Forward Discrete Probability Propagation

Rasit Onur Topaloglu

Ph.D. candidate

rot@ucsd.edu

SPICE Hierarchy and the Problem

ex: gm=2*k*ID

Level1

Level4

Level0 TLeffWefftox

Level2

Level3

0 NSUBms

CoxF

k Qdep

Vth

ID

gm rout Level5

•SPICE formulas are hierarchical; hence can progressively relate physical parameters to device parameters in connectivity graphs•Recent models attribute process variations to physical parameters

physical

electrical/mathematical

•Probability density functions can be input at Level0 independently

Probability Propagation

•Estimation of device parameters at highest level needed to examine effects of process variations•An analytic solution not possible since functions highly non-linear and Gaussian approximations not accurate in deep sub-micron

GOALS

Algebraic tractability : enable manual applicability by designers

Speed : be comparable or outperform Monte Carlo in quick estimation

•A method to propagate pdf’s to highest level necessary

Flexibility : be able to use non-standard densities to outperform parametric belief propagation

Shortcomings of Monte Carlo

•No algebraically tractability : No manual estimation by designers possible due to large number of iterations and random sampling

•Limited to standard distributions : Random number generators in CAD tools only provide certain distributions, hence a new module usually needs to be programmed

•Speed : No quick convergence to an estimate distribution due to random sampling unless a large number of costly

iterations employed

A Reminder on Applying Monte Carlo for Probability Propagation

gm

Level1

Level4

Level0 nVFBNSUBLW

Vth Cox

tox

ID

k Level2

Level3

•Pick independent samples from distributions of Level0 parameters

•Compute functions using these samples until highest level reached •Construct a histogram to approximate the distribution

•Repeat while desired accuracy is not yet reached:

Parametric Belief Propagation

•Each node receives and sends messages to parents and children until equilibrium•Parent to child () : causal information

•Parent to parent () : diagnostic information

Calculationshandledat each node:

Parametric Belief Propagation

•When arrows in the hierarchy tree indicate linear addition operations on Gaussians, analytic formulations possible•Not straightforward for other distributions or non-standard distributions

Implementing FDPP

•F (Forward) : Given a function, estimates the distribution of next node in the formula hierarchy using samples

•Q (Quantize) : Discretizes a pdf to operate on its samples

Analytic operation on continuous distributions difficult; instead work in discrete domain and convert back to continuous

domain at the end:

•Q-1 (De-Quantize) : Converts a discrete pdf back to continuous domain : implemented as an interpolation function

•B (Band-pass) : Decrements number of samples using a threshold on sample probabilities

•R (Re-bin) : Decreases number of samples by combining close samples together

T NSUB

PHIf

Necessary Operators (Q, F, B, R, Q-1) on a Connectivity Graph

Q Q

F

B

•F, B and R repeated until we acquire the distribution of a high level parameter; Q and Q-1 used just once

R

Q-1

TLeffWefftox 0 NSUBms

CoxF

k

Vth

ID

gmrout

Qdep

pdf(X)Q Operator

•QN band-pass filters pdf(X) and divides into bins

))(()( XpdfQX N N in QN indicates number of bins

spdf(X)=(X)

X

pdf(

X)

spdf

(X)

X

•Use N>(2/m), where m is maximum derivative of pdf(X), thereby obeying a bound similar to Nyquist •If quantizer uniform and small, quantization error

random variable Q is uniformly distributed, then2/ 22 2 2

12/ 2[ ] ( )Q E Q q pdf Q dq

Variance of quantization error:

•Increase number of bins to reduce quantization error

F Operator •F operator implements a function over spdf’s using

deterministic sampling

•Corresponding function in connectivity graph applied to deterministic pair-wise combination of impulse values to get the value of the new sample

•Heights of impulses (probabilities) multiplied to get probability of new sample

Effect of Non-linear Functions

•Application of functions cause accumulation in certain ranges

Band-pass and re-bin operations needed after F operation

Impulses after F, before B and R

•De-quantization would not result in a pdf•Increased number of samples would induce a computational burden

Band-pass, Be, Operator•Eliminate samples having values out of range (6): might

cut off tails of bi-modal or long-tailed distributions

Margin-based Definition:

Error-based Definition:•Eliminate samples having probabilities least likely to occur :

can also eliminate samples in useful range hence offers more computational efficiency

•Implementation : eliminate samples with probabilities less than 1/e times the sample with the largest probability

•e should be chosen such that it is smaller than the ratio of products of maximum and minimum probability

samples for nodes to which F is applied

Re-bin, RN, OperatorImpulses after F Resulting spdf(X)Unite into one bin

•Samples falling into the same bin congregated in one

jbjiji

ji pwmdi

)(:,

),(•Total distortion given by

2)(),( jiji wmwmd mi : center of i’th bin

can be used to select bin locations, where

Experimental Results

•Impulse representation for threshold voltage and transconductance are obtained through FDPP on the graph

(X) for gm(X) for Vth

•Matlab R12 used to evaluate FDPP method

•A close match is observed after interpolation

Monte Carlo – FDPP Comparison

solid : FDPP dotted : Monte Carlo

Pdf* of VthPdf* of ID

•Correlation error introduced by the independence assumption of F operator results in negligible error as R operator helps distribute this error over the pdf state space

Monte Carlo – FDPP Comparison with a Low Sample Number

•Monte Carlo inaccurate for moderate number of samples•Indicates FDPP can converge to an acceptable estimate with far less number of samples

solid : FDPP with 100 samples

Pdf* of FPdf* of F

noisy : Monte Carlo with 1000 samples

solid : FDPP with 100 samples

noisy : Monte Carlo with 100000 samples

•Edges define a linear sum, ex: n5=n2+n3

Monte Carlo – FDPP ComparisonPdf of n7Benchmark example

solid : FDPP dotted : Monte Carlo triangles:belief propagation

•Monte Carlo result is separated as FDPP and belief propagation neglect correlation

•When distributions at internal nodes n4, n5, n6 re-sampled using Monte Carlo, all methods converge

Faulty Application of Monte CarloPdf of n7Benchmark example

solid : FDPP dotted : Monte Carlo triangles:belief propagation

Conclusions•Forward Discrete Probability Propagation is introduced as

an alternative to Monte Carlo and parametric belief propagation methods for quick estimation :

•FDPP should be preferred to MC when a faster convergence to real distribution is necessary with limited number of samples

•FDPP provides an algebraic intuition due to deterministic sampling and manual applicability due to using less number of samples

•FDPP can account for non-standard pdf’s where parametric methods are limited to certain ones