In[]:=
CompoundExpression[
]
​​deploy
Wed 22 Mar 2023 15:27:06

Asymptotic Behavior

Asymptotic behavior of ∏𝑠𝑖=1(1−𝛼𝑥2𝑖)2 for Gaussian 𝑥𝑖
- https://stats.stackexchange.com/questions/610099/asymptotic-behavior-of-prod-i-1s-1-alpha-x2-i2-for-gaussian-x-i
- root finders: forum-gaussian-integrals.nb
​
Top-level section:
“Multiplicative Convergence + Kaczmarz” of Notes: Optimization
​
In[]:=
SeedRandom[Method->"MKL"];​​gaussianSampler[diag_]:=With{d=Length[diag]},​​Compile{{n,_Integer}},​​Module{vals,diagSqrt},​​diagSqrt=
diag
;​​vals=diagSqrt*#&/@RandomVariate[NormalDistribution[],{n,d}];​​​​Clear[step];​​​​d=1;​​h=ConstantArray[1.,d];​​h=h/Total[h];​​​​sampler=gaussianSampler[h];​​α=2.422;​​B=1001;(*batch*)​​s=100;(*simulationlength*)​​x0=sampler[10];​​step[w_,x_]=w-αxx.w;​​batchStep[W_]:=MapThread[step,{W,sampler[B]}];​​vals=NestList[batchStep,sampler[B],numSteps];(*numSteps+1xBxd*)​​sqNorms=Total[vals*vals,{3}];(*numSteps+1XB*)​​​​lastStep=Log@Last@sqNorms;​​​​medianNorms=Median/@sqNorms;​​meanNorms=Mean/@sqNorms;​​SF=StringForm;​​nf[val_]:=NumberForm[N@val,{3,3}];​​ListPlot[{Log[meanNorms],Log[medianNorms]},PlotLegends->{"mean","median"},PlotRange->All,AxesLabel->{"s","value"},PlotLabel->"Values of
s
∑
log(1-
2
x
i
2
)
"]​​SmoothHistogram[lastStep,PlotLabel->SF["s=``, α=``",numSteps,nf[α]]]​​Print[SF["At step=``",s]];​​​​​​Clear[expectation,a,x];​​pdfN01=1/Sqrt[2*Pi]*Exp[-x^2/2];​​term=2*Log[Abs[1-α*x^2]];​​expectation[a_]:=NIntegrate[term*pdfN01,{x,-Infinity,Infinity}];​​expectation2[a_]:=NIntegrate[
2
term
*pdfN01,{x,-Infinity,Infinity}];​​variance[a_]:=(expectation2[α]-expectation[α]);​​​​​​TableForm[​​{{nf@Mean[lastStep],nf@Variance[lastStep]},​​{nf[expectation[α]*s],nf[variance[α]*s]},​​{expectation[α],variance[α]}​​},​​TableHeadings->{{"Monte-Carlo","NIntegrate","NIntegrate at s=1"},{"mean","variance"}}]​​
Out[]=
mean
median
Out[]=
At step=100
Out[]//TableForm=
mean
variance
Monte-Carlo
-1.610
653.000
NIntegrate
0.040
646.000
NIntegrate at s=1
0.00039635
6.45951