Timing-aware Energy Optimization for Probabilistic Boolean Circuits

1. Timing-aware Energy/Correctness Optimization forProbabilistic BooleaN Circuits Ching-Yi Huang & Yung-Chun Hu & Black 2013/10/14

2. Outline • Introduction • Previous works • Issues & motivation • Problem formulation & key ideas • Proposed algorithm • Initial Replacing • Cell selection • Further Optimization • Future work

3. Introduction (1/3) Noise effect Lower VDD

4. Introduction (2/3) Energy per switching: Energy ratio p

5. Introduction (3/3) • Probabilistic operations • OR: ∨p • AND: ∧p • NOT: ¬p • Let probabilistic parameter p= 0.9 A 0.9 F B

6. Previous works Exact method MC method Formula-based method

7. Previous works • Strategies of correctness optimization Random assignment Observability-based assignment

8. However… • 1. Let us locate the position of MC method • Used for quick evaluation or verification? • How about hybrid with formula-based method? • 2. The observability-based correctness optimization • Did not consider the energy consumption/reduction • Did not propose how to choose the probabilityp for p-gate • Did not consider the circuit delay

9. Issues • Supply voltage vs. probability (on 45nm) • Other influences of voltage-scaling • Cell characterization • Multiple supply voltages

10. V-P & E-P • PTM 45nm bulk • Vtn 0.466 ; Vtp 0.412 • Wn 0.415 ; Wp 0.63; L 0.5 • Standard deviation 0.24 0.22 0.20 0.18 • Cell function: INV, NAND

11. V-P p p 11 Vdd (V) Sd=0.24, max_p = 0.9857 Sd=0.22, max_p = 0.9911 Vdd (V) p p Vdd (V) Vdd (V) Sd=0.20, max_p = 0.9951 Sd=0.18, max_p = 0.9977

12. P-E Energy (normalized) Energy (normalized) 1.1 V 0.8 V 0.5V p Sd=0.24, max_p = 0.9857 p Sd=0.22, max_p = 0.9911 Energy (normalized) Energy (normalized) Sd=0.20, max_p = 0.9951 Sd=0.18, max_p = 0.9977 p p

13. Other influence of voltage-scaling • Supply voltage • Delay • Driving strength • Leakage power (static power) • Switching power (dynamic power) • The “energy” we care = power × delay P = VDDIDDQ+COUTVDD2 f ?

14. Cell characterization • Why cell characterization is necessary • timing-aware • power-aware • Arcs • CCS, ECSM, NLDM • Synopsys – Liberty NCX

18. Multiple supply voltages • How many? • Should be as reasonable as possible • What values? • > both Vth of pmos and nmos

19. Outline • Introduction • Previous works • Issues & motivation • Problem formulation & key ideas • Proposed algorithm • Initial Replacing • Cell selection • Further Optimization • Future work

20. Problem formulation Objective 1 • Given • A deterministic circuit (every p=1) • Correctness constraint (rate/magnitude) • Derive • Energy optimized PBC • Considering circuit delay (little suffer) Objective 2 • Given • A deterministic circuit (every p=1) • Energy constraint (% of reduction) • Derive • Correctness optimized PBC • Considering circuit delay (little suffer) Timing-aware !!!!!

21. Key ideas • Timing • Keep intact (no timing violation or little suffer) • Probability selection • Scalable enough algorithm to deal with different # of p • P-gate selection -> cell selection • Energy optimization • Correctness optimization

22. Energy optimization 22 • As many P-gates as possible • Lower observability first • Avoid placing at critical paths • Correctness is not proportional to the # of P-gates constraint

23. Correctness optimization 23 • Lower observability first • Avoid placing at critical paths • The energy reduction should be proportional to the # of P-gates constraint Normalized energy consumption Ratio of probabilistic gates (%)

24. Trade-off between p and E • p • Correctness -> # of P-gates • Energy -> total energy ? -> Need observation • Initial replacing • Based on the observation, determine where to change p • Iterative cell selection algorithm • Similar to gate sizing

25. Evaluation • In algorithm • Correctness • Formula-based method • Delay & Power • Simple calculation • Estimation based on the information of cells • Not considering false paths • Final check • PrimeTime/PrimePower • Proposed MC method

26. Comparison • 1. Random assignment • 2. Previous work (observability-based assignment) • 3. Design Compiler

27. Outline • Introduction • Previous works • Issues & motivation • Problem formulation & key ideas • Proposed algorithm • Initial Replacing • Cell selection • Further Optimization • Future work

28. Proposed algorithm • Replace lower observability first • Avoid placing at critical paths • Select voltage (probability) • correctness sensitivity estimation • Estimate correctness/energy after every placing • Final check with a little tuning Initial replacing Cell selection Further optimization

29. Proposed algorithm Initial replacing Cell selection Selection algorithm Further optimization

30. Proposed algorithm Initial replacing Cell selection • Approximate PBC minimization • Calculate observability • Find near 0 or 1 signals • Evaluate correctness • Remove near-redundant wire according to observability and the diff of correctness • Final check Further optimization

31. Redundant Removal • In deterministic circuit, we can inject SA faults and remove redundant wires. SA fault Observable? No Remove the wire

32. Near-redundant removal • In PBC, we may regard signals which have high probability to be 0 or 1 as stuck-at-faults. E A 0.0375 w 0.9 0.9625 0.3 B 0.075 Correctness=0.99375 0.25 C D If every PI has probability of 0.5 to be 1, w has probability of 0.9625 to be 0.

33. Near-redundant removal • In PBC, we may regard signals which have high probability to be 0 or 1 as stuck-at-faults. 0 Correctness=0.977 The node reduction is 66%.

34. Approximate PBC minimization Start Identify near-redundant wires Select the node with lowest observability Recalculate observabilty & Re-simulate circuit Check the correctness suffering Yes Remove the near redundant wire Pass ? No Yes Other candidates No Final check & restore Report energy consumption Report correctness Derive final PBC Finish

35. Correctness definition with respect to applications • Correctness rate vs. correctness magnitude • Others’ definition • Relative error = POorig / POapprox×100% • Error magnitude = (|POorig-POapprox|)/|POmax| ×100% • Error rate (ER) : the percentage of vectors for which the values at a set of outputs deviate from the error free responses, during normal operation. • Error significance (ES) for a set of outputs : defined as the maximum amount by which the numerical value at the outputs of an imperfect circuit version can deviate from the corresponding value for the perfect version • RS = ER*ES

36. Application (1/2)

37. Application (2/2) Assume each PO has same weight, the average correctness is 0.9.

38. Future work • Learn Liberty NCX • Cell characterization and observation • Correctness redefinition • What technology? (45nm, 90nm, 130nm, 250nm) • More applications (benchmarks referenced by papers) • Merge near-identical signals?