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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
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