Variance Reduction via Simultaneous Importance Sampling and Control Variates Techniques Using Vegas

Referee 1

1) The authors should consider adding more references at certain places, in particular concerning ML methods for integration in the introduction, where currently only Ref. [6] is cited. There are way more papers that should be quoted here.

We thank the referee for the suggestion, and have updated the references in various places, including the introduction. As a result, the number of references has increased from 24 to 51.

2) Also in the introduction the authors state that the 'change of variables ... is difficult, if not impossible, to accomplish ... ' Its certainly difficult but this is precisely the target normalising flows or in general INN have been applied for recently and prove to be quite versatile. This should clearly be mentioned here.

We thank the referee for the comment. We actually agree with the referee and have completely rewritten that paragraph to avoid any misunderstanding.

3) Similarly, in the conclusion the authors should quote papers regarding the statements about inference techniques. Furthermore, just quoting Ref. [24] for using domain knowledge about the integrand is by no means adequate. This is a standard technique used in ALL event generators and not just MadGraph!

We thank the referee for the suggestion, and have updated the references.

Referee 2

As said in the above report, the requested changes are mainly suggestion to the authors on the point that I would have found interesting for them to comment.

1. Given the importance in our field of event generator code (Pythia, MadGraph, Sherpa, ...) on which many reader will think when reading this paper, it would be important to comment on the impact on event generation and in particular un-weighting efficiency.

We thank the referee for the suggestion, we now added a comment in the second paragraph of the conclusions (the method is not meant to improve event generation).

2. Maybe it would be good to give more details on reference #10 and #11 in the introduction in order to provide a better picture on how novel your approach is compare to those two papers.

Those references (25 and 26 in the revised version) are not targeting a high energy physics audience, and are primarily concerned with theoretical and mathematical aspects like convergence and asymptotic bounds. Our focus is on providing a practical tool which leverages the existing VEGAS infrastructure and can be used immediately by particle physicists (as well as other users of VEGAS).

3. I have some doubt on the validity of Equation 19 if p(x) and g(x) are not normalised to one. My issue is to understand why the last term is within the integral and not outside of the integral. While not critical for the paper, I would kindly ask If he authors could double check that formula.

Equation (19) is correct. Note that being a probability distribution, p(x) is unit-normalized, as explicitly written in eqn (6). g(x) does not need to be unit-normalized for eqn (19) to be valid; it follows from the fact that the integral of g(x) equals the expected value of g(x)/p(x) for x sampled from p.

4. Maybe it would be interesting to comment on either section 2.3 or section 2.5 on the situation if the CV function is constant (which in section 2.5 is identical to the case when the grid is converging)

A constant is not a good CV, since constants do not covary (Cov(f,constant)=0) and therefore there is no improvement per eqs. (17,18).

5. From the toy function used in the paper, I would claim that only two class of functions are providing significant result for the case with one CV, namely the Gaussian case and the Polynomial case. I would have like the authors to comment/investigate on the underlying reason for those absence (or presence) of such gain. My (maybe naive) guess is that this directly related to the fact that the function is (or not) separable dimension by dimension.

We are actually not quite sure as to the underlying reason for the different performance gain observed in the toy experiments. Any comments from our side would be pure speculation.

6. On table 1, the authors do not comment on the weird number on the RMS for the gaussian case with 8 dimension. In this case the RMS is significantly larger with CV, but according to table 2 the variance is reduced by 20%. Could the author check if they are a typo in one of the table or explained the reason of such apparent miss-match.

We thank the referee for their careful reading of the paper. We have double checked for typos. The two tables show errors that are computed differently. In Table 1 we compare the estimates to the true answer for the integral in order to compute the normalized rms error via eqn (31). In Table 2 we use the variance of the estimated values as the performance metric. It is possible for an estimator with lower variance to occasionally lead to a result which is farther away from the true value, simply due to statistical fluctuations. We think that this explains the rms error with CV being slightly larger than without CV for the case spotted by the referee.

7. On Figure 2, it is not clear what "normalised variance" means in the sense that it is not clear if it is the normalised variance with or without CV for that iteration. The author can also consider to put both on the plot which can be instructive to understand such figure (see next point). This should also help to comment more on the reason about the correlation observed in that figure which are also not explained.

We thank the referee for the suggestion, we rewrote the figure caption of figure 2 and defined precisely what is meant by "normalized variance".