Probabilistic Evaluation of Solutions in Variability ... · Azadeh Davoodi and Ankur Srivastava University of Maryland, College Park. Presenter: Vishal Khandelwal Probabilistic Evaluation

ISPD20061

Azadeh Davoodi and Ankur SrivastavaUniversity of Maryland,

College Park.

Presenter: Vishal Khandelwal

Probabilistic Evaluation of Solutions in Variability-Driven Optimization

ISPD20062

Outline• Motivation

– Challenge in probabilistic optimization considering process variations

• Pruning Probability– Metric for comparison of potential solutions

• Computing the Pruning Probability• Application

– Dual-Vth assignment considering process variations

• Results

ISPD20063

Motivation• Many VLSI CAD optimization problems rely

on comparison of potential solutions– To identify the solution with best quality, or to

identify a subset of potentially good solutions

• Any potential solution Si has a corresponding timing ri & cost ci:– e.g., A solution to the gate-sizing problem has:

• Timing: Delay of the circuit • Cost: Overall sizes of the gates

ISPD20064

Motivation

• Process variations randomize the timing and cost associated with a potential solution

0.9

1.0

1.1

1.2

1.3

1.4

0 5 10

20X

15 20Normalized Leakage Power

Nor

mal

ized

Fre

quen

cy

Source: Intel

30%

1)&( ≈≤≤⇔ jijij CCRRPSiS superior

jijij ccrrS ≤≤⇔ &iS superior

• A good solution is the one with better timing and cost

ISPD20065

Pruning Probability

∫ ∫∞ ∞

=≥≥0 0 , ),()0&0( drdccrfCRP CR

• Let and

fR,C : joint probability density function (jpdf) of random variables R and C

ij RRR −=ij CCC −=

1)&( ≈≤≤⇔ jijij CCRRPSiS superior

ISPD20066

Computing the Pruning Probability:Challenges

• Accuracy– Might not have an analytical expression for fR,C

– Might require numerical methods to compute the probability

• Fast computation – Necessary in an optimization framework– Makes the use of numerical techniques such as

Monte Carlo simulation impractical

∫ ∫∞ ∞

=≥≥0 0 , ),()0&0( drdccrfCRP CR

ISPD20067

• Based on analytical approximation of the jpdf( fR,C )– With a well studied jpdf– For which computing the probability integral is

analytically possible

• Using Conditional Monte Carlo simulation – Bound-based numerical evaluation of the probability– Potentially much faster than Monte Carlo

Computing the Pruning Probability:Methods

ISPD20068

Computing the Pruning Probability: Approximating jpdf by Moment Matching

R and C X and Y

Match the first few terms (moments)

CharacteristicFunction

ΦR,C ΦX,Y

fR,C fX,Y

• Approximate R,C with new random variables X,Y where the type of jpdf of X,Y is known

• Compute the first few terms of the characteristic functions (Fourier transform) of the two jpdfs (i.e., moments)

• Match the first few moments and determine the parameters of fX,Y

• Compute the pruning probability for X and Y


Calculate Probability for fX,Y

ISPD20069


∫∫ == ][),(,ji

YXji

ij YXEdxdyyxfyxm

...2

1

),(),(

20

21

012101

,)(

21,21

−−++=

=Φ ∫∫ +

mtmitmit

dxdyyxfett YXytxti

YX

R and C X and Y



ΦR,C ΦX,Y

fR,C fX,Y


Calculate Probability for fX,Y

ISPD200610


• Challenges:– Very few bivariate jpdfs have closed form expressions

for their moments– Integration of very few known jpdfs over the quadrant

are analytically possible

• Will study the example of bivariate Gaussian approximation given polynomial representation of R and C

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Example: Bivariate Gaussian jpdf for PolynomialsPolynomial representation of R and C

under process variations

• Can represent R and C as polynomials– By doing Taylor Series expansion of the R and C expressions in

terms of random variables representing the varying parameters due to process variations (e.g., Leff, Tox, etc.)

– Higher accuracy needs higher order of expansion– These r.v.s can be assumed to be independent

• Using Principal Component Analysis (PCA)

,...),(1 oxeff TLfR =,...),( 212 XXPolyC =,...),( 211 XXPolyR =

,...),(2 oxeff TLfC =PCA and Taylor Series Expansion

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Example: Bivariate Gaussian jpdf for Polynomials

• Assuming {X1,X2,…} are independent r.v.s with Gaussian density functions– The jpdf (fR,C) is approximated to be bivariate Gaussian – Using linear approximation of R and C

∑+≈= ii XrrXXPolyR 0211 ,...),( ∑+≈= ii XccXXPolyC 0212 ,...),(

22

222

21

22112

1

211

2221

,

)())((2)(

])1(2

exp[12

1

σµ

σσµµρ

σµ

ρρσπσ

−+

−−−

−=

−−

−=

xxxxz

zf YX

• Moments of bivariate Gaussian jpdf are related to

– Need to specify the values of these parameters using moment matching

ρσσµµ ,,,, 2121

ISPD200613

Example: Bivariate Gaussian jpdf for Polynomials

][1 RE=µ][ 22

12

1 RE=+ µσ][2 CE=µ][ 22

22

2 CE=+ µσ

][RCEyxyx =+ µµσρσ

22

222

21

22112

1

211

2221

,

)())((2)(

])1(2

exp[12

1

σµ

σσµµρ

σµ

ρρσπσ

−+

−−−

−=

−−

−=

xxxxz

zf YX

R and C X and Y



ΦR,C ΦX,Y

fR,C fX,Y


Calculate

Probability for fX,Y

∑+≈= ii XrrXXPolyR 0211 ,...),(∑+≈= ii XccXXPolyC 0212 ,...),(

Analytical expression for probability integral of bivariateGaussian jpdf is available (Hermite Polynomials)*

*[Vasicek 1998]

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Computing the Pruning Probability: Conditional Monte Carlo (CMC)

• CMC is similar to MC but:– Uses simple bounds that can evaluate the sign of R

and C for most of the MC samples• Evaluation of simple bounds are much more efficient than

polynomial expressions that are potentially of high order

– Only in the cases that the simple bounds can not predict the sign of R and C, the complicated polynomial expressions are evaluated

∫ ∫∞ ∞

=≥≥0 0 , ),()0&0( drdccrfCRP CR

ISPD200615Pruning Probability


Compute SimpleBounds for R and C:

ULUL CCRR ,,,

totalCR

#)0,0(# ≥≥

Generate SamplesBased on pdf ofthe Xi variables

ULUL CCRR ,,,Evaluate

0,00,0

><

><UL

UL

CCRR

Determine if R>0 & C>0from the bounds

Determine if R>0 & C>0by evaluating R and C

Update count of # R &C>0

Y N

or

Can predict sign of R and

C from its bounds

Can NOT predict sign of R and C from

its bounds

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• Accurately predicts the probability value

• Speedup is due to the following intuition:- Evaluation of simple bounds are much faster than high-order polynomials- If the bounds are accurate, they predict the sign of the polynomials very frequently resulting in significant speedup


ISPD200617

Example: Computing Bounds on Polynomials

• Bernstein coefficients define convex hull for any polynomial*

∑ ∑= =

=1

1

21

10 0

21,...,1 ......),...,(l

i

l

i

in

iiiin

n

n

n

nxxxaxxPoly

∑ ∑= =

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛

=1

111

0 0,...,

1

1

1

1

,...,

...

......

i

j

j

jii

n

n

n

n

ii

n

nnn

a

jl

jl

ji

ji

b);;...;( ,...,

1

11 nii

n

n bli

li

*[Cargo, Shisha 1966]

ISPD200618

Example: Computing Bounds on Polynomials

• Simple hyper-plane lower-bounds are defined for each polynomial from its Bernstein coefficients*

*[Garloff, Jansson, Smith 2003]

ISPD200619

Application:Dual-Vth Assignment for Leakage

Optimization Under Process VariationsVthL

VthH

VthL

VthH

VthLVthL

VthH

VthH

VthHVthL

VthL

• Assignment of either high or low threshold voltage to gates in a circuit (represented as nodes in a graph)

- Higher threshold (slow), lower threshold (leaky)• Under process variations the goal is:

-To minimize expected value of overall leakage (E[L])-Subject to bounding the maximum probability of violating a Timing Constraint (Tcons) at the Primary Output

ISPD200620

• Each solution:- Overall leakage at the node’s subtree:

- Arrival time of the node’s subtree: Approximated as a linear combination of parameters

...)()()(0 +++= ∑∑∑ ji

kiji

ki

kk XXlXllL

• Dynamic programming based formulation - Topological traversal from PIs to POs- Solution at a node:

-Vth assignment to sub-tree rooted at the node- Solution set at each node:

-Generated by combining solutions of a node’s children + node’s Vth possibilities-Pareto-optimal set identified & stored*

Dual-Vth Assignment for Leakage Optimization Under Process Variations

VthL

VthH

VthL

VthH

VthLVthL

Using the Pruning Probability

*[Davoodi, Srivastava ISLPED05]

ISPD200621

%Error in Estimation of Pruning Probability

For 2600 solution pairs from the dual-Vth framework

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Speedup in Computing the Pruning Probability

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Comparing Quality of Solution in Dual-Vth Assignment

Run Time (sec) E[I] in pA

Maximum allowed risk (probability) for violating the timing constraint: 0.3

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Conclusions

• Introduced pruning probability as metric to compare potential solutions in a variability-driven optimization framework

• Illustrated computing of pruning probability:– Using efficient jpdf approximation– Using accurate Conditional Monte Carlo

simulation– Both methods significantly faster the MC

ISPD200625

Thank You For Your Attention!

For details please contact the authors: {azade,ankurs}@eng.umd.edu

Documents

Probabilistic Evaluation of Solutions in Variability ... · Azadeh Davoodi and Ankur Srivastava University of Maryland, College Park. Presenter: Vishal Khandelwal Probabilistic Evaluation