Forward Discrete Probability Propagation

1
Forward Discrete Probability Propagation
Rasit Onur Topaloglu Ph.D. candidate rot_at_ucsd.edu
2
SPICE Hierarchy and the Problem
T
Leff
Weff
tox
?0
NSUB
Level0
?ms
physical
Level1
Cox
?
?F
Level2
k
Qdep
electrical/ mathematical
Level3
Vth
Level4
ID
Level5
rout
gm
ex gm?2kID

• SPICE formulas are hierarchical hence can
  progressively relate physical parameters to
  device parameters in connectivity graphs

• Recent models attribute process variations to
  physical parameters

• Probability density functions can be input at
  Level0 independently

3
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

• A method to propagate pdfs to highest level
  necessary

GOALS
Speed be comparable or outperform Monte Carlo
in quick estimation
Algebraic tractability enable manual
applicability by designers
Flexibility be able to use non-standard
densities to outperform parametric belief
propagation
4
Shortcomings of Monte Carlo

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

• 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

5
A Reminder on Applying Monte Carlo for
Probability Propagation
? n
VFB
NSUB
L
W
tox
Level0
Level1
Cox
Vth
k
Level2
ID
Level3
gm
Level4

• Repeat while desired accuracy is not yet reached
  

• Pick independent samples from distributions of
  Level0 parameters

• Compute functions using these samples until
  highest level reached

• Construct a histogram to approximate the
  distribution

6
Parametric Belief Propagation
Calculations handled at each node

• Each node receives and sends messages to parents
  and children until equilibrium

• Parent to child (?) causal information

• Parent to parent (?) diagnostic information

7
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

8
Implementing FDPP
Analytic operation on continuous distributions
difficult instead work in discrete domain and
convert back to continuous domain at the end

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

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

• 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

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

9
Necessary Operators (Q, F, B, R, Q-1) on a
Connectivity Graph
tox
T
Leff
Weff
?0
NSUB
?ms
Cox
?
?F
k
Qdep
Vth
ID
rout
gm

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

10
Q Operator
spdf(X)?(X)
pdf(X)
pdf(X)
spdf(X)
X
X
N in QN indicates number of bins

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

• Use Ngt(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, then

Variance of quantization error

• Increase number of bins to reduce quantization
  error

11
F Operator

• F operator implements a function over spdfs
  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

12
Effect of Non-linear Functions
Impulses after F, before B and R

• Application of functions cause accumulation in
  certain ranges

• De-quantization would not result in a pdf

• Increased number of samples would induce a
  computational burden

Band-pass and re-bin operations needed after F
operation
13
Band-pass, Be, Operator
Margin-based Definition

• Eliminate samples having values out of range
  (6?) might cut off tails of bi-modal or
  long-tailed distributions

• 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

14
Re-bin, RN, Operator
Resulting spdf(X)

• Samples falling into the same bin congregated in
  one

• Total distortion given by

can be used to select bin locations, where
mi center of ith bin
15
Experimental Results

• Matlab R12 used to evaluate FDPP method

?(X) for Vth

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

16
Monte Carlo FDPP Comparison
Pdf of Vth
Pdf of ID
solid FDPP dotted Monte Carlo

• A close match is observed after interpolation

• 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

17
Monte Carlo FDPP Comparison with a Low Sample
Number
Pdf of ?F
Pdf of ?F
solid FDPP with 100 samples
solid FDPP with 100 samples
noisy Monte Carlo with 1000 samples
noisy Monte Carlo with 100000 samples

• Monte Carlo inaccurate for moderate number of
  samples

• Indicates FDPP can converge to an acceptable
  estimate with far less number of samples

18
Monte Carlo FDPP Comparison
Pdf of n7
Benchmark example
solid FDPP dotted Monte Carlo
trianglesbelief propagation

• Edges define a linear sum, ex n5n2n3

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

19
Faulty Application of Monte Carlo
Pdf of n7
Benchmark example
solid FDPP dotted Monte Carlo
trianglesbelief propagation

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

20
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 pdfs where
  parametric methods are limited to certain ones