On the regularization of Wasserstein GANs
Since their invention, generative adversarial networks (GANs) have become a popular approach for learning to model a distribution of real (unlabeled) data. Convergence problems during training are overcome by Wasserstein GANs which minimize the distance between the model and the empirical distribution in terms of a different metric, but thereby introduce a Lipschitz constraint into the optimization problem. A simple way to enforce the Lipschitz constraint on the class of functions, which can be modeled by the neural network, is weight clipping. Augmenting the loss by a regularization term that penalizes the deviation of the gradient norm of the critic (as a function of the network*s input) from one, was proposed as an alternative that improves training. We present theoretical arguments why using a weaker regularization term enforcing the Lipschitz constraint is preferable. These arguments are supported by experimental results on several data sets.
 This paper is proposing a new formulation for regularization of Wasserstein Generative Adversarial models (WGAN). The original min/max formulation of the WGAN aim at minimizing over all measures, the maximal dispersion of expectation for 1-Lipschitz with the one provided by the empirical measure. This problem is often regularized by adding a "gradient penalty", .ie a penalty of the form ".lambda E_{z~.tau}}(||.grad f (z)||-1)^2" where .tau is the distribution of (tx+(1-x)y) where x is drawn according to the empirical measure and y is drawn according to the target measure. In this work the authors consider substituting the previous penalty by ".lambda E_{z~.tau}}(max( ||.grad f (z)||-1,0)^2". Overall the paper is too vague on the mathematical part, and the experiments provided are not particularly convincing in assessing the benefit of the new penalty. The authors have tried to use mathematical formulations to motivate their choice, but they lack rigorous definitions/developments to make their point convincing. They should also present early their model and their mathematical motivation: in what sense is their new penalty "preferable"? Presentation issues: - in printed black and white versions most figures are meaningless. - red and green should be avoided on the same plots, as colorblind people will not perceived any difference... - format for images should be vectorial (eps or pdf), not jpg or png... - legend/sizes are not readable (especially in printed version). References issues: - harmonize citations: if you add first name for some authors add them for all of them: why writing Harold W. Kuhn and C. Vilani for instance? - cramer->Cramer - wasserstein->Wasserstein (2x) - gans-> GANs - Salimans et al. is provided twice, and the second is wrong anyway. Specific comments: page 1: - "different more recent contributions" -> more recent contributions - avoid double brackets "))" page 2: - Please rewrite the first sentence below Definition 1 in a meaningful way. - Section 3: if .mu is an empirical distribution, it is customary to write it .mu_n or .hat .mu_n (in a way that emphasizes the number of observations available). - d is used as a discriminator and then as a distance. This is confusing... page 3: - "f that plays the role of an appraiser (or critic)...": this paragraph could be extended and possibly elements of the appendix could be added here. - Section 4: the way clipping is presented is totally unclear and vague. This should be improved. - Eq (5): as written the distribution of .tilde{x}=tx+(1-t)y is meaningless: What is x and y in this context? please can you describe the distributions in a more precise way? - Proof of Proposition 5 (cf. page 13): this is a sketch of proof to me. Please state precise results using mathematical formulation. - "Observation 1": real and generated data points are not introduced at this stage... data points are not even introduced neither! page 5: - the examples are hard to understand. It would be helpful to add the value of .pi^* and f^* for both models, and explaining in details how they fit the authors model. - in Figure 2 the left example is useless to me. It could be removed to focus more extensively on the continuous case (right example). - the the -> the page 6: - deterministic coupling could be discussed/motivated when introduced. Observation 3 states some property of non non-deterministic coupling but the concept itself seems somehow to appear out of the blue. page 10: - Figure 6: this example should be more carefully described in terms of distribution, f*, etc. page 14: - Proposition 1: the proof could be shorten by simply stating in the proposition that f and g are distribution... page 15: - "we wish to compute"-> we aim at showing? - f_1 is not defined sot the paragraph "the latter equation..." showing that almost surely x .leq y is unclear to me, so is the result then. It could be also interesting to (geometrically) interpret the coupling proposed. The would help understanding the proof, and possibly reuse the same idea in different context. page 16: - proof of Proposition 2 : key idea here is using the positive and negative part of (f-g). This could simplify the proof.
You are a member of the scientific community tasked with peer review. Review the following paper title and abstract.
### Title + Abstract
On the regularization of Wasserstein GANs
Since their invention, generative adversarial networks (GANs) have become a popular approach for learning to model a distribution of real (unlabeled) data. Convergence problems during training are overcome by Wasserstein GANs which minimize the distance between the model and the empirical distribution in terms of a different metric, but thereby introduce a Lipschitz constraint into the optimization problem. A simple way to enforce the Lipschitz constraint on the class of functions, which can be modeled by the neural network, is weight clipping. Augmenting the loss by a regularization term that penalizes the deviation of the gradient norm of the critic (as a function of the network*s input) from one, was proposed as an alternative that improves training. We present theoretical arguments why using a weaker regularization term enforcing the Lipschitz constraint is preferable. These arguments are supported by experimental results on several data sets.
user
<think>Okay, to do a decent peer review, I should start with summarizing what the work is about. This includes novelty, contributions, clarity, and impact. I might include suggestions to issues that I find critical to the usefulness of the work. Starting with novelty, there are different types of novelty. The first aspect of novelty is scope and scale. Is the work covering a broader perspective than previous works? The second aspect is methodology. Is the proposed methodology novel compared to previous methods? Do they provide any significant advantage over previous methodologies? The third aspect is results. Are the results different compared previous work? Does this lead to any novel theories or an extension of current theories? Next is clarity, I will be reading the work in depth and analyze if any sentences can be potentially ambiguous or misleading. If any sentences are ambiguous, I will state them and give a suggestion on how to rephrase such sentence or how to make more clear to the audience. Lastly, impact is important. I will attempt to predict how such work will impact the scientific community. This might include the applications of such work and how the work might transform the way we study a certain topic. Finally, I will might only mention issues that are critical. This is to keep the review as concise and helpful as possible.</think>



 This paper is proposing a new formulation for regularization of Wasserstein Generative Adversarial models (WGAN). The original min/max formulation of the WGAN aim at minimizing over all measures, the maximal dispersion of expectation for 1-Lipschitz with the one provided by the empirical measure. This problem is often regularized by adding a "gradient penalty", .ie a penalty of the form ".lambda E_{z~.tau}}(||.grad f (z)||-1)^2" where .tau is the distribution of (tx+(1-x)y) where x is drawn according to the empirical measure and y is drawn according to the target measure. In this work the authors consider substituting the previous penalty by ".lambda E_{z~.tau}}(max( ||.grad f (z)||-1,0)^2". Overall the paper is too vague on the mathematical part, and the experiments provided are not particularly convincing in assessing the benefit of the new penalty. The authors have tried to use mathematical formulations to motivate their choice, but they lack rigorous definitions/developments to make their point convincing. They should also present early their model and their mathematical motivation: in what sense is their new penalty "preferable"? Presentation issues: - in printed black and white versions most figures are meaningless. - red and green should be avoided on the same plots, as colorblind people will not perceived any difference... - format for images should be vectorial (eps or pdf), not jpg or png... - legend/sizes are not readable (especially in printed version). References issues: - harmonize citations: if you add first name for some authors add them for all of them: why writing Harold W. Kuhn and C. Vilani for instance? - cramer->Cramer - wasserstein->Wasserstein (2x) - gans-> GANs - Salimans et al. is provided twice, and the second is wrong anyway. Specific comments: page 1: - "different more recent contributions" -> more recent contributions - avoid double brackets "))" page 2: - Please rewrite the first sentence below Definition 1 in a meaningful way. - Section 3: if .mu is an empirical distribution, it is customary to write it .mu_n or .hat .mu_n (in a way that emphasizes the number of observations available). - d is used as a discriminator and then as a distance. This is confusing... page 3: - "f that plays the role of an appraiser (or critic)...": this paragraph could be extended and possibly elements of the appendix could be added here. - Section 4: the way clipping is presented is totally unclear and vague. This should be improved. - Eq (5): as written the distribution of .tilde{x}=tx+(1-t)y is meaningless: What is x and y in this context? please can you describe the distributions in a more precise way? - Proof of Proposition 5 (cf. page 13): this is a sketch of proof to me. Please state precise results using mathematical formulation. - "Observation 1": real and generated data points are not introduced at this stage... data points are not even introduced neither! page 5: - the examples are hard to understand. It would be helpful to add the value of .pi^* and f^* for both models, and explaining in details how they fit the authors model. - in Figure 2 the left example is useless to me. It could be removed to focus more extensively on the continuous case (right example). - the the -> the page 6: - deterministic coupling could be discussed/motivated when introduced. Observation 3 states some property of non non-deterministic coupling but the concept itself seems somehow to appear out of the blue. page 10: - Figure 6: this example should be more carefully described in terms of distribution, f*, etc. page 14: - Proposition 1: the proof could be shorten by simply stating in the proposition that f and g are distribution... page 15: - "we wish to compute"-> we aim at showing? - f_1 is not defined sot the paragraph "the latter equation..." showing that almost surely x .leq y is unclear to me, so is the result then. It could be also interesting to (geometrically) interpret the coupling proposed. The would help understanding the proof, and possibly reuse the same idea in different context. page 16: - proof of Proposition 2 : key idea here is using the positive and negative part of (f-g). This could simplify the proof.
assistant
 This paper is proposing a new formulation for regularization of Wasserstein Generative Adversarial models (WGAN). The original min/max formulation of the WGAN aim at minimizing over all measures, the maximal dispersion of expectation for 1-Lipschitz with the one provided by the empirical measure. This problem is often regularized by adding a "gradient penalty", .ie a penalty of the form ".lambda E_{z~.tau}}(||.grad f (z)||-1)^2" where .tau is the distribution of (tx+(1-x)y) where x is drawn according to the empirical measure and y is drawn according to the target measure. In this work the authors consider substituting the previous penalty by ".lambda E_{z~.tau}}(max( ||.grad f (z)||-1,0)^2". Overall the paper is too vague on the mathematical part, and the experiments provided are not particularly convincing in assessing the benefit of the new penalty. The authors have tried to use mathematical formulations to motivate their choice, but they lack rigorous definitions/developments to make their point convincing. They should also present early their model and their mathematical motivation: in what sense is their new penalty "preferable"? Presentation issues: - in printed black and white versions most figures are meaningless. - red and green should be avoided on the same plots, as colorblind people will not perceived any difference... - format for images should be vectorial (eps or pdf), not jpg or png... - legend/sizes are not readable (especially in printed version). References issues: - harmonize citations: if you add first name for some authors add them for all of them: why writing Harold W. Kuhn and C. Vilani for instance? - cramer->Cramer - wasserstein->Wasserstein (2x) - gans-> GANs - Salimans et al. is provided twice, and the second is wrong anyway. Specific comments: page 1: - "different more recent contributions" -> more recent contributions - avoid double brackets "))" page 2: - Please rewrite the first sentence below Definition 1 in a meaningful way. - Section 3: if .mu is an empirical distribution, it is customary to write it .mu_n or .hat .mu_n (in a way that emphasizes the number of observations available). - d is used as a discriminator and then as a distance. This is confusing... page 3: - "f that plays the role of an appraiser (or critic)...": this paragraph could be extended and possibly elements of the appendix could be added here. - Section 4: the way clipping is presented is totally unclear and vague. This should be improved. - Eq (5): as written the distribution of .tilde{x}=tx+(1-t)y is meaningless: What is x and y in this context? please can you describe the distributions in a more precise way? - Proof of Proposition 5 (cf. page 13): this is a sketch of proof to me. Please state precise results using mathematical formulation. - "Observation 1": real and generated data points are not introduced at this stage... data points are not even introduced neither! page 5: - the examples are hard to understand. It would be helpful to add the value of .pi^* and f^* for both models, and explaining in details how they fit the authors model. - in Figure 2 the left example is useless to me. It could be removed to focus more extensively on the continuous case (right example). - the the -> the page 6: - deterministic coupling could be discussed/motivated when introduced. Observation 3 states some property of non non-deterministic coupling but the concept itself seems somehow to appear out of the blue. page 10: - Figure 6: this example should be more carefully described in terms of distribution, f*, etc. page 14: - Proposition 1: the proof could be shorten by simply stating in the proposition that f and g are distribution... page 15: - "we wish to compute"-> we aim at showing? - f_1 is not defined sot the paragraph "the latter equation..." showing that almost surely x .leq y is unclear to me, so is the result then. It could be also interesting to (geometrically) interpret the coupling proposed. The would help understanding the proof, and possibly reuse the same idea in different context. page 16: - proof of Proposition 2 : key idea here is using the positive and negative part of (f-g). This could simplify the proof.