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