Stochastic Lower Complexity

Let us consider the followingsimplified problem

A e Rdxd

Tfcn Ar b b e Rd

THR ga Ak 1st Gpu

z A b exists

Assume il him E G EI Fm S o

K soo

Ii A o

Then Polyak and Tsypkin 1976

proved that when En k I

are i i d

Effy n Xin 3 total

where

V A's A JT

Remarks

i Proof needs some clarification

Ii There are algorithms that

attempt to estimate the

smallest eigenvalue of AThis is generally a bad idea

Can an algorithm be constructed

so that the eigenvalues of Aneed not be directly estimated

Yes

Xp Xm NPF Xm ED

XI It Xj Offline Avg

Theorem In addition to Cil S iii suppose

iii ca mini 0thmgnxj

OR Xj j i 2 are eigenvalues of A

b ya 70 Mk Nkt otherna

NHII HII E HE.IT FnJc a s

hold Then

RE Ik n dN o V

What does Condition Ciii b mean

I Is Mn Cz allowed

2 N ME O L x a 1

satisfies

Notice that fixed step is okayHowever