zfso.TT General p spin model Hulot the centered Gaussian with yN - - PDF document

SLIDE 1

Sherrington Kirkpatrick model

Hn

Ily

R

HN161

21

6 Wo

j

Wn GOEIN W La G1Gt

Gigi

N

• N10,1

Hw16

• ey yw

centered

Gaussian

process

ELHnldthd D

zfso.TT

General

p spin model

Hulot the

yN

centered Gaussian

with

ELHnlolH.de f

Ngc

EE

l I

347

ZzzCEXk

Sk

3G

Hz

Halo

Ez

LW

• ak

112

7 Wk

indep gaussian tensors

how

convex

many local

minima

maximize

Hnl 6

subj to

• c LI lb

Q can

we

solve this approx in poly time

input

NH

zz

Output

5 SEL 11

14N st

N 2 N

p

Ancoats Ci e

max

nHNlo

n g

SLIDE 2

H

IN

feet yN

Is

this

possible V E

• If

not

V E

C 70

Exact optimization PCH.vlohst

mgxth.to

21

Refutation

Upper bd

ALG

w

c IR

st

4

ALG W

mixHalo

2

P

ALG W

E

HE

m

Halo

7

Worst

case

Unless P

NP

no algorithm can do

ab Ao b

3

cmf

07

Typical value

Parisi's formula V

Lt

0,1

2K

• nondecr So'HHdt soo

zollt.xl

l 54H 20lt.xl HHG.roltxlTT

• Io Dx IR

016 7

1 1

lo

Rr

Igloo

I fo'tg lHHtIdt

i EHIdHH N3t

vt

The

OPT v

mix Hnk

firm

OPTN

info Pcr

The

ft

Pcr

is strictly convex

igf par

EU

achieved

at unique

8 EU

SLIDE 3

Interpretation of I

Consider

Gibbs

measure

ZypEP

Nb Mwp o

L

Se

• LILY

Halo

4 e mix Halo b

Mwp

Unit Se

E E Cf

• 02 Iw

Mwp Mwp

Hid

V

Pp N

law of

ko

Nsoo

N

Ppw

Up

• n
• D

t

line

Pip

CEO t

structure of r

I

Nooverlapgap r

strictly incr

• n 6,1

Il

Overlap gap

Fct ta EEo

at

rct

e srctntl

rlt.is Htt

is

t

if Ceo t

strictly increasing for

te lo g

constantabove

1197 of'Ese ko

left stts

a

SLIDE 4

ft

E

0,9 1

Algorithms

L

rico D Rao

113 rll w.co sottEl9D

Jo3 rfttodtJ 3 Htt

3 htt Htt

L Z U

r

nondeer fittlott soo f

U

L

n f

non dear

Thin

Assume

fifth

PCH is achieved

ThenHE

• f alg

with linear complexity St

PifHulot

infzRH

t

1

Rmt in Pan signaler

L

Pcr

intfift higfsinuf an.t fact

I

hear

• pt

Lem

If

no overlap gap

then

in.fi crl

iyfuPC8

g

OPT

Greoltry

If

no gap TE te

• f

linear time 1g

N 70

sit

ACH Isa's

4 e mixHNK

21

a

Conj

For SK

no overlapgap

Conjecture

If overlap gap

no poly time 21g V E

SLIDE 5

Unless P

NP

Proof

Construct

AMP algo

Sk

xt

W ftry

it

ftp.t.sfs.icxi

x

I

zt ftG

InCzt

zt zt

• 7

t E

• 8

7 8

I

I

8

20

stevolution

SDE

drift

choose coefficients of SDEopt

solving a stock opt contr

infzPL'T

B

E Exit

x

z Nco 8Id