474 y(The)69 b(curren)m(tly)g(b)s(est)g(learning)d
(algorithm)f(for)j(monotone)f Ft(m)p Fz(-term)h(DNF)257
3595 y(is)f(the)g(one)g(of)f(Corollary)1415 3595 y
SDict begin H.S end
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3595 a Fz(34)1512 3530 y
SDict begin H.R end
 1512 3530 a 1512 3595 a
SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (theorem.34) cvn H.B /ANN pdfmark end
 1512
3595 a 67 w Fz(due)h(to)g(Serv)m(edio)g([)2402 3595 y
SDict begin H.S end

2402 3595 a Fz(40)2499 3530 y
SDict begin H.R end
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SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.S04) cvn H.B /ANN pdfmark end

2499 3595 a Fz(])g(whic)m(h)g(runs)h(in)e(time)257 3715
y Fy(p)s(oly)r(\()p Ft(n;)17 b Fy(\()p Ft(m)g Fy(log)g
Ft(n)p Fy(\))961 3679 y Fw(')p Fx(\()p Fw(m)p Fx(\))1129
3715 y Fy(\))46 b Fz(where)i Ft(')p Fy(\()p Ft(m)p Fy(\))k(=)g(log)17
b Ft(m)p Fz(.)86 b(It)47 b(w)m(ould)g(b)s(e)g(v)m(ery)h(in)m(teresting)
257 3835 y(to)31 b(impro)m(v)m(e)g(this)g(result)g(to)g(an)h(algorithm)
c(ha)m(ving)j Fm(\034xe)-5 b(d)33 b(p)-5 b(ar)g(ameterize)g(d)32
b(c)-5 b(omplexity)257 3956 y Fz(in)32 b(the)h(sense)h(of)f(Do)m(wney)h
(and)f(F)-8 b(ello)m(ws)31 b([)1830 3956 y
SDict begin H.S end
 1830 3956
a Fz(11)1928 3891 y
SDict begin H.R end
 1928 3891 a 1928 3956 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.DF99) cvn H.B /ANN pdfmark end
 1928 3956
a Fz(].)257 3981 y
SDict begin H.S end
 257 3981 a 257 3981 a
SDict begin 14.5 H.A end
 257 3981 a 257
3981 a
SDict begin [ /View [/XYZ H.V] /Dest (theorem.39) cvn H.B /DEST pdfmark
end
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b(e)63 b(monotone)f Ft(m)p Fm(-term)h(DNF)g(formulas)f(le)-5
b(arnable)62 b(in)g(time)257 4265 y Fy(p)s(oly)r(\()p
Ft(n;)17 b(')p Fy(\()p Ft(m)p Fy(\)\))p Fm(,)35 b(wher)-5
b(e)34 b Ft(')h Fm(do)-5 b(es)34 b(not)h(dep)-5 b(end)33
b(on)i Ft(n)p Fm(?)257 4522 y Fs(Jun)m(tas)257 4586 y
SDict begin H.S end

257 4586 a 257 4586 a
SDict begin 14.5 H.A end
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SDict begin [ /View [/XYZ H.V] /Dest (section*.7) cvn H.B /DEST pdfmark
end
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a 121 x Fz(As)c(w)m(e)g(sa)m(w)h(in)d(Section)1176 4707
y
SDict begin H.S end
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SDict begin H.R end
 1301 4642 a 1301 4707
a
SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (subsection.4.2) cvn H.B /ANN pdfmark end
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4707 y
SDict begin H.S end
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SDict begin H.R end
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4707 a
SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (theorem.39) cvn H.B /ANN pdfmark end
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4827 y(has)24 b(b)s(een)g(solv)m(ed)g(b)m(y)g(Bshout)m(y)h(and)e(T)-8
b(amon)23 b([)1954 4827 y
SDict begin H.S end
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y
SDict begin H.R end
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SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.BT96) cvn H.B /ANN pdfmark end
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i(algorithm)c(running)257 4959 y(in)46 b(time)f Fy(p)s(oly)r(\()p
Ft(n;)17 b(')p Fy(\()p Ft(k)s Fy(\)\))46 b Fz(for)g Ft(')p
Fy(\()p Ft(k)s Fy(\))51 b(=)g Ft(k)1814 4866 y Fn(p)p
1873 4866 39 3 v 57 x Fw(k)1962 4959 y Fz(\(cf.)85 b(Corollary)2630
4959 y
SDict begin H.S end
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SDict begin H.R end
 2727 4895 a 2727
4959 a
SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (theorem.28) cvn H.B /ANN pdfmark end
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y(question)33 b(is)f(whether)i(this)e(can)h(b)s(e)g(extended)h(to)f
(general)f(jun)m(tas.)257 5099 y
SDict begin H.S end
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a
SDict begin 14.5 H.A end
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SDict begin [ /View [/XYZ H.V] /Dest (theorem.40) cvn H.B /DEST pdfmark
end
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b Fm(A)n(r)-5 b(e)28 b Ft(k)s Fm(-juntas)h(le)-5 b(arnable)28
b(in)g(time)g Fy(p)s(oly)r(\()p Ft(n;)17 b(')p Fy(\()p
Ft(k)s Fy(\)\))p Fm(,)30 b(wher)-5 b(e)28 b Ft(')g Fm(do)-5
b(es)28 b(not)257 5388 y(dep)-5 b(end)34 b(on)h Ft(n)p
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SDict begin H.S end
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SDict begin H.R end
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SDict begin [ /View [/XYZ H.V] /Dest (page.30) cvn H.B /DEST pdfmark
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b(e)47 b(w)m(an)m(t)h(to)e(emphasize)h(that)g(Problem)2084
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SDict begin H.S end
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SDict begin H.R end
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a
SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (theorem.40) cvn H.B /ANN pdfmark end
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SDict begin H.S end
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870 y
SDict begin H.R end
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SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
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SDict begin H.S end
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Fz(37)2898 870 y
SDict begin H.R end
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SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.MODS03) cvn H.B /ANN pdfmark end
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(Mossel)h(et)257 1657 y(al.)33 b([)421 1657 y
SDict begin H.S end
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a Fz(37)518 1593 y
SDict begin H.R end
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SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.MODS03) cvn H.B /ANN pdfmark end
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(in)f(time)f Ft(O)s Fy(\()p Ft(n)1499 1621 y Fw(\013k)1586
1657 y Fy(\))i Fz(for)e(some)i Ft(\013)29 b(<)g Fy(0)p
Ft(:)p Fy(7)p Fz(.)46 b(So)33 b(ev)m(en)i(an)f(impro)m(v)m(emen)m(t)257
1777 y(to)f(an)f Ft(\013)c(<)g Fy(0)p Ft(:)p Fy(5)k Fz(w)m(ould)h(b)s
(e)f(a)h(signi\034can)m(t)e(progress.)257 1797 y
SDict begin H.S end
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1797 a 257 1797 a
SDict begin 14.5 H.A end
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SDict begin [ /View [/XYZ H.V] /Dest (theorem.41) cvn H.B /DEST pdfmark
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x Fs(Problem)36 b(41.)42 b Fm(A)n(r)-5 b(e)34 b Ft(k)s
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Fy(\()p Ft(n)2389 1948 y Fw(\013k)2476 1984 y Fy(\))g
Fm(for)g(some)f Ft(\013)28 b(<)g Fy(0)p Ft(:)p Fy(5)p
Fm(?)404 2191 y Fz(F)-8 b(or)25 b(the)i(sp)s(ecial)e(case)i(of)f
(symmetric)f(jun)m(tas,)k(Mossel)d(et)h(al.)e([)2709
2191 y
SDict begin H.S end
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SDict begin H.R end
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2191 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.MODS03) cvn H.B /ANN pdfmark end
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2311 y(v)m(on)35 b(zur)g(Gathen)f(and)h(Ro)s(c)m(he)f([)1476
2311 y
SDict begin H.S end
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SDict begin H.R end
 1574 2247 a 1574
2311 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.GR97) cvn H.B /ANN pdfmark end
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Ft(n)3010 2275 y Fw(\013k)3133 2311 y Fz(for)e(some)257
2431 y Ft(\013)c(<)g Fy(2)p Ft(=)p Fy(3)p Fz(.)43 b(The)34
b(analogue)e(of)h(Problem)1781 2431 y
SDict begin H.S end
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2367 y
SDict begin H.R end
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SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (theorem.41) cvn H.B /ANN pdfmark end
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(jun)m(tas)i(has)f(b)s(een)h(solv)m(ed)257 2552 y(b)m(y)41
b(Lipton)d(et)h(al.)63 b([)1036 2552 y
SDict begin H.S end
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2487 y
SDict begin H.R end
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SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.LMMV05) cvn H.B /ANN pdfmark end
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(viding)e(an)h(algorithm)d(with)j(running-time)e Ft(O)s
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2672 y Fz(for)d Ft(\013)d Fy(=)f(3)p Ft(=)p Fy(31)p Fz(.)47
b(This)34 b(b)s(ound)g(has)h(b)s(een)f(subsequen)m(tly)j(impro)m(v)m
(ed)d(to)f Ft(O)s Fy(\()p Ft(n)3111 2636 y Fw(k)r(=)11
b Fx(log)i Fw(k)3342 2672 y Fy(\))34 b Fz(in)257 2792
y([)284 2792 y
SDict begin H.S end
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SDict begin H.R end
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382 2792 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.KMM05) cvn H.B /ANN pdfmark end
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3228 y
SDict begin H.S end
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SDict begin 14.5 H.A end
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SDict begin [ /View [/XYZ H.V] /Dest (section*.8) cvn H.B /DEST pdfmark
end
 257
3228 a 257 3190 a
SDict begin H.S end
 257 3190 a 257 3190 a
SDict begin 13.6 H.A end
 257 3190 a 257
3190 a
SDict begin [ /View [/XYZ H.V] /Dest (cite.BFH94) cvn H.B /DEST pdfmark
end
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b(Bartlett,)i(P)-8 b(.)27 b(Fisc)m(her,)g(and)f(K.-U.)g(Ho\033gen.)33
b(Exploiting)27 b(random)g(w)m(alks)f(for)f(learn-)447
3455 y(ing.)41 b(In)30 b Fe(Pr)-5 b(o)g(c.)65 b(7th)33
b(A)n(nnual)f(A)n(CM)f(Confer)-5 b(enc)g(e)34 b(on)e(Computational)g(L)
-5 b(e)g(arning)33 b(The-)447 3568 y(ory)p FE(,)d(pages)h(318\025327,)g
(1994.)257 3587 y
SDict begin H.S end
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SDict begin 13.6 H.A end
 257 3587 a 257
3587 a
SDict begin [ /View [/XYZ H.V] /Dest (cite.Bec75) cvn H.B /DEST pdfmark
end
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b(Inequalities)39 b(in)g(Fourier)g(analysis.)65 b Fe(A)n(nnals)40
b(of)f(Mathematics)p FE(,)j(102:)447 3834 y(159\025182,)31
b(1975.)257 3853 y
SDict begin H.S end
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SDict begin 13.6 H.A end
 257 3853 a 257
3853 a
SDict begin [ /View [/XYZ H.V] /Dest (cite.BL89) cvn H.B /DEST pdfmark
end
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(Linial.)48 b(Collectiv)m(e)35 b(coin)e(\035ipping.)48
b(In)32 b(S.)g(Micali,)i(editor,)g Fe(R)-5 b(an-)447
4099 y(domness)32 b(and)g(Computation)p FE(,)e(pages)g(91\025115.)i(A)m
(cademic)g(Press,)e(1989.)257 4118 y
SDict begin H.S end
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a
SDict begin 13.6 H.A end
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SDict begin [ /View [/XYZ H.V] /Dest (cite.BFJKMR94) cvn H.B /DEST
pdfmark end
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b(A.)20 b(Blum,)j(M.)e(F)-8 b(urst,)23 b(J.)e(Jac)m(kson,)j(M.)d
(Kearns,)i(Y.)d(Mansour,)j(and)f(S.)e(Rudic)m(h.)26 b(W)-8
b(eakly)447 4365 y(learning)27 b(DNF)e(and)h(c)m(haracterizing)i
(statistical)e(query)f(learning)i(using)f(Fourier)g(anal-)447
4478 y(ysis.)33 b(In)25 b Fe(Pr)-5 b(o)g(c.)30 b(26th)f(A)n(CM)f(Symp)
-5 b(osium)28 b(on)g(The)-5 b(ory)29 b(of)f(Computing)p
FE(,)f(pages)g(253\025262,)447 4591 y(1994.)257 4592
y
SDict begin H.S end
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SDict begin 13.6 H.A end
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SDict begin [ /View [/XYZ H.V] /Dest (cite.BL97) cvn H.B /DEST pdfmark
end
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a 303 4744 a FE([5])49 b(A.)33 b(Blum)g(and)h(P)-8 b(.)34
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