GPR-Based Framework for Statistical Analysis of Gate Delay under NBTI and Process Variation Effects

Round 1

Reviewer 1 Report

The overall presentation of the paper is good.

The paper addresses the process variation (PV) to the normal NBTI failure mechanisms and performs a monte-carlo simulation to the circuit.

The authors propose a solution for reducing the time to process a known simulation problem. They used machine learning to cut down on a large amount of heavy computations to solve the problem.

The paper is written very well and is worthy of publication although it is not clear how much of the readership may be interested.   It is certainly publication worthy.  

Author Response

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Reviewer 2 Report

In this paper, a variation of gate delay by jointly considering Negative Bias Temperature Instability (NBTI) and Process Variation (PV) effects is analyzed. A Statistical Gate Delay Extraction (SGDE) framework is proposed, by using a machine learning technique.

The general idea is interesting, and the overview of the related work and background theory is adequate. 

The paper contribution is presented mostly in subsections 3.3 and 3.4. Verification of the framework, given in subsection 4.2.3, convinced me of the framework's accuracy. A comparison with the existing solutions is also given in Table 5. 

The application of the ML model to fast Monte-Carlo simulation seems to be very useful. Therefore, it should be useful if this part of the paper can be extended with some additional explanations.

The technical preparation of the paper should be improved before the publication, as a lot of minor incorrectness  are visible:

 - table should be placed on one page, not divided into two pages (please, see line 177)

 - a figure caption should not be given on a different page than the figure itself (please, see line 357);

 - when a sentence is finished with an equation, a dot should be placed after the equation (please see Eqns. (3), (5), (6), (7),...);

 - the word "where" in line 300 should not be capitalized, and the first words in table captions should be capitalized (lines 177 and 190)

 - all references should be written according to the journal template.

Please, correct the technical incorrectness very carefully!

Author Response

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Reviewer 3 Report

This paper investigates the statistical analysis of the gate delay by means of a surrogate model built via the Gaussian Process Regression (GPR). The proposed application of the GPR is rather interesting. However, the overall novelty of the proposed modeling scheme is rather limited. Indeed, different from what it is stated at the end of Sec. II, the idea of applying the GPR or Kriging to the uncertainty quantification and reliability analysis in circuital and electronic applications has been already presented in several works available in the literature.

See as an example:

- J. N. Tripathi, H. Vaghasiya, D. Junjariya and A. Chordia, "Machine Learning Techniques for Modeling and Performance Analysis of Interconnects," in IEEE Open Journal of Nanotechnology, vol. 2, pp. 178-190, 2021, doi: 10.1109/OJNANO.2021.3133325.

- T. Nguyen et al., "Comparative Study of Surrogate Modeling Methods for Signal Integrity and Microwave Circuit Applications," in IEEE Transactions on Components, Packaging and Manufacturing Technology, vol. 11, no. 9, pp. 1369-1379, Sept. 2021, doi: 10.1109/TCPMT.2021.3098666.

- R. Trinchero and F. Canavero, "Machine Learning Regression Techniques for the Modeling of Complex Systems: An Overview," in IEEE Electromagnetic Compatibility Magazine, vol. 10, no. 4, pp. 71-79, 4th Quarter 2021, doi: 10.1109/MEMC.2021.9705310.

Additional comments:

• How did you compute the PDFs in fig. 1? Did you use a MC simulation?
• I do not completely understand the aim of Sec. 3.2. It seems that the conclusion is that there is an unknown and possible non-linear map g(.) between the considered PV and NBTI variables, and the gate delay.
• The GPR is not explained at all. How does the GPR work? How is the prior used within the GPR training? What is the role of the kernel (covariant function)? The mathematical background of the proposed regression technique should be clearly presented in Sec. 3.3.
• How do you define the composite kernel? Is it a sum of RBF, RQ and linear kernel or it is the product among them? Why did you choose Matern12 (i.e., the absolute exponential kernel) instead of the most interesting Matern 32 and Matern 52? Indeed, for last two, the resulting covariance functions provide by their self a composite kernel since they can be seen as product of an exponential and a polynomial. Additional details are provided in:“C. E. Rasmussen and C. K. I. Williams, Gaussian Processes for Machine Learning. Cambridge, MA, USA: MIT Press, 2008”.
• What do you mean with linear (power law) model? Is it standard polynomial expansion? How did you choose the polynomial degree?
• Are you tuning the hyperparameters of the SVM regression, i.e., the regularizer and the width of the epsilon insensitive zone?
• If I understood well, you are considering 3 different testcases (i.e., the INV gate, NAND2 gate and NOR2 gate). How many input parameters are you considering for each example?
• Could you show the scatter plots for each of the proposed techniques?

Author Response

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Round 2

Reviewer 2 Report

Now the manuscript can be accepted as it is. 

Author Response

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Reviewer 3 Report

I would like to thank the Authors for partially addressing my comments. Indeed, I had the feeling that some of my comments were not taken seriously, since they have not implemented in the revised version of the manuscript.

Specifically:

- I have noticed that the brief mathematical introduction for the GPR is almost “equivalent” to the one provided in:

Trinchero and F. Canavero, "Machine Learning Regression Techniques for the Modeling of Complex Systems: An Overview," in IEEE Electromagnetic Compatibility Magazine, vol. 10, no. 4, pp. 71-79, 4th Quarter 2021,

I would like to point out that: (i) such paper has not been properly cited; (ii) there are notation issues (e.g., vectors should be in bold) an some variables are not properly defined (e.g., X_j, and x_*); (iii) there is a typo in the definition of K_* which should be K_*=[k(x_*,x_1),…,k(x_*,x_L)].

- In Response 5, what does it mean: “we have also experimented Matern32 and Matern52 kernel, but the performances of those are close to Matern12’s. So all of them don’t perform better than our proposed kernel.” ? How close are they? Can you provide the error?

- In response 6: “We use the defualt LR in scikit library. We’ve deleted the “(power law)” in our paper. Thanks for your remind.” Could you explain which kind of model are you using? What does it mean default LR in scikit library?

- Response 7. For the SVM with RBF kernel there are 3 hyperparameters to tune. Are you tuning all of them?

- The information provided in Response 8 should be implemented within the manuscript.

- I don’t understand your answer to Response 9. The complexity of scatter plots should be independent from the number of points, usually we are interested in the correlation between the model prediction and the “golden” model, i.e., if the data points form a straight line.

Author Response

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Author Response File: Author Response.pdf

Round 3

Reviewer 3 Report

The Authors addressed all my comments. However, I would like to point out that a comparison with a SVM model trained without tuning the epsilon parameter is quite unfair.

Author Response

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