Venue: NeurIPS 2021 Conference

Reviews:

final_score: rating: 8: Top 50% of accepted NeurIPS papers, clear accept
review_content: summary: This work investigates optimization in infinite dimensional functionals of probability measures. The authors first define a notion of gradient of functional using Wasserstein gradient flows. Then, the authors propose a Frank-Wolfe procedure in which gradient steps can be computed using finite dimensional and convex optimization methods. The authors show that their algorithm is guaranteed to converge under some assumptions by choosing proper step sizes. Finally, the authors apply their method to two non-parametric estimation problems.
main_review: The paper aims to optimize a functional of probability measures. This problem is very important and has a lot of applications in machine learning. The paper is well written and has a nice logical flow. The claims in the paper are also well supported. The notion of Wasserstein differentiability introduced to define the gradient of functional is novel, and the proposed Frank-Wolfe procedure is promising. In particular, I like their “primal-dual” algorithm for solving the Frank-Wolfe step which can implicitly maintain a primal-feasible iterate and also make progress on the primal-dual gap.

Some questions:

(1) The notion of Wasserstein differentiability is defined with respect to Wasserstein distance of order 2. I am curious whether this definition can be easily extended to $p$-order Wasserstein distance (e.g., p=1). Similarly, if considering 1-Wasserstein distance, does the algorithm (also the convergence guarantees) still apply?

(2) In Definition 1, the authors define the gradient of a functional in terms of cotangent bundle. If I understand correctly, the element in cotangent bundle should be a mapping from $\mathbb{R}^d\rightarrow \mathbb{R}^d$. Then in Section 4.2, the authors claim that the Wasserstein derivative of squared MMD is the difference between the mean embedding of $\mu$ and $\nu$, i.e., $f_{\mu}^*(\cdot)$. However $f_{\mu}^*(\cdot)$ is a mapping from $\mathbb{R}^m\rightarrow \mathbb{R}$. Isn't this contradictory? Can the authors clarify this?

(3) All assumptions and also Theorem 1 assume a geodesically convex set S. I am interested in what S would be like in practice. Is there any functional of probability measures such that S would be the whole space $\mathcal{P}_2(\mathbb{R}^d)$? Moreover, the authors claim that the Frank-Wolfe problem (15) exhibits a dual form given by (18). Can the authors please provide some references for this dual result?

Minor comments:

(1) The authors should comply with the NeurIPS 2021 style where there is no "Broader impact" Section and the checklist should be after the References Section.

(2) There are some typos, e.g.,
- In Line 108, "a selections"-> "a selection"
- In Line 116, "a aesthetic"-> "an aesthetic"
- In Line 260, $\nu$ should be $\mu$
- In line 163 and (15), $W$ should be $\mathcal{W}$.
- What is $\bar{\mathbb{R}}$?
limitations_and_societal_impact: The original Frank-Wolfe algorithm proposed by the authors might be computationally expensive for high-dimensional cases as it need a large number of ascent step for $\lambda$ in computation of gradient steps. To address this issue, the authors provide a uniform strategy for $\lambda$  which can reduce computational costs. The authors claim that they do not foresee any negative societal impact as the contributions of their work are theoretical.
review_title: null

final_score: rating: 9: Top 15% of accepted NeurIPS papers, strong accept
review_content: summary: The paper introduces Frank-Wolfe algorithms for minimizing differentiable functionals over probability measures. It can be applied to a wide range of applications from several fields and shed light on distributionally robust optimization problems over ambiguity sets defined through the Wasserstein metric.
main_review: The methods and perspectives are novel. Potential applications of the proposed algorithms are wide are and remain open. 
limitations_and_societal_impact: More discussions on how to handle other types of Wasserstein metrics are necessary.
review_title: null

final_score: rating: 5: Marginally below the acceptance threshold
review_content: summary: This paper proposes an algorithm for the minimization of functionals of probability distributions. The authors derive a convergence rate in the case of a smooth and PL function (under the geometry given by the Wasserstein distance of order 2). They also run some synthetic experiments of non-parametric estimation
main_review: ## Main Points
### Originality and significance: 
I am not an expert in the area of Distributionally Robust Optimization (DRO) and Variational method which are the areas addressed in the related work of this paper. However, I can comment on the originality of this paper with respect to the optimization literature.
It seems to me that the connection between the main algorithm (alg. 1) and the Frank-Wolfe algorithm is not clear. Algorithm 1 seems to be a steepest descent method [Boyd, 2004] with the Wasserstein distance: at each timestep an estimation of the gradient is computed and then a minimizer of the inner product with the gradient within a delta ball is computed in (15). 

Moreover, the Frank-Wolfe algorithm is a constrained convex optimization method. The problem considered in (1) is unconstrained. It is not clear how the set S is handled and if S is actually a constraint or not. All this to say that I do not see any clear connection with the Frank-Wolfe algorithm. Can the authors comment on the following question: to what extent your algorithm is related to Frank-Wolfe and steepest descent method? What is the novelty of your analysis with regards to a standard analysis of steepest descent method under PL and smoothness assumptions?

### Clarity: 
The topic of this paper is quite technical and I think this paper does an OK job at introducing the notions. However, one missing key point in this paper is the algorithm to compute the approximate gradient oracle (called ‘Frank Wolfe step’ in the paper.. Though it is more a steepest descent step in my opinion). I looked at the appendix quite extensively and it seems that one should run algorithm 3 to get a $\lambda$, sample $x \sim \mu$  and then return $m_\lambda(x)$ see (99). It would be useful to have a pseudo algorithm summarizing this in the main paper.
 
### Soundness of the experiments: 
Are the experiments within the assumptions? It is not mentioned or justified that the functions considered are smooth and PL. 
A significant part of the related work is dedicated to DRO? Since this work cannot handle constrained optimization (J has to be smooth) how is this work related to DRO?
About the Set S: All your assumptions mention a set $S$: If the set $S$ is not the whole space, how do you make sure that the iterates stay in $S$? What are non-trivial examples of sets $S$?



## Minor details:
- L29: DRO in mentioned before being defined
- L129 Gauteaux
- L135 What is ‘the supremum”
- L660 a ‘\hat’ is missing 
- Eq (54), (65) and (58) it seems that a square is missing on $g_{i-1}$ (thankfully since you need it for the second line of (58))

## References:
Boyd, Stephen, Stephen P. Boyd, and Lieven Vandenberghe. Convex optimization. Cambridge university press, 2004.

### I have read the authors' responses and the other reviews. I stand with the points of reviewer pe3w so I will maintain my score. 
limitations_and_societal_impact: Yes
review_title: null

final_score: rating: 5: Marginally below the acceptance threshold
review_content: summary: This paper provides a first-order iterative method based on Local Linear Minimization Oracle (LLMO) to solve smooth optimization problems in probability spaces. It also analyses its convergence rate under a variety of localized relaxation of classical structural assumptions (local Holder smoothness or Lojasiewicz type of inequality) and approximate oracles (gradient and LLMO).

** The answer to the author's answers is in the comment. I raise my score to 5.
main_review: The paper originally proposes an algorithm based on LLMO in the context of Probability space. However, I find the paper hard to read with many inaccurate/incomplete statements, missing references, and non-explicit facts preventing the reader from fully appreciating the authors' contribution. A striking example of inaccuracy is that the algorithm is not the "Frank-Wolfe" method adapted to Probability Space. Therefore, I believe that this paper requires a major revision. Hence, although the contribution is appreciable, I strongly recommend rejecting this paper. I now provide a detailed review which will hopefully be helpful for the major revision.

Major comments.


- Algorithm description. Algorithm 3 is a non-standard first order method that iteratively solve linear minimization oracle on a shrinking Wasserstein neighborhood of the current iterates.
The size of the neighborhood is a trade-off between the progression made at each iteration and the computational tractability of the linear minimization subproblems.
The algorithm is itself very interesting but is almost not explained nor compared/related to existing Frank-Wolfe methods.
The explanations are confusing and too much space is attributed to speculative discussion about futur work rather that explaning the algorithm (see, e.g., Line 202-220).\
Also, a first obvious remark is that it should not be called "Frank-Wolfe for (3)" (or the paper be called "Frank-Wolfe on Probability Space"), it is absolutely not the classical form of the original Frank-Wolfe algorithm, for which every subproblems is a linear minimization oracle over the initial constraint domain.\
The algorithm uses  Local Linear Minimization Oracle (LLMO) very reminiscent to the LLMO use in the conditional gradient methods of [Garber2013] and following works. Unfortunately, the authors do not mention any of these previous papers using such LLMO.\
Finally, there are also some notational problems that make it hard to read, e.g. $s$ as the stopping criterion with the norm of the gradient, in (15) $\nu$ should belong to $B_{\mathcal{W}}(\mu_{i-1}, \delta)\cap \mathcal{P}_2(\mathbb{R}^d)$ or the choice of $\delta$ that can be confused with that of (2).

- Theorem 1 is not properly stated. What does an "appropriate choice of input parameters" mean? In Assumption 1, $S$ is not specified to be geodesically convex; is it in Theorem 1? What is the nature of $S$? In the Holder definition, the interplay between $S$ and $\Delta$ could gain from being clarified. Do the authors assume $J$ to be geodesically convex?\
Is $\mu_0$ belonging to $S$? Are there guarantees that (15) maintains the iterates in $S$? The questions are rhetoric here to outline the riddle the reader has to go through to understand the significance of the results.\
Also, Assumption 2 is not discussed in conjunction with $S$. To draw a parallel with the normed space setting, if $S$ is an open neighborhood in the compact constraint set $\mathcal{C}$, vanilla Frank-Wolfe converge linearly when the function is smooth and strongly convex (and the convergence results of FW with Holder-smoothness and Lojasiewicz inequality are discussed in [Ker18]).\
As currently written, it is impossible to understand the significance of Theorem 1. (not to mention that numerical results do not illustrate the theorem).

- Unfair or unclear devaluating of related work.
In "However, [39] narrowly considers the Sinkhorn barycenter problem [...]" the wording "narrowly" is inappropriate and inaccurate. There are seemingly no significant difficulty at extending the results of [Liuse2019] to functional $J$ that satisfy the smoothness assumption proposed in [Liuse2019], i.e., outside the Sinkhorn barycenter problem. Also, example 4.1. relies on [Liuse2019] to establish the Wasserstein smoothness of the objective function. Yet, the authors only mention it in Section 4.
However, it is important to relate it in the introduction since it outlines that a main challenge of optimization in probability spaces is to establish the structural properties satisfied by the problems and not only prove the convergence behavior under these structural properties (the current focus of the authors). This fact appears very clearly in the paper of [Liuse2019], where the main technical challenge seems to establish these structural assumptions rather than extending the FW convergence results to probability spaces.\
Also, it would be better to clarify the distinction between the algorithm in [Liuse2019] and Algorithm 1: indeed, the algorithm of [Liuse2019] is much closer to the "Frank-Wolfe in Probability Spaces". For instance, in [Liuse2019], they solve a similar type of problem with smooth objective function only that they solve the constraint problem on $\mathcal{M}_+(\mathbb{R}^d)$ rather than on $\mathcal{P}_2(\mathbb{R}^d)$. Is that difference important?\
So, the term "narrowly" is confusing and a  bit impolite. Finally, see the list of minor comments for more examples of inadequate treatments of the related works.

- Importance of Assumption 2. From the previous point, if both methods are a Frank-Wolfe on probability space, then how do the $\mathcal{O}(1/T)$ of [Liuse2019] is to be compared with the linear convergence rates of Theorem 1 (when the function is smooth and the Lojasiewicz inequality is global with parameter $\theta=1/2$)? Is the Frank-Wolfe of [Liuse2019] simply less efficient?\
Actually, I tend to believe that Assumption $2$ plays a central role in the distinction. To draw a comparison with known convergence rates of the Frank-Wolfe methods in normed spaces, the type of rates obtained in Theorem 1 corresponds to the settings when the optimum is in the interior of the constraint set. If the comparison holds, then the $1/T$ convergence rate in [Liuse2019] may stem from the fact that "the domain $\mathcal{M}_+^1(\mathcal{X})$ has empty interior in the space of finite signed measure." [Liuse2019] such that the comparison cannot be drawn. In this paper, we do not know to what the rate should be compared: Is the choice of "S" such that the solution is in the interior of $\mathcal{P}_2(\mathcal{X})$? Does it even make sense? What is the cost of an LLMO iteration? Is it comparable to a proximal step or that of an LMO? Overall, it seems very tangential to see Algorithm 1 as a Frank-Wolfe method.

- Comparison of Theorem 1 with known results in "similar" settings. How does the convergence rates compares with known convergence regimes of Frank-Wolfe methods? The authors seems to compare their convergence result with that of proximal methods. Is it because the authors consider the LLMO as kind of proximal steps? This might be the case, since it is not really comparable with a LMO on the initial constraint domain.

- Lojasiewicz conditions. The current presentation of the convergence results with objective functions satisfying Lojasiewicz inequality seems shallow since the class of functions satisfying Loja in that setting is not discussed, no examples are given, and references are missing. I tend to believe that the paper would be clearer without extension to Lojasiewicz inequalities in the main paper but discussion about the meaning of Assumption 2. 1) Lojasiewicz inequalities seem not particularly interesting if they do not come with the theory that explains for which problems' structures they hold generically; 2) missing references about the uses of Lojasiewicz type condition with Frank-Wolfe algorithms, i.e., [Ker2019,Rinaldi2020]; 3) the author present it as a contribution (remark 4) to provide these Lojasiewicz type of inequality, while it is very straightforward or already explored in the context of probability spaces [Bolte2016,Hauer2017]; 4) no examples are given in the numerical experiment part with $\theta\neq 1/2$; 6) no discussion about the nature of the subset $S$ on which it is satisfied.\
For point 1), the crux of the power of Lojasiewicz inequalities is that they are known to hold generically for some classes of functions, e.g., subanalytical functions, see [Bolte2007] for references and extension of the classes of function on which it holds generically. However, the authors do not discuss how this applies in their setting, which I believe can be challenging. Important references exploring these directions are missing, e.g. (but probably others), [Bolte2016,Hauer2017].\
Line 40 "Even in the case where the cost in the squared Euclidean norm $c(x,y)=\|x-y\|^2$ (the case of primary interest [...]". Also, in the numerical experiment, the authors only consider (but do not prove it) strongly convex objective functions. So, the Lojasiewicz extension seems relatively shallow. It is not motivated by practical examples, nor is guaranteed by the theory to hold generically.\

- Numerical Experiments. There are no log plots, and it appears that the convergence is sublinear and not even linear in the asymptotic. So experiments do not support the theory?
The authors do not discuss how the given examples satisfy the Lojasiewicz assumption, and I believe they are only strongly convex.  In line 278, the authors write, "our method reaches the global optima very quickly". This is not very professional; we expect a link with the theoretical results.

- Important references are missing.\
The Kernel Herding methods with Frank-Wolfe [Bach2012]. This is an important reference as it deals with the difficulty of obtaining a faster rate of convergence on Frank-Wolfe in the infinite-dimensional setting. These papers [Bach2012,Lacoste2015] could be discussed in a related work paragraph on Frank-Wolfe methods.\
Missing ref about Lojasiewic in metric spaces, e.g. [Bolte2016,Hauer2017] and others.\
There are specific papers that provide a complete description on the convergence of Frank-Wolfe methods under Holder-smoothness and Lojasiewicz type of inequalities [Ker18,Ker19,Rinaldi2020,Bomze2021].\
Other papers about infinite dimensional FW in probability spaces like [Denoyelle2019,Mensch2019] and others.\
[Garber2013] for Local Linear Minimization Oracle.\
[Guelat1986] for linear convergence of Frank-Wolfe with only smoothness and strong convexity when the optimum is in the interior of the constraint set.\



Others semi-major comments:

- The narrative about the origin of the FW scheme is unclear. Line 29 "development of our FW method is inspired by efforts in DRO [42,5,24,52]". This is confusing because the authors ultimately solve another problem (3). And the confusion is reinforced by the $\delta$ parameter in Algorithm 1 that does not correspond to (2).
It rather seems that [42,5,24,52] arguably inspired the paper to focus on problems such as (2), but where is the link with Frank-Wolfe? How did these works inspire the authors' algorithmic development with the Frank-Wolfe algorithms? It is rather the paper from [Liuse2019] (and [Mensch2019] for which the authors missed the reference although they are indicated in the Liuse paper) that inspired the Frank-Wolfe scheme for that type of problem.\
In Line 32-33, "The form (2), itself, immediately suggests the basis of an infinite-dimensional Frank-Wolfe procedure since it provides a "linear" objective [...]". This explanation is not convincing and looks like an attempt to write another narrative about the origin of using Frank-Wolfe (maybe the reader becomes overly suspicious); in page 4 of [Liuse2019], the comment "[...] in this work we adopt the Frank-Wolfe we adopt the Frank-Wolfe (FW) algorithm. Indeed, one key advantage of this method is that it is formulated in terms of directional derivatives along with feasible directions [...]" is a more convincing reason for using FW algorithms specifically for these types of problems.\
There are also issues with the paragraph "Variational methods" in Line 66. "[...] which draws similar inspiration from finite-dimensional Frank-Wolfe procedures", it is unclear and might lead the reader to believe that the authors deal with the infinite-dimensional setting while other works did not; which is false since e.g. [Liuse2019] already did. The authors do not mention (in the related work paragraph, although they do it in the numerical experiment) also that an important contribution of [Liuse2019] is to prove smoothness of their functional w.r.t. the total variation norm, and the authors use these results in section 4.1.

-  Unclear claims: in Line 53-56, "Unfortunately, the techniques offered by these efforts require assumptions which are too restrictive for this work." For the reader, this statement becomes unclear given the lack of details in Theorem 1, or the authors not discussing when their Lojasiewicz types of inequality are available nor discussing either how restrictive the local richness assumption is.\
In Line 136-138, "To resolve this issue, and to simplify our treatment, this work utilizes the existence of an oracle for the computation of a Wasserstein gradient." The wording "to simplify our treatment" is doubtful: isn't it a necessity (which is perfectly fine but triggers doubt during reading)?
Besides, it is unclear how such an assumption does not limit the result of this work to the scope of the previous works, which were deemed not to be general enough. The authors could expand on that. It is always uncomfortable to read claims about generality with respect to previous works and then see the results with restrictive assumptions that are not discussed.

- First-order methods and Frank-Wolfe. It is relatively well-known that Frank-Wolfe methods have different rates of convergence than other first-order methods. However, in several places, the authors keep on referring to non-projection-free first-order methods, e.g., "we recover sublinear rates that are to be expected from finite-dimensional analogs [34]" (Line 27-28) or "A common approach to establishing improved iteration complexities for convex [...]" (Line 210-211). \
The reference to [34] is strange; they do not deal with Frank-Wolfe algorithms. The linear convergence rates of Frank-Wolfe are sometimes compared with that of other first-order methods, e.g., when the unconstrained optimum is in the interior of the constraint set. An in-depth discussion of this assumption in the context of (2) would be interesting. Several papers seem to suggest that the assumptions that lead to accelerated convergence results on Frank-Wolfe are relatively similar in finite-dimensional or infinite-dimensional settings or in normed spaces versus metric spaces. Here, the authors do not connect with these types of results.

- The title is too generic with respect to the content of the paper. It is also improper; it is far from being a canonical Frank-Wolfe algorithm. The current title is detrimental to previous and future works on the topic.

- The introduction brings some confusion as to what problem is being solved. It is only via the caption of Algorithm 1 that the reader understands that (3) that is being solved.

Minor comments.
- Line 9: "reasonable assumptions"? "mild"?

- Line 20-21: "problems in the form of (1) provide access to rich infinite dimensional structure that sidesteps brittle artifacts of finite dimensional formulations". Could you add references and details of specific examples of "brittle artifacts"?

- Line 36-37 "such works have failed to consider (2) within the context of a general variational method for (1)." the wording "have failed" is violent and unfair.

- Line 62-65, "In that work, smoothness of the objective in (2) is used to, qualitatively, argue that [..]. In contrast, we provide quantification of the level of robustness required to achieve this tractability and demonstrate [...]": here, there is no reference as to where the authors deal with that in the paper; also, the wording "quantitatively" and "demonstrate" are ambiguous: do the authors provide a theoretical result? Or rather provide numerical schemes to better chose the level of smoothness? In the absence of forward reference, it is hard to understand.

- Line 69: "(with quantitative bounds on complexity and convergence)": I would drop "quantitative".

- Line 97 "is a Polish spaces [57]": not defined.

- Line 107-108 "gradients must be defined in terms of a selections in an appropriation cotangent bundle space". Typo on selections. But most importantly, these important technical details are not sufficiently explained. The authors allow space for speculative discussions as in Line 209-220 or long sentences to explain common sense (e.g., Line 48-50).

- Line 107-108 "In particular, as $P_2(R)$ is curved under W (see Appendix A), gradients must be defined in terms of selections in an appropriate cotangent bundle". In Appendix A, the proof of the statement states in Line 602 "The final bullet [the space curvature results] is a direct restatement of the results in Section 8.4. in [1]". So, that would be nice in the main to add (along with the forward reference to the appendix) directly the reference to [1]. Otherwise, it provides the wrong impression that it is the authors' contribution (no one read the appendices).

- Line 109 "is essentially": I don't know what "essentially" refers to in that setting.

- Line 125 "the curious reader": a bit annoying when the reader is already annoyed by the lack of clarity.

- Line 156: "In finite dimensions, iterative, gradient-based ..." There exists infinite-dimensional work already, so why suggesting that this work provide a novel view on infinite dimensional settings?
Also, at first sight, the paper does not describe all gradient based methods but Frank-Wolfe methods, so why keeping it general to first order methods?

- Line 168-178: No mention of the analogous results in normed spaces and no mention of what structure to assume on $J$ to ensure the existence of Lojasiewicz inequality with a non-trivial parameter. This is however the main appeal of Lojasiewicz inequalities.

- Line 180: In algorithm, the second line in the for loop: I found it strange to ask to compute s (and actually, the instruction is even less clear since it asks to compute an inequality which makes no sense), why not just directly taking the norm of your approximate gradient as the stopping criterion?

- Line 180: The choice of $\delta$ in Algorithm 3 is confusing since it could refers to Equation (2) while the algorithm seek to solving Equation (3).

- Line 191-195: Please provide the reference as to where the proof is in the appendix.

- Line 209-220: I find it too speculative; this is the type of discussion that should be in the conclusion. Also, it is at odd to elaborate on a futur results when the algorithm itself has not been properly introduced. Besides, the elaboration concern techniques of acceleration for proximal methods, not conditional gradient methods (that are different, see e.g. corrective version of FW or [Comb2020]), so hardly understandable in that context.

Additional references.

[Bach2012] Francis Bach, Simon Lacoste-Julien, Guillaume Obozinski. On the Equivalence between Herding and Conditional Gradient Algorithms. ICML 2012.
	
[Bolte2007] Bolte, J.; Daniilidis, A. and Lewis, A. The {\L}ojasiewicz inequality for nonsmooth subanalytic functions with applications to subgradient dynamical systems. 2007.

[Bolte2016] J. Bolte, A. Blanchet. A family of functional inequalities: Lojasiewicz inequalities and displacement convex functions. 2016

[Bomze2020] I Bomze, F Rinaldi, D Zeffiro. Frank-Wolfe and friends: a journey into projection-free first-order optimization methods. 2020

[Comb2020] Combettes, C. and Pokutta, S. (2020). Boosting Frank-Wolfe
by chasing gradients. ICML.

[Denoyelle2019] Denoyelle, Q.; Duval, V.; Peyré, G. and Soubies, E. The sliding Frank-Wolfe algorithm and its application to super-resolution microscopy. 2019.

[Garber2013] Dan Garber, Elad Hazan. A linearly convergent conditional gradient algorithm with applications to online and stochastic optimization.

[Guelat1986] J. GuéLat; P. Marcotte. Some comments on Wolfe's away step.

[Hauer2017] D. Hauer, J. Mazon. Kurdyka-Lojasiewicz-Simon inequality for gradient flows in metric spaces. 2017.

[Ker18] T. Kerdreux, A. d'Aspremont, S. Pokutta. Restarting Frank-Wolfe: Faster Rates Under Hölderian Error Bounds. 

[Ker19] T. Kerdreux, A. d'Aspremont, S. Pokutta. Restarting Frank-Wolfe. AISTATS 2019.

[Lacoste2015] S. Lacoste-Julien, F. Lindsten, F. Bach. Sequential Kernel Herding: Frank-Wolfe Optimization for Particle Filtering. ICML 2015.

[Liuse2019] G. Luise, S. Salzo, M. Pontil and C. Ciliberto. Sinkhorn barycenters with free support via frank-wolfe methods. NeurIPS 2019.

[Mensch2019] Arthur Mensch, Mathieu Blondel, Gabriel Peyré. Geometric Losses for Distributional Learning. ICML 2020.

[Rinaldi2020] F. Rinaldi, D. Zeffiro. A unifying framework for the analysis of projection-free first-order methods under a sufficient slope condition.
limitations_and_societal_impact: yes.
review_title: null

Meta-review: I carefully read the interactions between the reviewers and the authors as initially there was some strong split between the reviewers. Some of the things have been clarified in the meantime, so that the overall assessment seems to be that this paper is a good paper. Certain issues regarding notions and review of related methods etc have been raised however. I would like the authors to give the reviews appropriate considerations when preparing their revision. The authors answers have been taking into consideration.
Decision: Accept (Poster)