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

Probabilistic Evaluation of Solutions in Variability-Driven Optimization Azadeh Davoodi and Ankur Srivastava University of Maryland, College Park. Presenter: Vishal Khandelwal 1 ISPD2006

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 2 ISPD2006

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 S i has a corresponding timing r i & cost c i : – e.g., A solution to the gate-sizing problem has: • Timing: Delay of the circuit • Cost: Overall sizes of the gates 3 ISPD2006

Motivation • A good solution is the one with better timing and cost ⇔ ≤ ≤ S & S r r c c superior i j i j i j • Process variations randomize the timing and cost associated with a potential solution ⇔ ≤ ≤ ≈ S ( & ) 1 S P R R C C superior i j i j i j Source: Intel 1.4 Normalized Frequency 1.3 1.2 30% 1.1 20X 1.0 0.9 0 5 10 15 20 4 ISPD2006 Normalized Leakage Power

Pruning Probability ⇔ ≤ ≤ ≈ S ( & ) 1 S P R R C C superior i j i j i j = − = − R R R • Let C C C and j i j i ∞ ∞ ∫ ∫ ≥ ≥ = P ( R 0 & C 0 ) f ( r , c ) drdc R , C 0 0 f R,C : joint probability density function ( jpdf ) of random variables R and C 5 ISPD2006

Computing the Pruning Probability: Challenges ∞ ∞ ∫ ∫ ≥ ≥ = P ( R 0 & C 0 ) f ( r , c ) drdc R , C 0 0 • Accuracy – Might not have an analytical expression for f R,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 6 ISPD2006

Computing the Pruning Probability: Methods • Based on analytical approximation of the jpdf ( f R,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 7 ISPD2006

Computing the Pruning Probability : Approximating jpdf by Moment Matching R and C • Approximate R,C with new X and Y random variables X,Y where the f R,C f X,Y type of jpdf of X,Y is known • Compute the first few terms Characteristic Characteristic of the characteristic functions Function Function (Fourier transform) of the two Φ R,C Φ X,Y jpdf s (i.e., moments) • Match the first few moments Match the first few and determine the parameters terms (moments) of f X,Y • Compute the pruning Calculate probability for X and Y Probability for f X,Y 8 ISPD2006

Computing the Pruning Probability : Approximating jpdf by Moment Matching R and C X and Y f R,C f X,Y Characteristic Characteristic ∫∫ Φ = + i ( t x t y ) Function Function ( t , t ) e f ( x , y ) dxdy 1 2 X , Y 1 2 X , Y Φ R,C Φ X,Y 2 t = + + − − 1 1 it m it m m ... 1 10 2 01 20 2 Match the first few ∫∫ = = i j i j m x y f ( x , y ) dxdy E [ X Y ] terms (moments) ij X , Y Calculate Probability for f X,Y 9 ISPD2006

Computing the Pruning Probability : Approximating jpdf by Moment Matching • Challenges: – Very few bivariate jpdf s have closed form expressions for their moments – Integration of very few known jpdf s over the quadrant are analytically possible • Will study the example of bivariate Gaussian approximation given polynomial representation of R and C 10 ISPD2006

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., L eff , T ox , etc.) – Higher accuracy needs higher order of expansion – These r.v.s can be assumed to be independent • Using Principal Component Analysis (PCA) R = R = f ( L eff T , ,...) Poly ( X , X ,...) 1 ox 1 1 2 C = C = f ( L eff T , ,...) Poly ( X , X ,...) 2 ox 2 1 2 PCA and Taylor Series Expansion 11 ISPD2006

Example: Bivariate Gaussian jpdf for Polynomials ∑ = ≈ + ∑ = ≈ + R Poly ( X , X ,...) r r i X C Poly ( X , X ,...) c c i X 1 1 2 0 i 2 1 2 0 i • Assuming {X 1 ,X 2 ,…} are independent r.v.s with Gaussian density functions – The jpdf ( f R,C ) is approximated to be bivariate Gaussian – Using linear approximation of R and C 1 z = − f exp[ ] − ρ X , Y πσ σ − ρ 2 2 2 ( 1 ) 2 1 1 2 − µ ρ − µ − µ − µ 2 2 ( ) 2 ( )( ) ( ) x x x x = − + 1 1 1 1 2 2 2 2 z σ σ σ σ 2 2 1 2 1 2 • Moments of bivariate Gaussian jpdf are related to µ µ σ σ ρ , , , , 1 2 1 2 – Need to specify the values of these parameters using moment matching 12 ISPD2006

Example : Bivariate Gaussian jpdf for Polynomials 1 z = − ∑ = ≈ + f exp[ ] R Poly ( X , X ,...) r r i X − ρ X , Y πσ σ − ρ 2 2 2 ( 1 ) 1 1 2 0 i 2 1 ∑ = ≈ + 1 2 C Poly ( X , X ,...) c c i X 2 1 2 0 i − µ ρ − µ − µ − µ 2 2 ( x ) 2 ( x )( x ) ( x ) = − + 1 1 1 1 2 2 2 2 z σ σ σ σ 2 2 1 2 1 2 R and C X and Y µ = f R,C σ + µ = f X,Y E [ C ] 2 2 2 E [ R ] 2 1 1 µ = σ + µ = 2 2 2 E [ R ] E [ C ] Characteristic Characteristic 1 2 2 Function Function ρσ σ + µ µ = [ RC ] E x y x y Φ R,C Φ X,Y Analytical expression for probability integral of bivariate Match the first few terms (moments) Gaussian jpdf is available (Hermite Polynomials)* Calculate Probability for f X,Y 13 ISPD2006 * [Vasicek 1998]

Computing the Pruning Probability : Conditional Monte Carlo (CMC) ∞ ∞ ∫ ∫ ≥ ≥ = P ( R 0 & C 0 ) f ( r , c ) drdc R , C 0 0 • 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 14 ISPD2006

Computing the Pruning Probability : Conditional Monte Carlo (CMC) Compute Simple Generate Samples Bounds for R and C: Based on pdf of the Xi variables L U L U R , R , C , C Evaluate L U L U , , , R R C C Can predict Can NOT sign of R and predict sign of C from its < > L U R and C from R 0 , R 0 bounds its bounds or < > L U C 0 , C 0 N Y Determine if R>0 & C>0 Determine if R>0 & C>0 by evaluating R and C from the bounds Update count of # R &C>0 Pruning Probability ≥ ≥ # ( 0 , 0 ) R C 15 ISPD2006 # total

Computing the Pruning Probability : Conditional Monte Carlo (CMC) • 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 16 ISPD2006

Example : Computing Bounds on Polynomials l l ∑ ∑ 1 n = i i i Poly ( x ,..., x ) ... a x 1 x 2 ... x n 1 n i ,..., i 1 2 n 1 n = = i 0 i 0 1 n • Bernstein coefficients define convex hull for any polynomial* ⎛ ⎞ ⎛ ⎞ i i ⎜ ⎟ ⎜ ⎟ i i 1 n ... ⎜ ⎟ ⎜ ⎟ n b 1 ( ;...; ; ) ⎝ ⎠ ⎝ ⎠ i j j j ∑ ∑ 1 n = i ,..., i 1 n l l b ... a 1 n ⎛ ⎞ ⎛ ⎞ ,..., ,..., i i i i 1 n l l 1 n 1 n = = ⎜ ⎟ ⎜ ⎟ j 0 j 0 1 n 1 n ... ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ j j 1 n 17 ISPD2006 * [Cargo, Shisha 1966]

Example : Computing Bounds on Polynomials • Simple hyper-plane lower-bounds are defined for each polynomial from its Bernstein coefficients* 18 ISPD2006 * [Garloff, Jansson, Smith 2003]

Application: Dual-V th Assignment for Leakage Optimization Under Process Variations VthH VthL VthL VthL VthL VthH VthL VthL VthH VthH VthH • 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 (T cons ) at the Primary Output 19 ISPD2006