__ Summary and Contributions__: A soft-max function has two main efficiency measures, approximation and smoothness. Authors goal is to identify the optimal approximation-smoothness tradeoffs for different measures of approximation and smoothness. They introduce a soft-max function, called piece-wise linear soft-max, with optimal tradeoff between approximation measured in terms of worst-case additive approximation, and smoothness measured with respect to l𝑞 -norm. The worst-case approximation guarantee of the piece-wise linear mechanism enforces sparsity in the output of our soft-max function, a property that is known to be important in Machine Learning applications and is not satisfied by the exponential mechanism. Finally, they investigate another soft-max function, called power mechanism, with optimal tradeoff between expected multiplicative approximation and smoothness with respect to the Rényi Divergence, which provides improved theoretical and practical results in differentially private submodular optimization.

__ Strengths__: The PLSOFTMAX in Section 4 is quite interesting. Since it's a piece-wise linear function and achieves the best Lipschitz constant. This result might inspire us to seek some piece-wise linear function rather than smooth function when facing similar situation. During the proof, this is another interesting thing, the norm $l_r$ in Theorem B.6 is the dual norm of norm $l_p$. The dual norm also appears in Theorem 4.4 for norms $l_\infty$ and $l_1$.
From the applications in Section 6, we can see that the soft-max function could be used to different areas. That stands out the meaning of `piece-wise linear' of the function PLSOFTMAX, since it gives us another idea to find proper functions when dealing with different situations.

__ Weaknesses__: The result is quite good.

__ Correctness__: For Theorem 3.2, from the proof we can see that $c$ could be equal to $ \frac{\log d-2}{2}$, then the result in the statement of Theorem 3.2 should be $c\ge \frac{\log d-2}{2}$
In the Appendix B, for the definition of $\bm{m}_i^{(k,d)}$ and $\bm{s}_i^{(k,d)}$, using words to describe is enough. The dashes in illustration (B.6) and (B.7) are redundant. In line 173(Appendix), the Riemann zeta function should be an infinite sum rather than the finite sum appears here. Therefore, line 174 should be edited(this will not influence the result).

__ Clarity__: Main paper:
Both `soft-max function' and `softmax function' (in Section 1) appear.
Renyi, line 48.
$y \to \bm{y}$, line 98, 102.
x \to \bm{x}$, line 98, 102, 108, 110, 145, 148, 150.
`the following inequality' $\to$ `if the following inequality', line 250.
`soft maximum', line 265.
Supplementary Material:
$x \to \bm{x}$, line 7.
$f \to \bm{f}$, line 80.
$\bm{x}$ should be $\bm {x}_a$, line 13.
Missing full stop at the end of line 14, 15, 89-92, 95, 143.
Missing comma, line 99.
`forbitten', above line 251.

__ Relation to Prior Work__: They show a detailed comparison with the most-commonly used soft-max function, exponential mechanism. And authors get new results on some common problems via a new way -- construct a piece-wise linear function, which is creative without repeating old methods.

__ Reproducibility__: Yes

__ Additional Feedback__: The feedback answers my question.

__ Summary and Contributions__: This paper studies approximation and smoothness properties of soft-max functions. The authors show that the commonly used exponential function has optimal tradeoff between expected additive approximation and smoothness measured with respect to Renyi divergence. However, once smoothness is measured with represent to l_p-norms, the exponential function is no longer the optimal one. The authors present a new the piecewise linear function which is the optimal one for norms, and besides the expected additive approximation, they also study its worst-case approximation. For multiplicative approximation, they present the power mechanism, which is optimal for Renyi Divergence. The paper also discusses many different applications of the new functions to mechanism design, sub modular optimisation, and deep learning.

__ Strengths__: The paper is well-written, and the result seems to be quite important with many possible applications in many different topics. I think that the contribution of the paper will be appreciated by multiple sub-communities of NeurIPS.
Post rebuttal: Thank you for your response.

__ Weaknesses__: Approximate incentive compatibility is a rather weak concept in mechanism design, so I don't find this application very interesting. It would be nice to include proofs in the main text.

__ Correctness__: Could not verify the claimed statements: no proofs are presented in the main text. However, the results seem plausible.

__ Clarity__: The paper is quite well-written and easy to follow, which I really appreciate. The related literature could be discussed in more detail, and I think it would be nice to have a section with open problems/directions for future work.

__ Relation to Prior Work__: The contribution of the paper is quite clear, but the related work could be discussed in more detail.

__ Reproducibility__: No

__ Additional Feedback__: -- Why is O(log(d)/delta) a "constant"? It depends on the input d, which can be as large as it wants, right?
-- piece-wise --> piecewise
-- line 77: take have been --> remove "take"
-- line 86: usually, the unit simplex is denoted by Delta^{d-1}, not Delta_{d-1}

__ Summary and Contributions__: The paper proposed two soft-max functions: one denoted as PLSoftMax, for piece-wise linear soft max, and the other denoted as POW, for power mechanism.
The paper investigated the theoretical properties of these two functions on the two aspects: (1) approximation and (2) smoothness, with intensive theoretical development.
Further the paper demonstrated in empirical experiments how PLSoftMax can be used for solving differentially private submodular optimization problem on the data of DBLP computer scientist co-authorship networks; the result indicates that by comparing the exponential soft max functions, the proposed POW achieves better smoothness and approximation.
Lastly, the paper claimed without empirical studies that (1) the PLSoftMax has good properties in the context of multi-class classification problems and (2) the PLSoftMax can be used to design an incentive compatible mechanism.

__ Strengths__: (1) the paper claimed contributions to a wide spectrum of applications from mechanism design, differential privacy, and supervised learning
(2) the paper provided comprehensive review of the theoretical development.

__ Weaknesses__: (1) The motivation of why a better tradeoff between approximation and smoothness is clear; but the motivation of how the proposed soft-max functions are designed is unclear. Further, in addition to the theoretical proof, some more discussions or intuitions would be helpful for readers to understand better why empirically the proposed methods should be used.
(2) No empirical support is provided for two of the three applications as claimed in the paper.

__ Correctness__: Most of the claims are correct; the empirical methodology for sub-modular optimization is valid.
The theoretical claims are not fully reviewed due to reviewer's time constraints.
The claims on the applications on ML and mechanism design are not supported empirically.

__ Clarity__: Yes, I enjoyed reading the paper a lot.

__ Relation to Prior Work__: Yes.

__ Reproducibility__: Yes

__ Additional Feedback__: Other minor comments:
(1) Line 135: some discussion would be nice to introduce why we want to switch perspective from (l_p, D_\inf)-Lipschitz to (l_p, l_q)-Lipschitz. Is it because (l_p, l_q)-Lipschitz is of particular interest to some applications or because that's the only setup where the approximation guarantee can be derived?
(2) Line 346, the sentence is not well grounded by saying "useful in Machine Learning". More precise wording is needed, for example, "smoothness is preferred for gradient calculation in commonly adopted stochastic gradient descent algorithms".
(3) In the appendix line 513, it'd be good to also report the standard deviation in addition the mean in the figure; this will help confirm the statistical significance visually.