Probabilistic Evaluation of Solutions in Variability-Driven - - PowerPoint PPT Presentation

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Azadeh Davoodi and Ankur Srivastava University of Maryland, College Park. Presenter: Vishal Khandelwal

Probabilistic Evaluation of Solutions in Variability-Driven Optimization

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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

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Motivation

• Many VLSI CAD optimization problems rely
• n 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

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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 5 10

20X

15 20

Normalized Leakage Power Normalized Frequency

Source: Intel

30%

1 ) & ( ≈ ≤ ≤ ⇔

j i j i j

C C R R P S

i

S superior

j i j i j

c c r r S ≤ ≤ ⇔ &

i

S superior

• A good solution is the one with better timing

and cost

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Pruning Probability

∫ ∫

∞ ∞

= ≥ ≥

,

) , ( ) & ( drdc c r f C R P

C R

• Let

and

fR,C : joint probability density function (jpdf) of

random variables R and C

i j

R R R − =

i j

C C C − = 1 ) & ( ≈ ≤ ≤ ⇔

j i j i j

C C R R P S

i

S superior

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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

∫ ∫

∞ ∞

= ≥ ≥

,

) , ( ) & ( drdc c r f C R P

C R

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• 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

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Computing the Pruning Probability:

Approximating jpdf by Moment Matching

R and C X and Y

Match the first few terms (moments) Characteristic Function

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
• f the characteristic functions

(Fourier transform) of the two jpdfs (i.e., moments)

• Match the first few moments

and determine the parameters

• f fX,Y
• Compute the pruning

probability for X and Y Characteristic Function Calculate Probability for fX,Y

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Computing the Pruning Probability:

Approximating jpdf by Moment Matching

∫∫

= = ] [ ) , (

, j i Y X j i ij

Y X E dxdy y x f y x m

... 2 1 ) , ( ) , (

20 2 1 01 2 10 1 , ) ( 2 1 ,

2 1

− − + + = = Φ

∫∫

+

m t m it m it dxdy y x f e t t

Y X y t x t i Y X

R and C X and Y

Match the first few terms (moments) Characteristic Function

ΦR,C ΦX,Y

fR,C

fX,Y

Characteristic Function Calculate Probability for fX,Y

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Computing the Pruning Probability:

Approximating jpdf by Moment Matching

• 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

• f R and C

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Example: Bivariate Gaussian jpdf for Polynomials

Polynomial 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

• x

eff T

L f R =

,...) , (

2 1 2

X X Poly C = ,...) , (

2 1 1

X X Poly R =

,...) , (

2

• x

eff T

L f C =

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

∑

+ ≈ =

i iX

r r X X Poly R

2 1 1

,...) , (

∑

+ ≈ =

i iX

c c X X Poly C

2 1 2

,...) , (

2 2 2 2 2 2 1 2 2 1 1 2 1 2 1 1 2 2 2 1 ,

) ( ) )( ( 2 ) ( ] ) 1 ( 2 exp[ 1 2 1 σ µ σ σ µ µ ρ σ µ ρ ρ σ πσ − + − − − − = − − − = x x x x z z f

Y X

• Moments of bivariate Gaussian jpdf are related to

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

ρ σ σ µ µ , , , ,

2 1 2 1

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Example: Bivariate Gaussian jpdf for Polynomials

] [

1

R E = µ

] [

2 2 1 2 1

R E = + µ σ

] [

2

C E = µ

] [

2 2 2 2 2

C E = + µ σ

] [RC E

y x y x

= + µ µ σ ρσ

2 2 2 2 2 2 1 2 2 1 1 2 1 2 1 1 2 2 2 1 ,

) ( ) )( ( 2 ) ( ] ) 1 ( 2 exp[ 1 2 1 σ µ σ σ µ µ ρ σ µ ρ ρ σ πσ − + − − − − = − − − = x x x x z z f

Y X

R and C X and Y Match the first few terms (moments) Characteristic Function

ΦR,C ΦX,Y

fR,C fX,Y

Characteristic Function Calculate Probability for fX,Y

∑

+ ≈ =

i iX

r r X X Poly R

2 1 1

,...) , (

∑

+ ≈ =

i iX

c c X X Poly C

2 1 2

,...) , (

Analytical expression for probability integral of bivariate Gaussian 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

∫ ∫

∞ ∞

= ≥ ≥

,

) , ( ) & ( drdc c r f C R P

C R

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Pruning Probability

Computing the Pruning Probability:

Conditional Monte Carlo (CMC)

Compute Simple Bounds for R and C:

U L U L

C C R R , , ,

total C R # ) , ( # ≥ ≥

Generate Samples Based on pdf of the Xi variables

U L U L

C C R R , , ,

Evaluate

, , > < > <

U L U L

C C R R

Determine if R>0 & C>0 from the bounds Determine if R>0 & C>0 by evaluating R and C Update count

• f # R &C>0

Y N

• r

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 Computing the Pruning Probability:

Conditional Monte Carlo (CMC)

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Example: Computing Bounds on Polynomials

• Bernstein coefficients define convex hull for any polynomial*

∑ ∑

= =

=

1 1 2 1 1

2 1 ,..., 1

... ... ) ,..., (

l i l i i n i i i i n

n n n n

x x x a x x Poly

∑ ∑

= =

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ =

1 1 1 1

,..., 1 1 1 1 ,...,

... ... ...

i j j j i i n n n n i i

n n n n

a j l j l j i j i b

) ; ;...; (

,..., 1 1

1 n

i i n n b

l i l i

*[Cargo, Shisha 1966]

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Example: Computing Bounds on Polynomials

• Simple hyper-plane lower-bounds are defined for each

polynomial from its Bernstein coefficients* *[Garloff, Jansson, Smith 2003]

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Application:

Dual-Vth Assignment for Leakage Optimization Under Process Variations

VthL VthH VthL VthH VthL VthL VthH VthH VthH VthL 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

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• Each solution:
• Overall leakage at the node’s subtree:
• Arrival time of the node’s subtree: Approximated as a linear

combination of parameters

...

) ( ) ( ) (

+ + + =

∑∑ ∑

j i k ij i k i k k

X X l X l l L

• 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 VthL VthL

Using the Pruning Probability

*[Davoodi, Srivastava ISLPED05]

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%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

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Thank You For Your Attention!

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