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Sherrington Kirkpatrick modelHn Ily R
HN161 21 6 Wo j Wn GOEIN
W La G1Gt Gigi N o N10,1Hw16 oey yw centered Gaussian process
ELHnldthd D zfso.TTGeneral p spin model
Hulot the yN centered Gaussian with
ELHnlolH.de f Ngc EEl I
347 ZzzCEXk Sk 3G Hz
Halo Ez LW oak112 7 Wk indep gaussian tensors
how convex
many local minima
maximize Hnl6subj to o c LI lb
Q can we solve this approx in poly timeinput NH zz Output 5 SEL 11 14N st
N 2 Np Ancoats Ci e max nHNlo n g
H IN feet yNIs this possible V E oIf not V E C 70
Exact optimizationPCH.vlohst mgxth.to 21
Refutation Upper bdALG w c IR st4 ALG W mixHalo2 P ALG W E HE m Halo 7
Worst case Unless P NP no algorithm can doab Ao b 3 cmf 07
Typical value Parisi's formula
V Lt 0,1 2K o nondecr So'HHdt soozollt.xl l 54H 20lt.xl HHG.rolt xlTT oIo Dx IR 016 7 1 1 loRr Igloo I fo'tg lHHtIdt i EHIdHH N3tvtThe OPT v mix Hnk
firm OPTN info Pcr
The ft Pcr is strictly convex igf parEUachieved at unique 8 EU
Interpretation of IConsider Gibbs measure
ZypEP NbMwp o L
Se o LILY Halo 4 e mix Halo bMwp Unit Se E E Cfo 02 Iw Mwp Mwp
Hid
VPp N law of ko
Nsoo N
Ppw Up on o Dt line PipCEO t
structure of rI Nooverlapgap r strictly incr on 6,1Il Overlap gap Fct ta EEo at
rct e srctntl rlt.is Htt
is tif Ceo t strictly increasing forte lo g constantabove
1197 of'Ese ko left stts a
ft E 0,9 1AlgorithmsL rico D Rao 113 rll w.co sottEl9D Jo3 rfttodtJ3 Htt 3 htt HttL Z U r nondeer fittlott soo fU L n f non dear
Thin AssumefifthPCH is achieved ThenHE of alg
with linear complexity St
PifHulot infzRH t 1Rmt in Pan signaler
L Pcr
intfift higfsinuf an.t factIhearoptLem If no overlap gap then
in.fi crl iyfuPC8g OPT
Greoltry If no gap TE te of linear time 1gN 70sit ACH Isa's 4 e mixHNK 21 a
Conj For SK no overlapgapConjecture If overlap gap no poly time 21g V E
Unless P NP
Proof Construct AMP algo Skxt W f try it ftp.t.sfs.icxi x Izt ftG
InCzt zt zt o7
t E o 8 7 8 I
I 8 20stevolution SDE driftchoose coefficients of SDEoptsolving a stock opt contrinfzPL'T B
E Exitx z Nco 8 Id