Download - Statistical Analysis and Modeling for Error Composition in Approximate Computation Circuits
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Statistical Analysis and Modeling for Error Composition in Approximate
Computation Circuits
Wei-Ting Jonas Chan1, Andrew B. Kahng1, Seokhyeong Kang1, Rakesh Kumar2, and John Sartori3
1VLSI CAD LABORATORY, UC San Diego2PASSAT GROUP, Univ. of Illinois
3Univ. of Minnesota
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Threats to traditional IC design approach ...Extreme variations / Reliability issues / Cost:
Approximate Computation:Relaxing the requirement of correctness can dramatically reduce costs of the design
Why Approximate Computation? Threats to traditional IC design approach ...
Extreme variations: PVT variation uncertainty leads to design overheadReliability issues: Hard errors (NBTI, latchup), Soft errors (α-particle)Cost: Cost (power/performance) of perfect accuracy is too high!
Approximate ComputationRelaxing the requirement of correctness can dramatically reduce costs of the design
What is the square root of 10 ?
“a little more than three”
“3.162278....”
Approximation could be faster and more powerful
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Reduce Design Cost with Approximations
Simplified critical paths but with errors
Accurate hardware
Approximate hardware
Approach: insert approximate hardware modules on critical paths
What is the output quality of this circuit?
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Building Blocks: Approximate Hardware Modules
Zhu et al. TVLSI 2010 ETAI : accurate part + inaccurate part Reduce error size Error rate is high
ETAIIM : limited carry-chain run-length Extra protection hardware Reduce error rate and significance
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(c)~(f) have 50% power of accurate adder (b), BUT……
Result Quality Estimation of Approximate Computation
Image smoothing(Addition operations executed by different approximate adders)(a) Original image(b) Accurate adder(c) ACA(d) ETAI(e) ETAII(f) LU
(a) (b) (c)
(d) (e) (f)
How can system designers estimate result quality metrics for circuits containing approximate adders?
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Problem: Result Quality Estimation
Correct results Approximate results
Arithmetic hardware replacement
Accurate hardware
Approximate
hardware
Given: Input statistical propertiesHardware configurationsTopologies of circuits
Output:Estimated error metrics
Goal: quantify degradation of result accuracy after approximate hardware modules are inserted
How to compose errors at circuit level?Solution from this work:
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Outline Related Work Problem Modeling and Proposed Approaches Results and Conclusions
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Related Work
[HuangLR12] Propagates error metrics Improves estimation accuracy and
runtime
Our work
Category Gate level RoundingApproxima
te Arithmetic
VDD scaling
Manipulated Elements Logic cell Arithmetic Arithmetic Multiple Levels
Error Source Appx. HW Rounding Appx. HW Over-scaled VDD
Probabilistic Errors N N N Y
Intensively characterize error distributions over different intervals
Propagate distributions with interval arithmetic
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Related Work [HuangLR12] Intensively characterize error distributions over different intervals Single intervals represent multiple values in log scale
quantization inaccuracy
Positive ErrorsNegative Errors
abs(log(Probability))
PDF PMF
2min, 2max-2max, -2min
If the inputs are out of range, there will be extra inaccuracy
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Related Work [HuangLR12] Source of estimation inaccuracy: quantization errors from
interval representation Accuracy does not scale with characterization runtime
10 100 1000 100001000000.10
1.00
10.00
0.23 0.260.39
1.78
Sample Size (K)
Tot
al C
hara
cter
iza-
tion
Run
time
(hr)
For better accuracy, alternative approach is required
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Error Metrics for Quality Estimation
• Error rate (ER): measures the frequency of error occurrences
• Error significance (ES): measures the magnitude of errors• Average relative error significance (ARES): measures the ratio
between error magnitude and signal magnitude
• Mean square error (MSE): common metric in signal processing • Signal to Noise Ratio (SNR): common metric for quality of
image processing
• Max error (MAXE): measure the upper bound of errors
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Outline Related Work Problem Modeling and Proposed Approaches Results and Conclusions
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Our Quality Estimation Approach
Traverse the design to propagate
statistical property
Look up EMin
in pre-characterized library
Compute EM at output by propagations
Pre-characterized STD tables
Pre-characterized EMin tables
Stage 1:Hardware characterization
Stage 2: Composition of EMs
Statistical property
Information of EMs
STD: standard deviationEMin: intrinsic error metric
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Our Quality Estimation Approach
Traverse the design to propagate
statistical property
Look up EMin
in pre-characterized library
Compute EM at output by propagations
Pre-characterized STD tables
Pre-characterizedEMin tables
Stage 1:Hardware characterization
Stage 2: Composition of EMs
Statistical property
Information of EMs
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Hardware Characterization: Observation #1 Observation #1: EMs of approximate hardware depend
on input patterns
Input patterns decide whether carry chain will lose bits
ETAIIM
CLA
RCA
CLA
RCA
CLA
RCA
CLA
RCA
‘0’
k guard blocks for MSBMSB
{A, B} {A, B} {A, B} {A, B}
EMin = f( k, STDA, STDB )
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Hardware Characterization: Observation #2 Observation #2: EMs in ETAIIM-type adders depend on
input distribution and hardware configuration
k = # of guard blocks to mitigate errors
Log(ES) vs. input STDs ER vs. input STDs
k = 1k = 2
k = 3k = 4
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Hardware Characterization: Our Solution Generate lookup tables to store pre-characterized EMs
Generate libraries
STDZ tables
STD A
STDB
Hardware configurations
EMin tablesST
D A
STDB
Hardware configurations
EMs vs. input STDs
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Our Quality Estimation Approach
Traverse the design to propagate
statistical property
Look up EMin
in pre-characterized library
Compute EM at output by propagations
Pre-characterized STD tables
Pre-characterized EMin tables
Stage 1:Hardware characterization
Stage 2: Composition of EMs
Statistical property
Information of EMs
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Composition of EMs: Error Propagation
EMin: EM generated by approximate hardware
{STD{A,B}, EM{A,B}}: propagated standard deviations / EMs from previous stages
{EMZ, STDZ}: EMs and STDs at output nodes
{STDA, EMA} {STDB, EMB}
{STDz, EMZ}
EMin
+* +*
+*
+*: approximate adders
Key issue: enable error propagation in circuit topology
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Composition of EMs: Observation Observation: EM (e.g., rate, magnitude) at a node
depends on both intrinsic and propagated EMs
ERA ERB
ERZ
ERin
+* +*
+*
ESA ESB
ESZ
ESin
+* +*
+*
ESZ = ESin + ESA + ESB(assume no cancellations between all error sources)
Pass RatePass Rate
Pass Rate
ERZ = 1-(1-ERin)⋅(1-ERA)⋅(1-ERB)
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Composition of EMs: Our Method Our method:
– Traverse the circuit and propagate STDs in its topology– EMs are looked up in the pre-characterized libraries
A B C D E F
Function= ((A+B)+(C+D))+(E+F)ERZ = 1−(1−ERin) · (1−ERA) · (1−ERB)EMZ = EMin + EMA + EMB
For each node, EMs are propagated as follows:
Traverse and propagate
(for EMs other than ER)
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Outline Related Work Problem Modeling and Proposed Approaches Results and Conclusions
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Results: Table-Lookup Approach
• Testcase: 5-node adder tree
• Input distributions: zero mean normal distribution with different STDs
• Different configurations of ETAIIMs
• Compared with Monte Carlo simulation
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Experimental Results: FIR Filter
NET11
NET1
C1 = 0.1
NET2
C2 = 0.2
NET3
C3 = 0.3
NET4
C4 = 0.4
NET10
NET9
NET5 NET6 NET7 NET8
Net Type Error Estimation Inaccuracy (%)ER ES ARES MSE SNR MAXE
NET9ETAII
M IN 0.3% 6.4% 17.0% 6.4% 19.1% 0.0%
NET10ETAII
M IN 1.3% 2.6% 61.9% 3.3% 10.7% 0.0%
NET11ETAII
M IN 1.0% 6.3% 419.6% 6.2% 6.1% 0.0%
NET11ETAII
M P13.4
% 5.8% 692.3% 5.8%436.4
% 0.7%
Approximate FIR Adders are approximate Multipliers are accurate
Estimation inaccuracies at each node for different error metrics
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Experimental Results: MAC
C0
A0 A1
level 1
C1 Ci
Ai
level i
Output
...
...
Approximate MAC(multiply-accumulate) Adders are approximate Multipliers are accurate 14 levels of MAC are tested20 testcases for each #level
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MAC: Comparison with HuangLR12
[HuangLR12] Relative inaccuracy = 109
beyond the lower bound of characterization
ES ER
Our method interpolates continuously changing EM in lookup table
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MAC: Speedup and Accuracy Improvement
Speedup= 8.4x Accuracy improvement = 3.75x
Faster runtime allows designer to evaluate more design combinationsBetter accuracy reduce the iterations due to mis-prediction
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Conclusions We propose an approach for output quality
estimation of approximate designs Our approach achieves 8.4× runtime
improvement for error composition and 3.75× average accuracy improvement for ES compared to previous (DAC-2012) work of Huang et al.
We demonstrate results on FIR filter and MAC circuits with up to 30 nodes
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Future Work Improve accuracy of EM estimation for relative
error metrics (e.g., ARES and SNR) Extend our approach to other approximate
modules, including multipliers Develop a synthesis flow for approximate
circuits using our EM analysis approach Generalize our approach to arbitrary input
distributions
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Thank You!
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Backup Slides
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Experiment and ResultsApproximate circuit: Random-generated circuits Netlists are randomly generated with accurate multipliers and different ETAIIM approximate adders
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Regression study of EM Composition
We also tried to generalize our propagation model with parameter regression
General form of error propagation models:
Simulated EM results from different hardware configurations and input distributions/EMs are used for regression
Parameters in the models are fitted with simulation data
are regression parameters
𝐸𝑅𝑍=1−10𝛼𝐶 ∙ (1−𝐸𝑅𝑖𝑛 )𝛼 𝑖𝑛 ∙((1−𝐸𝑅𝐴 ) ∙ (1−𝐸𝑅𝐵 ))α 𝑃
)
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Regression study of EM Composition
Results of parameter regression
Regression Parameters
ER ES ARES MSE SNR MAXE
1.03E+00 1.00E+00 2.42E-02 1.00E+00 3.46E-01 9.40E-01
1.26E+00 9.98E-01 9.76E-01 1.00E+00 7.15E-02 7.98E-01
-5.85E-03 5.74E-08 -5.92E-03 -5.55E-09-1.27E+00 8.65E-05Estimation Inaccuracy
w/o Reg. 4.15E-02 7.77E-02 8.38E+02 1.08E-01 1.35E+02 1.28E-01
with Reg. 7.40E-03 5.55E-01 2.09E+02 4.44E+04 4.04E-01 1.88E+01
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Experimental Results: FIR Filter
Approximate FIR Adders are approximate Multipliers are accurate
Estimation inaccuracies at each node for different error metrics NET11
NET1
C1 = 0.1
NET2
C2 = 0.2
NET3
C3 = 0.3
NET4
C4 = 0.4
NET10
NET9
NET5 NET6 NET7 NET8
Net Type Error Estimation Inaccuracy (%)ER ES ARES MSE SNR MAXE
NET9ETAII
M IN 0.3% 6.4% 17.0% 6.4% 19.1% 0.0%
NET10ETAII
M IN 1.3% 2.6% 61.9% 3.3% 10.7% 0.0%
NET11ETAII
M IN 1.0% 6.3% 419.6% 6.2% 6.1% 0.0%
NET11ETAII
M P13.4
% 5.8% 692.3% 5.8%436.4
% 0.7%