The Numerics of GANs
In this paper, we analyze the numerics of common algorithms for training Generative Adversarial Networks (GANs). Using the formalism of smooth two-player games we analyze the associated gradient vector field of GAN training objectives. Our findings suggest that the convergence of current algorithms suffers due to two factors: i) presence of eigenvalues of the Jacobian of the gradient vector field with zero real-part, and ii) eigenvalues with big imaginary part. Using these findings, we design a new algorithm that overcomes some of these limitations and has better convergence properties. Experimentally, we demonstrate its superiority on training common GAN architectures and show convergence on GAN architectures that are known to be notoriously hard to train.

This paper presents a novel analysis of the typical optimization algorithm used in GANs (simultaneous gradient ascent) and identifies problematic failures when the Jacobian has large imaginary components or zero real components. Motivated by these failures, they present a novel consensus optimization algorithm for training GANs. The consensus optimization is validated on a toy MoG dataset as well as CIFAR-10 and CelebA in terms of sample quality and inception score.
I found this paper enjoyable to read and the results compelling. My primary concern is the lack of hyperparameter search when comparing optimization algorithms and lack of evidence that the problems identified with simultaneous gradient ascent are truly problems in practice.
Major concerns:
1. The constant discriminator/generator loss is very weird. Any ideas on what is going on? Does this hold when you swap in different divergences, e.g. WGAN?
2. Which losses were you using for the generator/discriminator? From Fig 4, it looks like you’re using the modified generator objective that deviates from the minimax game. Do your convergence results hold for non-zero sum games when f and g are different functions, i.e. f != -g?
3. Experiments: You note that simultaneous and alternating gradient descent may require tiny step sizes for convergence. In your experiments did you try tuning learning rates? Additional optimization hyperparameters? Without sweeping over learning rates it seems unfair to say that one optimization technique is better than another. If consensus optimization is more robust across learning rates then that should be demonsrated empirically.
4. Empirical validation of the failures of simultaneous gradient ascent: are zero real components and large imaginary parts really the problem in practice? You never evaluate this. I expected to see empirical validation in the form of plots showing the distribution of eigenvalues of the Jacobian over the course of optimization for simultaneous gradient ascent.
Minor concerns:
1. From the perspective of consensus optimization and \gamma being a Lagrange multiplier, I’d expect to see \gamma *adapt* over time. In particular, ADMM-style algorithms for consensus optimization have dynamics on \gamma as well. Have you explored these variants?
2. Many recent approaches to optimization in GANs have proposed penalties on gradients (typically the gradient of the discriminator w.r.t. Its inputs as in WGAN-GP). Are these related to the approach you propose, and if so is another explanation for the success of consensus optimization just that you’re encouraging a smooth discriminator? One way of testing this would be to remove the additional \nabla L(x) term for the generator update, but keep it for the discriminator update.
3. Figure 5 (table comparing architectures and optimizers) is a nice demonstration of the stability of Consensus relative to other approaches, but did you sweep over learning rate/optimization hyperparameters? If so, then I think this would make a strong figure to include in the main text.
4. I had a hard time following the convergence theory section and understanding how it relates to the results we see in practice.


Fast Rates for Bandit Optimization with Upper-Confidence Frank-Wolfe
We consider the problem of bandit optimization, inspired by stochastic optimization and online learning problems with bandit feedback. In this problem, the objective is to minimize a global loss function of all the actions, not necessarily a cumulative loss. This framework allows us to study a very general class of problems, with applications in statistics, machine learning, and other fields. To solve this problem, we analyze the Upper-Confidence Frank-Wolfe algorithm, inspired by techniques for bandits and convex optimization. We give theoretical guarantees for the performance of this algorithm over various classes of functions, and discuss the optimality of these results.

This is a very interesting paper at the intersection of bandit problems and stochastic optimization. The authors consider the setup where at each time step a decision maker takes an action and gets as a feedback a gradient estimate at the chosen point. The gradient estimate is noisy  but the assumption is that we have control over how noisy the gradient estimate is.  The challenge for the decision maker is to make a sequence of moves such that the average of all iterates has low error  compared to the optima. So the goal is similar to the goal in stochastic optimization, but unlike stochastic optimization (but like bandit problems) one gets to see limited information about the loss function.
For the above problem setting the authors introduce an algorithm that combines elements of UCB along with Frank-Wolfe updates. The FW style updates means that we do not play the arm corresponding to the highest UCB, but instead play an arm that is a convex combination of the UCB arm and a probability distribution that comes from historical updates. The authors establish stochastic optimization style bounds for a variety of function classes.  The paper is failry well written and makes novel contributions to both stochastic opt and bandit literature. I have a few comments
1. It is not clear to me why the authors consider only Frank-Wolfe algorithm. Is there a deep motivation here or can the arguments made in this paper be translated to even work with other optimization algorithms such as stochastic gradient descent etc...?
2. In lines 214-219, it is not clear to me why one needs to invoke KKT condition. It was not clear to me what the relevance and importance of this paragraph is. Can the authors shine some light here?
3. Definition of strong convexity in line 230 is incorrect.
4. It would perhaps be useful to moved the lines 246-249 to the start of the paper. In this way it will be more clear why and how the problem considered here is different from standard stochastic optimization algorithms.
5. In line 257 what is \sigma_max?


Cost efficient gradient boosting
Many applications require learning classifiers or regressors that are both accurate and cheap to evaluate. Prediction cost can be drastically reduced if the learned predictor is constructed such that on the majority of the inputs, it uses cheap features and fast evaluations. The main challenge is to do so with little loss in accuracy. In this work we propose a budget-aware strategy based on deep boosted regression trees. In contrast to previous approaches to learning with cost penalties, our method can grow very deep trees that on average are nonetheless cheap to compute. We evaluate our method on a number of datasets and find that it outperforms the current state of the art by a large margin. Our algorithm is easy to implement and its learning time is comparable to that of the original gradient boosting. Source code is made available at http://github.com/svenpeter42/LightGBM-CEGB.

1. Summary of paper
This paper introduced a technique to learn a gradient boosted regression tree ensemble that is sensitive to feature costs and the cost of evaluating splits in the tree. Thus, the paper is similar to the work of Xu et al., 2012. The main differences are the fact that the feature and evaluation costs are input-specific, the evaluation cost depends on the number of tree splits, their optimization approach is different (based on the Taylor expansion around T_{k-1}, as described in the XGBoost paper), and they use best-first growth to grow the trees to a maximum number of splits (instead of a max depth). The authors point out that their setup works either in the case where feature cost dominates or evaluation cost dominates and they show experimental results for these settings.
2. High level subjective
The paper is clearly written. Apart from introducing a significant amount of notation, I find it clear to follow.
The contribution of the paper seems a bit small given the work of XGBoost and GreedyMiser.
The paper seems to have sufficient experiments comparing the method with prior work and showing how model parameters affect performance. I have one confusion which may warrant an additional experiment, which I describe below.
I think this model could be interesting for practitioners interested in classification in time-constrained settings.
3. High level technical
One thing that is unclear is in Figures 2a and 2b, how was the cost measured for each method to obtain the Precision versus Cost curves? It seems to me that for CEGB cost is measured per-input and per-split, but for the other methods cost is measured per-tree. If this is the case this seems a bit unfair. It seems to me that the cost of each method should be evaluated in exactly the same way in order to compare each method fairly, even if the original papers evaluated cost in a different way. For this reason it's unclear to me how much better CEGB is than the other methods. If the costs are measured differently I think it would be really helpful to modify Figure 2 so that all of the costs are measured the same way.
Personally I think the paper would improve if Figure 1 was removed. I think it's pretty clear what you mean by breadth-first vs. depth-first vs. best-first. I would replace this figure with one that shows the trees generated by CEGB, GreedyMiser and BudgetPrune for the Yahoo! Learning to Rank dataset. And it shows in detail the features and the costs at each split. Something that gives the reader a bit more insight as to why CEGB is outperforming the other methods.
I think it would also be nice to more clearly emphasize what improvements over GreedyMiser you make to arrive at CEGB. For instance, it would be nice to write out GreedyMiser in the same notation and show the innovations (best-first, not limited-depth, different optimization procedure) that lead to CEGB. Then you could have a figure showing how each innovation improves the Precision vs. Cost curve.
In Section 5.2 it's not completely clear to me why GreedyMiser or BudgetPrune cannot be applied here. Why is the cost of feature responses considered an evaluation cost and not a feature cost?
4. Low level technical
- In equation 7, why is \lambda only in front of the first cost term and not the second?
- In equation 12, should the denominators of the first 3 terms have a term with \lambda added, similar to the XGBoost paper?
- What do the 'levels' refer to in Figure 4b?
- Line 205: I would recommend redefining \beta_m here as readers may forget its definition. Similarly with \alpha on Line 213.
5. Summary of review
While this paper aims to introduce a new method for cost efficient learning that (a) outperforms the state-of-the-art and (b) works in settings where previous methods are unsuitable, it is unclear to me if these results are because (a) the cost is measured differently and (b) because a cost is defined as an evaluation cost instead of a feature cost. Until these confusions are resolved I think the paper is slightly below the acceptance threshold of NIPS.


State Aware Imitation Learning
Imitation learning is the study of learning how to act given a set of demonstrations provided by a human expert. It is intuitively apparent that learning to take optimal actions is a simpler undertaking in situations that are similar to the ones shown by the teacher. However, imitation learning approaches do not tend to use this insight directly. In this paper, we introduce State Aware Imitation Learning (SAIL), an imitation learning algorithm that allows an agent to learn how to remain in states where it can confidently take the correct action and how to recover if it is lead astray. Key to this algorithm is a gradient learned using a temporal difference update rule which leads the agent to prefer states similar to the demonstrated states. We show that estimating a linear approximation of this gradient yields similar theoretical guarantees to online temporal difference learning approaches and empirically show that SAIL can effectively be used for imitation learning in continuous domains with non-linear function approximators used for both the policy representation and the gradient estimate.

I've read the feedback, thanks for the clarifications.
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This paper deals with imitation learning. It is proposed to maximize the posterior (policy parameters given the expert state-action pairs). The difference with a classic supervised learning approach is that here the fact that the policy acts on a dynamical system is taken into account, which adds the likelihood of states given policy (the states should be sampled according to the stationary distribution induced by the policy). This additional term is addressed through a temporal difference approach inspired by the work of Morimusa et al. [12], other terms are handled in the classic way (as in supervised learning). The proposed approach is experimented on two domains, a small on and a much larger one, and compared mainly to the supervised baseline (ignoring the dynamics).
This paper proposes an interesting, sound and well motivated contribution to imitation learning. I found the paper rather clear, but I think it could be clarified for readers less familiar with the domain. For example:
* in Eq. 3, the reverse probability appears. It would help to at least mention it in the text (it is not that classical)
* the constraint (l.168, expected gradient null) comes simply from the fact that the probabilities should sum to one. It would help a lot to say it (instead of just giving the constraint)
* the function f_w is multi-output (as many components as policy parameters), it could help to highlight this in the text
* the off-policy approach should be made explicit (notably, as it is a reversed TD approach, due to the reversed probabilities behind, do the importance weights apply to the same terms?)
Section 2 is rather complete, but some papers are missing. There is a line of work that tries to do imitation as a supervised approach while taking into account the dynamics (which is also what is done in this paper, the dynamics inducing a stationary distribution for each policy), for example :
* "Learning from Demonstration Using MDP Induced Metrics" by Melo and Lopes (ECML 2010)
* "Boosted and Reward-regularized Classification for Apprenticeship Learning" by Piot et al. (AAMAS 2014)
The second one could be easily considered as a baseline (it is a supervised approach whit an additional regularization term taking into account the dynamics, so the used supervised approach in the paper could be easily adapted)
Regarding the experiments:
* it would have been interesting to vary the number of episodes collected from the expert
* for the off-policy experiment, it would be good to compare also to (on-policy) SAIL, as a number of gathered trajectories
* for the walker, giving the number of input/output neurons would also help (28 in/4 out for the policy, 28 in/3200 out for the gradient estimator? so a 28:80:80:3200 network?) This makes big output networks, it could be discussed.
* still for the walker, what about off-policy learning?
Regarding appendix A, the gradient of the log-policy plays the role of the reward. Usually, rewards are assumed to be uniformly bounded, which may not be the case here (eg, with a linear softmax policy based on a tabular representation, the probabilities could be arbitrary close to zero, even if ever strictly positive). Isn't this a problem for the proof?


Sparse Embedded k-Means Clustering
The k-means clustering algorithm is a ubiquitous tool in data mining and machine learning that shows promising performance. However, its high computational cost has hindered its applications in broad domains. Researchers have successfully addressed these obstacles with dimensionality reduction methods. Recently, [1] develop a state-of-the-art random projection (RP) method for faster k-means clustering. Their method delivers many improvements over other dimensionality reduction methods. For example, compared to the advanced singular value decomposition based feature extraction approach, [1] reduce the running time by a factor of min{n, d} 2 log(d)/k for data matrix X ∈ R n×d with n data points and d features, while losing only a factor of one in approximation accuracy. Unfortunately, they still require O( ndk 2 log(d) ) for matrix multiplication and this cost will be prohibitive for large values of n and d. To break this bottleneck, we carefully build a sparse embedded k-means clustering algorithm which requires O(nnz(X)) (nnz(X) denotes the number of non-zeros in X) for fast matrix multiplication. Moreover, our proposed algorithm improves on [1]'s results for approximation accuracy by a factor of one. Our empirical studies corroborate our theoretical findings, and demonstrate that our approach is able to significantly accelerate k-means clustering, while achieving satisfactory clustering performance.

The authors proposed a sparse embedded k-means clustering algorithm to improve the running time of matrix multiplication of current proposed randomization projection method under sparse setting of data matrix in the literature. In particular, they demonstrated that their algorithms achieve the calculation time of matrix multiplication of order proportional to the number of non-zeroes entries of data matrix.
I think the sparse embedded k-means clustering algorithm is rather interesting; however, it is not a very surprising improvement given the current results from the paper of C. Boutsidis et al. (2015). More specifically, both papers try to approximate low dimension solution for the original solution of K-means problem. To do that, C. Boutsidis et al. (2015) proposed to multiply the data matrix X with a random matrix having entries $1/\sqrt{d'}$ or $-1/\sqrt{d'}$ where $d'$ is the dimension of the approximation solution. However, their proposed solution does not work work well with sparse setting of data matrix as the calculation of matrix multiplication from their method greatly depends on the number of data points and the number of clusters. To account for that drawback, the authors of current paper proposed to multiply the data matrix with sparse random matrix inspired from diagonal Rademacher matrix. At the high level, it is unsurprisingly will improve the computation of matrix multiplication from C. Boutsidis et al.'s work under sparse setting of data matrix. At the setting of rather dense data matrix, I think both methods may differ slightly in terms of accuracy and matrix multiplication time. In summary, even though the paper is rather interesting, I think the contribution of the paper is not very surprising.
Some minor comments:
(1) Will the condition in Definition 2 hold for any matrix $D$?
(2) It may be better for readers if the authors replace notation $\widehat{D}$ in Lemma 1 with less confusing notation. At the moments, the notations in Lemma 1 look rather similar.
(3) I do not see the clustering accuracy with Random projection method from C. Boutsidis et al.'s work with RCV1 data in the experiment section. I wonder what may happen with that accuracy result?


Streaming Robust Submodular Maximization: A Partitioned Thresholding Approach
We study the classical problem of maximizing a monotone submodular function subject to a cardinality constraint k, with two additional twists: (i) elements arrive in a streaming fashion, and (ii) m items from the algorithm's memory are removed after the stream is finished. We develop a robust submodular algorithm STAR-T. It is based on a novel partitioning structure and an exponentially decreasing thresholding rule. STAR-T makes one pass over the data and retains a short but robust summary. We show that after the removal of any m elements from the obtained summary, a simple greedy algorithm STAR-T-GREEDY that runs on the remaining elements achieves a constant-factor approximation guarantee. In two different data summarization tasks, we demonstrate that it matches or outperforms existing greedy and streaming methods, even if they are allowed the benefit of knowing the removed subset in advance.

This paper studies the problem of submodular maximization in a streaming and robust setting. In this setting, the goal is to find k elements of high value among a small number of elements retained during the stream, where m items might be removed after the stream. Streaming submodular maximization has been extensively studied due to the large scale applications where only a small number of passes over the data is reasonable. Robustness has also attracted some attention recently and is motivated by scenarios where some elements might get removed and the goal is to obtain solutions that are robust to such removals. This paper is the first to consider jointly the problem of streaming and robust submodular maximization.
The authors develop the STAR-T algorithm, which maintains O((m log k + k )log^2 k) elements after the stream and achieves a constant (0.149) approximation guarantee while being robust to the removal of m elements. This algorithm combines partitioning and thresholding techniques for submodular optimization which uses buckets. Similar ideas using buckets were used in previous work for robust submodular optimization, but here the buckets are filled in a different fashion. It is then shown experimentally that this algorithm performs well in influence maximization and movie recommendation applications, even against streaming algorithms which know beforehand which elements are removed.
With both the streaming and robustness aspects, the problem setting is challenging and obtaining a constant factor approximation is a strong result. However, there are several aspects of the paper that could be improved. It would have been nice if there were some non-trivial lower bounds to complement the upper bound. The writing of the analysis could also be improved, the authors mention an “innovative analysis” but do not expand on what is innovative about it and it is difficult to extract the main ideas and steps of the analysis from the proof sketch. Finally, the influence maximization application could also be more convincing. For the experiments, the authors use a very simplified objective for influence maximization which only consists of the size of the neighborhood of the seed nodes. It is not clear how traditional influence maximization objectives where the value of the function depends on the entire network for any set of seed nodes could fit in a streaming setting.


Convergence Analysis of Two-layer Neural Networks with ReLU Activation
In recent years, stochastic gradient descent (SGD) based techniques has become the standard tools for training neural networks. However, formal theoretical understanding of why SGD can train neural networks in practice is largely missing. In this paper, we make progress on understanding this mystery by providing a convergence analysis for SGD on a rich subset of two-layer feedforward networks with ReLU activations. This subset is characterized by a special structure called "identity mapping". We prove that, if input follows from Gaussian distribution, with standard O(1/ √ d) initialization of the weights, SGD converges to the global minimum in polynomial number of steps. Unlike normal vanilla networks, the "identity mapping" makes our network asymmetric and thus the global minimum is unique. To complement our theory, we are also able to show experimentally that multi-layer networks with this mapping have better performance compared with normal vanilla networks. Our convergence theorem differs from traditional non-convex optimization techniques. We show that SGD converges to optimal in "two phases": In phase I, the gradient points to the wrong direction, however, a potential function g gradually decreases. Then in phase II, SGD enters a nice one point convex region and converges. We also show that the identity mapping is necessary for convergence, as it moves the initial point to a better place for optimization. Experiment verifies our claims.

This paper proves the convergence of a stochastic gradient descent algorithm from a suitable starting point to the global minimizer of a nonconvex energy representing the loss of a two-layer feedforward network with rectified linear unit activation. In particular, the algorithm is shown to converge in two phases, where phase 1 drives the iterates into a one-point convex region which subsequently leads to the actual convergence in phase 2.
The findings, the analysis, and particularly the methodology for proving the convergence (in 2 phases) are very interesting and definitely deserve to be published.
The entire proof is extremely long (including a flowchart of 15 Lemmas/Theorems that finally allow to show the main theorem in 25 pages of proofs), and I have to admit that I did not check this part.
I have some questions on the paper and suggestions to further improve the manuscript:
- Figure 1 points out the different structure of the considered networks, and even the abstract already refers to the special structure of "identity mappings". However, optimizing for W (vanilla network) is equivalent to optimizing for (W+I), such that the difference seems to lie in different starting points only. If this is correct, I strongly suggest to emphasize the starting point rather than the identity mapping. In this case, please shorten the connection to ResNet.
- There is a fundamental difference between the architecture shown in figure 1 (which is analyzed in the proofs), and the experiments in sections 5.1, 5.4, and 5.5. In particular, the ResNet trick of an identity shortcut behaves fundamentally different if the identity is used after a nonlinearity (e.g. BatchNorm in Figure 6 and ReLu in the ResNet Paper), opposed to an equivalent reformulation of the original network in Figure 1. Moreover, sections 5.1, 5.4, and 5.5 consider deep networks that have not been analyzed theoretically. Since your theoretical results deal with the 2-layer architecture shown in Figure 1, I suggest to reduce the numerical experiments that go well beyond the theoretically studied case, and instead explain the latter in more detail. It certainly contains enough material. In particular, it would be interesting to understand how close real-world data is to satisfying the assumptions of Theorem 3.1.
- For a paper of the technical precision such as the one presented here, I think one should not talk about taking the derivative of a ReLU function without a comment. The ReLU is not differentiable whenever any component is zero - so what exactly is \nabla L(W)? In a convex setting one would consider a subgradient, but what is the right object here? A weak derivative?
- Related to the previous question, subgradient descent on a convex function already requires the step size to go to zero in order to state convergence results. Can you comment on why you can get away with constant step sizes?
- The plots in Figures 7 and 8 are difficult to read and would benefit from more detailed explanations, and labeled axes.
- I recommend to carefully check the manuscript with respect to third person singular verbs and plural nouns. Also, some letters are accidently capitalized after a comma.


Toward Goal-Driven Neural Network Models for the Rodent Whisker-Trigeminal System
In large part, rodents "see" the world through their whiskers, a powerful tactile sense enabled by a series of brain areas that form the whisker-trigeminal system. Raw sensory data arrives in the form of mechanical input to the exquisitely sensitive, actively-controllable whisker array, and is processed through a sequence of neural circuits, eventually arriving in cortical regions that communicate with decisionmaking and memory areas. Although a long history of experimental studies has characterized many aspects of these processing stages, the computational operations of the whisker-trigeminal system remain largely unknown. In the present work, we take a goal-driven deep neural network (DNN) approach to modeling these computations. First, we construct a biophysically-realistic model of the rat whisker array. We then generate a large dataset of whisker sweeps across a wide variety of 3D objects in highly-varying poses, angles, and speeds. Next, we train DNNs from several distinct architectural families to solve a shape recognition task in this dataset. Each architectural family represents a structurally-distinct hypothesis for processing in the whisker-trigeminal system, corresponding to different ways in which spatial and temporal information can be integrated. We find that most networks perform poorly on the challenging shape recognition task, but that specific architectures from several families can achieve reasonable performance levels. Finally, we show that Representational Dissimilarity Matrices (RDMs), a tool for comparing population codes between neural systems, can separate these higherperforming networks with data of a type that could plausibly be collected in a neurophysiological or imaging experiment. Our results are a proof-of-concept that DNN models of the whisker-trigeminal system are potentially within reach.

* Summary
  The authors of the paper "Toward Goal-Driven Neural Network Models
  for the Rodent Whisker-Trigeminal System" train networks for object
  detection on simulated whisker data. They compare several
  architectures of DNNs and RNNs on that task and report on elements
  that are crucial to get good performance. Finally they use a
  representational dissimilarity matrix analysis to distinguish
  different high performing network architectures, reasoning that this
  kind of analysis could also be used on neural data.
* Comments
  What I like most about the paper is direction that it is
  taking. Visual datasets and deep neural networks have shown that
  there are interesting links between artificial networks and
  neuroscience that are worthwhile to explore. Carrying these
  approaches to other modalities is a laudable effort, in particular
  as interesting benchmarks often have advanced the field
  (e.g. imagenet). The paper is generally well written, and the main
  concepts are intuitively clear. More details are provided in the
  supplementary material, and the authors promise to share the code
  and the dataset with the community.
  I cannot judge the physics behind simulating whiskers, but the
  authors seem to have taken great care to come up with a realistic
  and simulation, and verify that the data actually contains
  information about the stimulus. I think a bit more nuisance
  variables like slight airflow, as well as active whisking bouts
  would definitely make the dataset even more interesting, but I can
  see why it is a good idea to start with the "static" case first.
  With regard to benchmarking, the authors made an effort to evaluate
  a variety of different networks and find interesting insights into
  which model components seem to matter. I would have wished for more
  details on how the networks are setup and trained. For example, how
  many hyperparameters were tried, what was the training schedule, was
  any regularization used, etc? The code will certainly contain these
  details, but a summary would have been good.
  The part about the representation dissimilarity matrix analysis is a
  bit short and it is not clear to me what can be learned from it. I
  guess the authors want to demonstrate that there are detectable
  differences in this analysis that could be used to compare neural
  performance to those networks. The real test for this analysis is
  neural data, of course, which the paper does not cover.
  In general, I see the main contribution of the paper to be the
  dataset and the benchmarking analysis on it. Both together are a
  useful contribution to advance the field of neuroscience and machine
  learning.


Certified Defenses for Data Poisoning Attacks
Machine learning systems trained on user-provided data are susceptible to data poisoning attacks, whereby malicious users inject false training data with the aim of corrupting the learned model. While recent work has proposed a number of attacks and defenses, little is understood about the worst-case loss of a defense in the face of a determined attacker. We address this by constructing approximate upper bounds on the loss across a broad family of attacks, for defenders that first perform outlier removal followed by empirical risk minimization. Our approximation relies on two assumptions: (1) that the dataset is large enough for statistical concentration between train and test error to hold, and (2) that outliers within the clean (nonpoisoned) data do not have a strong effect on the model. Our bound comes paired with a candidate attack that often nearly matches the upper bound, giving us a powerful tool for quickly assessing defenses on a given dataset. Empirically, we find that even under a simple defense, the MNIST-1-7 and Dogfish datasets are resilient to attack, while in contrast the IMDB sentiment dataset can be driven from 12% to 23% test error by adding only 3% poisoned data.

The paper studies an interesting topic and is overall nice to read. However, I have doubts about the correctness of the main theorem (Proposition 1), which makes two statements: one that bounds the max-min loss M from below and above, and a second that further bounds the upper bound. While the second claim is proven in the appendix; the first is not. The upper bound follows from the convexity of U and Equation 5. (It might be helpful for the reader to explicitly state this finding and argue why D_p reduces to a set of \epsilon n identical points.) However, I can't see why the lower bound holds. Computing M requires the solution of a saddle point problem. If one knew one part of this saddle point solution, one could easily bound M from above and/or below. However, Algorithm 1 only generates interim solutions that eventually converge to the saddle point. Concretely, in (6), D_p is not (necessarily) part of the saddle point solution; and \hat theta is neither the minimizer of L w.r.t. D_c \cup D_p nor part of the saddle point solution. So why is this necessarily a lower bound?
It's hard to judge the quality of candidate set D_p. One could argue that it would be better to actually run the algorithm till convergence and take the final data pair (x, y) \epsilon n times. Alternatively, one could take the last \epsilon n data points before the algorithm converges. Why the first \epsilon n update steps give a better candidate set is unclear to me. Whether the resulting model after \epsilon n update steps is any close to an optimum is unclear to me, too. Overall, there is little theoretical justification for the generation of the candidate attack in Alg. 1; it is for sure not the solution of (4). Hence, it's hard to conclude something like "even under a simple defense, the XYZ datasets are resilient to attack". And since the attack is model-based - it is constructed with a certain model (in this case, a linear model), loss function, and regularization scheme in mind - it is also hard to conclude "the ABC dataset can be driven from 12% to 23% test error"; this might be true for the concretely chosen model, but a different model might be more robust to this kind of poising attack.
To me the value of the approach is that it allows to study the sensitivity of a concrete classifier (e.g., SVM) w.r.t. a rather broad set of attacks. In case of such sensitivity (for a concrete dataset), one could think about choosing a different defense model and/or classifier. However, in the absence of such sensitivity one should NOT conclude the robustness of the model.
In Section 3.2, the defenses are defined in terms of class means rather than centroids (as in (2)).
Eq. (8) requires F(\pi), though, F is defined over a set of data points. I don't fully understand this maximization term and I would like the authors to provide more details how to generalize Alg. 1 to this setting.
In the experimentation section, some details are missing to reproduce the results; for example, the algorithm has parameters \roh and \eta, the defense strategies have thresholds. It is neither clear to me how those parameters are chosen, nor which parameters were used in the experiments.
After authors response:
Regarding the correctness of correctness of Prop. 1: \hat {theta} is the minimizer of L w.r.t. D_c \cup D_p; but D_c \cup D_p is not simultaneously a maximizer of L. M states the value of L at a saddle point, but neither \hat{theta} nor D_c \cup D_p is necessarily part of a saddle point. However, it became clear to me that this is not required. We can replace
max_D min_\theta L(D, \theta)
by
max_D L(D, \hat
{\theta}_D)
where \hat{theta}_D is the minimizer w.r.t. D. Of course, this quantity is always smaller than M := L(\hat{D}, \hat{\theta}_\hat{D}) for any maximizer \hat{D} and saddle point (\hat{D}, \hat{\theta}_\hat{D}). I got confused from the two meanings of D_p, which is used as a free variable in the maximization in (4) and as the result of Alg. 1.


Higher-Order Total Variation Classes on Grids: Minimax Theory and Trend Filtering Methods
We consider the problem of estimating the values of a function over n nodes of a d-dimensional grid graph (having equal side lengths n 1/d ) from noisy observations. The function is assumed to be smooth, but is allowed to exhibit different amounts of smoothness at different regions in the grid. Such heterogeneity eludes classical measures of smoothness from nonparametric statistics, such as Holder smoothness. Meanwhile, total variation (TV) smoothness classes allow for heterogeneity, but are restrictive in another sense: only constant functions count as perfectly smooth (achieve zero TV). To move past this, we define two new higher-order TV classes, based on two ways of compiling the discrete derivatives of a parameter across the nodes. We relate these two new classes to Holder classes, and derive lower bounds on their minimax errors. We also analyze two naturally associated trend filtering methods; when d = 2, each is seen to be rate optimal over the appropriate class.

This paper considers graph-structured signal denoising  problem. The particular structure enforced involves a total variation type penalty involving higher order discrete derivatives. Optimal rates for penalized least-squares estimator is established and minimax lower bound is established. For the grid filtering case the upper bounds are established under an assumed conjecture.
The paper is well written. The main proof techniques of the paper are borrowed from previous work but there are some additional technicalities required to establish the current results. I went over the proofs  (not 100% though) and it seems there is no significant issues with it. Overall I find the results of the paper as interesting contributions to trend filtering literature and I recommend to accept the paper.
Minor: it is not clear why lines 86-87 are there in the supplementary material.


How regularization affects the critical points in linear networks
This paper is concerned with the problem of representing and learning a linear transformation using a linear neural network. In recent years, there is a growing interest in the study of such networks, in part due to the successes of deep learning. The main question of this body of research (and also of our paper) is related to the existence and optimality properties of the critical points of the mean-squared loss function. An additional primary concern of our paper pertains to the robustness of these critical points in the face of (a small amount of) regularization. An optimal control model is introduced for this purpose and a learning algorithm (backprop with weight decay) derived for the same using the Hamilton's formulation of optimal control. The formulation is used to provide a complete characterization of the critical points in terms of the solutions of a nonlinear matrix-valued equation, referred to as the characteristic equation. Analytical and numerical tools from bifurcation theory are used to compute the critical points via the solutions of the characteristic equation.

The authors of this article consider an infinite-dimensional version of a deep linear network. They propose to learn the weights using an objective function that is the sum of a square loss and a Tikhonov regularizer, controlled by a regularization parameter lambda. The authors derive analytic characterizations for the critical points of the objective, and use them to numerically show that, even for arbitrarily small lambda, local minima and saddle points can appear, that do not exist when there is no regularization.
I am not an expert of this subject, but it seems to me that looking at deep linear networks in an infinite-dimensional setting, instead of the usual finite-dimensional one, is a good idea. It seems to make computations simpler or at least more elegant, and could then allow for more in-depth analysis.
Using methods from optimal control to study this problem is original, and apparently efficient.
I also found the article well-written: it was pleasant to read, despite the fact that I am very unfamiliar with optimal control.
However, I am not sure that the picture that the authors give of the critical points in the limit where lambda goes to zero is clear enough so that I understand what is surprising in it, and what algorithmic implications it can have.
Indeed, when lambda is exactly zero, there are a lot of global minima. If these global minima can be grouped into distinct connected components, then, when lambda becomes positive, it is "normal" that local minima appear (intuitively, each connected component will typically give rise to at least one local minimum, but only one connected component - the one containing the global solution that minimizes the regularizer - will give rise to a local minimum that is also global). It seems to me that this is what the authors observe, in terms of local minima.
The appearing of saddle points is maybe more surprising, but since they seem to come in "from infinity", they might not be an issue for optimization, at least for small values of lambda.
Also, the authors focus on normal solutions. They are easier to compute than non-normal ones, but they are a priori not the only critical points (at least not when lambda=0). So maybe looking only at the normal critical points yields a very different "critical point diagram" than looking at all the critical points. If not, can the authors explain why?
Finally, this is less important but, after all the formal computations (up to Theorem 2), I was slightly disappointed that the appearing of local minima and saddle points was only established for precise examples, and through numerical computations, and not via more rigorous arguments.
Minor remarks:
- It seems to me that the regularization proposed by the authors is essentially "weight decay", that is well-known among the people that train deep networks; it should be said.
- Proposition 1: it is to me that it is not explicitly said that it only holds for positive values of lambda.
- The covariance is defined as \Sigma_0 in the introduction but subsequently denoted by \Sigma.
- l.129: it is said that the "critical points" will be characterized, but the proof uses Proposition 1, that deals with "minimizers", and not "critical points".
- l. 141: "T" is missing in the characteristic equation.
- Theorem 2 - (i): plural should probably be used, since there are several solutions.
- Proof of theorem 2: the proof sketch seems to suggest that the authors assume with no proof that there always exists a normal solution when lambda > 0. The appendix actually proves the existence, but maybe the sketch could be slightly reformulated, so as not to give a wrong impression?
- It seems to me that the authors never show that values of C that satisfy Equation (18) are critical points of (4) (the proposition says the opposite: critical points of (4) satisfy (18)). So how can they be sure that the solutions they compute by analytic continuation are indeed critical points of (4)?
- I am not sure I understand the meaning of "robust" in the conclusion.
Added after author feedback: thank you for the additional explanations. I am however slightly disappointed with your response to my main two comments.
I agree that the emergence of saddle points, as well as the normal / non-normal difference are somewhat surprising. However, it seems to me that it is not really what the article presents as a surprise: if you think that these two points are the most surprising contribution of the article, it should probably be better highlighted and discussed.
For the normal vs non-normal solutions, I do understand that studying non-normal critical points looks very difficult, and I am fine with it. However, I think that the abstract and introduction should then be clear about it, and not talk about general critical points, while the article only describes normal ones.


Spherical convolutions and their application in molecular modelling
Convolutional neural networks are increasingly used outside the domain of image analysis, in particular in various areas of the natural sciences concerned with spatial data. Such networks often work out-of-the box, and in some cases entire model architectures from image analysis can be carried over to other problem domains almost unaltered. Unfortunately, this convenience does not trivially extend to data in non-euclidean spaces, such as spherical data. In this paper, we introduce two strategies for conducting convolutions on the sphere, using either a spherical-polar grid or a grid based on the cubed-sphere representation. We investigate the challenges that arise in this setting, and extend our discussion to include scenarios of spherical volumes, with several strategies for parameterizing the radial dimension. As a proof of concept, we conclude with an assessment of the performance of spherical convolutions in the context of molecular modelling, by considering structural environments within proteins. We show that the models are capable of learning non-trivial functions in these molecular environments, and that our spherical convolutions generally outperform standard 3D convolutions in this setting. In particular, despite the lack of any domain specific feature-engineering, we demonstrate performance comparable to state-of-the-art methods in the field, which build on decades of domain-specific knowledge.

Paper summary:
The authors propose to separate the surface of a sphere using the equiangular cubed sphere representation of Ronchi et al.  (1996) in order to do a spherical convolution. This convolution has the advantage of being almost invariant to rotation of the volume and should have application in earth science, astronomy and in molecular modeling. The authors demonstrate the usefulness of their approach on practical tasks of this later field.

Strengths and weaknesses:
The paper is clearly written and easy to read. Also, I appreciate the amount of work that has been put in this paper as molecular modeling is a field with a steep learning curve, where the data is hard to pre-process, and where clear prediction task is sometime hard to define.

Although I appreciate the importance of spherical convolution on spherical data and the comparison with a purely dense model, there are other comparisons I believe would be of interest. Namely, I would like to know if empirical comparison with 3D convolution (ex: https://keras.io/layers/convolutional/#conv3d) in a cube is possible. For example, in the case of molecular modeling, the cube could be centered on the atom of interest. Also, it would also be of interest to compare the spherical convolution using different coordinate system: the equiangular cubed sphere (as proposed, Figure 1b) and the standard equiangular spacing (Figure 1a).

Finally, it would be nice if the conclusion contained ways to improve the proposed approach.

Quality:
I found the paper well written and easy to understand. I found few minor typos that are highlighted at the end of this review.

Clarity:
The figures are well done and help to understand the concepts that are important to the comprehension of the paper. I liked that the contribution of the authors is well-defined and that the authors didn’t try to oversell their ideas.

The authors should define the Q20 prediction score at line 229.

On line 245, the author state "Although there are a number of issues with our approach …". I appreciate that the authors were honest about it but I would like them to elaborate on the said issues.

Originality:
I found the paper interesting and refreshing. I liked that the authors have identified a niche problem and that they addressed it simply.

Significance:
The authors were honest about their work being preliminary and still at an early stage, to which I agree. However, I believe that their work could have a significant impact in domains where data are in a spherical format or when 3D data is acquired from a fixed point sensor.

I would like to emphasize that the impact of this paper will be greatly reduced if the author do not make the source code for the spherical convolution available online. In addition, having the source code to reproduce the results presented in the paper is good practice. For these reasons I strongly encourage the authors to make their source code available if the paper is accepted.

Errors / typos:
Line 68: "... based on the on the cubed …"
Line 101: "... we can make the make the …"
Line 117: The sentence seems unfinished or has a missing piece.


Deep Reinforcement Learning from Human Preferences
For sophisticated reinforcement learning (RL) systems to interact usefully with real-world environments, we need to communicate complex goals to these systems. In this work, we explore goals defined in terms of (non-expert) human preferences between pairs of trajectory segments. We show that this approach can effectively solve complex RL tasks without access to the reward function, including Atari games and simulated robot locomotion, while providing feedback on less than 1% of our agent's interactions with the environment. This reduces the cost of human oversight far enough that it can be practically applied to state-of-the-art RL systems. To demonstrate the flexibility of our approach, we show that we can successfully train complex novel behaviors with about an hour of human time. These behaviors and environments are considerably more complex than any which have been previously learned from human feedback.

Overall I find this paper is generally interesting, clearly presented, and technically sound. My concerns are that the contributions of this paper seems rather incremental when compared to previous work. Also, some of the experiments would benefit from further analysis. Let me elaborate below:
Summary: This paper advocates a preference-based approach for teaching an RL agent to perform a task. A human observes two trajectories generated by different policies and indicates which one is better performing the desired task. Using these indications, the agent infers a reward signal that is hopefully consistent with the preferences of the human, and then uses RL to optimize a policy that maximizes the inferred rewards.
My main hesitation for accepting this paper is that the method presented is not sufficiently different from previous work on preference-based RL (such as Wirth '16). The authors acknowledge that the main contribution of this work is to scale up preference-based RL to more complex domains by using deep reinforcement learning. I am less enthusiastic about this paper if the main algorithmic contribution is just using a NN to approximate the policy and extending to Mujoco/Atari.
Finally, I am concerned with the fact that synthetic queries are outperforming RL using the true reward function. It seems to me that the synthetic query examples should do strictly worse than having the actual reward function as they are just eliciting preferences based on sub-trajectories of the real rewards. The fact that they are in some cases solidly outperforming standard RL is highly unexpected and, in my opinion, deserves a more rigorous analysis. I don't find the authors explanation of the learned reward function being smoother a satisfying one.


On the Consistency of Quick Shift
Quick Shift is a popular mode-seeking and clustering algorithm. We present finite sample statistical consistency guarantees for Quick Shift on mode and cluster recovery under mild distributional assumptions. We then apply our results to construct a consistent modal regression algorithm.

In this paper the authors submit proofs regarding various notions of consistency for the Quick Shift algorithm. They also introduce adaptions of the Quick Shift algorithm for other tasks paired with relevant consistency results.
Quick Shift is a mode finding algorithm where one chooses a segmentation parameter \tau, constructs a kernel density estimator, and assigns a mode, m, at samples where the KDE magnitude at m is larger than the KDE magnitude evaluated at all samples within a \tau-ball of m. In this setting a true mode are those points in the sample space where the density is maximized within a \tau ball. Quick shift can also be used to assign samples to a mode by ascending a sample to the nearest sample where the KDE magnitude is larger and repeating until a mode is reached. The authors present proofs of consistency for both of these tasks: mode recovery and mode assignment.
The authors also propose an adaptation of Quick Shift for cluster tree recovery and modal regression for which they also prove consistency.
The proofs presented in this paper are dense and technical; I cannot claim to have to have totally validated their correctness. That being said I found no glaring errors in the authors' reasoning. My largest concern with this paper is reference [1]. [1] plays a central role in the proofs for individual mode recovery rates. [1] is an "anonymous  "unpublished manuscript" on Dropbox. I spent a fair amount of time searching for any other reference to this manuscript and I could find none. The authors give no explanation for this manuscript. I'm inclined to think that I cannot reasonably accept a paper whose results rely upon such a dubious source.
Additionally there are many other smaller issues I found in the paper. Some of them are listed here by line:
99: period after "we" should be removed
101-103: The sentence "For example, if given an input X the response y have two polarizing tendancies, modal regression would be able to adapt to these tendancies," is very strange and unclear. Also "tendencies" is misspelled.
103: "indepth" is not a word. Replace with "in-depth" or "indepth"
106: "require" should be "requires"
112: Why is the "distribution \script{F}" introduced? This is never mentioned elsewhere. We can simply proceed from the density.
113: I'm guessing that the "uniform measure" is the same thing as the "Lebesgue measure." "Lebesgue measure" is more standard terminology in my opinion.
115: Why is the norm notation replaced with absolute value notation? This occurs later as well elements which are clearly vectors, line 137 and Lemma 2 for instance.
139: Remark 3 should have a citation.
145: C is never defined.
150: Your style changes here. Assumption 3 uses parentheses and here Assumption 4 uses square brackets.
158: The final "x" in the set definition should be "x'"
242: "f(x)" should be to the right of "inf"
280: "conditiona" should be "conditional"
286: What exactly is meant by "valid bandwidth choices?"
Update: I've bumped the paper by two points. I still have concerns with the numerous small technical errors I found


Geometric Descent Method for Convex Composite Minimization
In this paper, we extend the geometric descent method recently proposed by Bubeck, Lee and Singh [1] to tackle nonsmooth and strongly convex composite problems. We prove that our proposed algorithm, dubbed geometric proximal gradient method (GeoPG), converges with a linear rate (1 − 1/ √ κ) and thus achieves the optimal rate among first-order methods, where κ is the condition number of the problem. Numerical results on linear regression and logistic regression with elastic net regularization show that GeoPG compares favorably with Nesterov's accelerated proximal gradient method, especially when the problem is ill-conditioned.

Summary: The paper extends the recent work of Bubeck et al. on the geometric gradient descent method, in order to handle non-smooth (but “nice”) objectives. Similar work can be found in [10]; however, no optimal rates are obtained in that case (in the sense of having a square root condition number in the retraction factor). The core ideas and motivation origin from the proximity operators used in non-smooth optimization in machine learning and signal processing (see e.g., lasso).
Quality: The paper is of good technical quality - for clarity see below. The results could be considered “expected” since similar results (from smooth to non-smooth convex objectives with proximal operators) have been proved in numerous papers the past decade; especially under convexity assumptions.
Technical correctness: I believe that the results I read in the main are correct.
Impact: The paper might have a good impact on the direction of demystifying Nesterov’s accelerated routines like FISTA (by proposing alternatives). I can see several ramifications based on this work, so I would say that the paper has at least some importance.
Clarity: My main concern is the balance of being succinct and being thorough. The authors tend towards the former; however, several parts of the paper are no clear (especially the technical ones). Also, the whole idea of this line of work is to provide a better understanding of accelerated methods; the paper does not provide much intuition on this front (update: there is a paragraph in the sup. material - maybe the authors should consider moving that part in the main text).
Comments:
1. Neither here, nor in the original paper of Bubeck it is clear how we get some basic arguments: E.g., I “wasted” a lot of time to understand how we get to (2.2) by setting y = x^* in (2.1). Given the restricted review time, things need to be crystal clear, even if they are not in the original paper, based on which this work is conducted.
Maybe this can be clarified by the rest of the reviewers. Assuming y = x++, this gives the bound x* \in B(x++,  2/\alpha^2 * || \nabla f(x) ||_2^2 - 2/\alpha (f(x) - f(x++))), but f(x*) \leq f(x++), so we cannot naively substitute it there. Probably I’m missing something fundamental.
Overall: There is nothing strongly negative about the paper - I would say, apart from some clarity issues and given that proofs work through, it is a “weak accept”.


DropoutNet: Addressing Cold Start in Recommender Systems
Latent models have become the default choice for recommender systems due to their performance and scalability. However, research in this area has primarily focused on modeling user-item interactions, and few latent models have been developed for cold start. Deep learning has recently achieved remarkable success showing excellent results for diverse input types. Inspired by these results we propose a neural network based latent model called DropoutNet to address the cold start problem in recommender systems. Unlike existing approaches that incorporate additional content-based objective terms, we instead focus on the optimization and show that neural network models can be explicitly trained for cold start through dropout. Our model can be applied on top of any existing latent model effectively providing cold start capabilities, and full power of deep architectures. Empirically we demonstrate state-of-the-art accuracy on publicly available benchmarks. Code is available at https://github.com/layer6ai-labs/DropoutNet.

Excellent paper, presenting an good idea with sufficient experiments demonstrating its efficiency. Well written and with a good survey of related work.
The paper addresses the following problem: when mixing id level information (user id of item id) with coarser user and item features, any model has the tendency to explain most of the training data with the fine grained id features, using coarse features only to learn the rarest ids. As a result, the generated model is not good at inference when only coarser features are available (cold-start cases, of a new user or a new item). The paper proposes a dial to control how much the model balances out the fine grained and coarser information via drop-out of fine grained info while training.
The loss function in Equation (2) has the merit of having a label value for all user-item pairs, by taking the output of the low rank model U_u V_v ^T as labels. My concern with such a loss is that the predictions from the low rank model have very different levels of accuracy: they are very accurate for the most popular users and the most popular items (users and items with most data), but can be quite noisy in the tail, and the loss function in Equation (2) does not account for the confidence in the low rank model prediction. Replacing the sum by a weighted sum using as weights some simple estimates of the confidence in each low rank model prediction could improve the overall model. But this can be done in future work.
Question about experiments on CiteULike: Figure 2 and Table 1 well support the point the author is making about the effect of dropout on recall for both, warm start cases and cold start cases. However, it is surprising that the cold start recall numbers are better than the warm start recall numbers. Can the authors explain this phenomenon?
Minor details:
Line 138: f_\cal{U} -> f_\cal{V}
Line 349: is able is able -> is able


A Scale Free Algorithm for Stochastic Bandits with Bounded Kurtosis
Existing strategies for finite-armed stochastic bandits mostly depend on a parameter of scale that must be known in advance. Sometimes this is in the form of a bound on the payoffs, or the knowledge of a variance or subgaussian parameter. The notable exceptions are the analysis of Gaussian bandits with unknown mean and variance by Cowan et al. [2015]  and of uniform distributions with unknown support . The results derived in these specialised cases are generalised here to the non-parametric setup, where the learner knows only a bound on the kurtosis of the noise, which is a scale free measure of the extremity of outliers.

Rebuttal
--------
I read the author feedback. There was no discussion, but due to the consensus, the paper should be accepted.
Summary
-------
The article considers a variant of the Upper Confidence Bound (UCB) algorithm for multi-armed bandit that is using the notion of kurtosis as well as median of means estimates for the mean and the variance of the considered distributions.
Detailed comments
-----------------
1) Table 1: I think the work by Burnetas and Katehakis regarding bandit lower bounds for general distributions deserves to be in this table.
2) It is nice to provide Theorem 7 and make explicit the lower bound in (special cases of) this setting.
3) The algorithm that is derived looks very natural in view of the discussion regarding the kurtosis.
One may observe that a bound kappa_0 on kappa is currently required by the algorithm, and I suggest you add some paragraph regarding this point:
Indeed on the one hand, a (naive) criticism is that people may not know a value kappa_0 in practice and thus one would now like to know what can be achieved when the kurtosis is unknown and we want to estimate it; This would then call for another assumption such as on higher order moments, and one may then continue asking similar questions for the novel assumption, and so on and so forth.
On the other hand, the crucial fact about kurtosis is that unlike mean or variance, for many classes of distributions all distributions of a certain type share the same kurtosis. Thus this is rather a "class" dependent quantity than a distribution dependent quantity, and using the knowledge of kappa_0 \geq \kappa can be considered to be a weak assumption. I think this should be emphasized more.
4) Proof of Lemma 5: The proof is a little compact. I can see the missing steps; but for the sake of clarity, it may be nice to reformulate/expand slightly. (Applying Lemma 4 twice, first to the random variables Y_s, then to (Y_s-mu)^2, etc).
5) Proof of Lemma 6: end of page 4, first inequality: from (c), you can improve the factor 2 to 4/3  (1/ (1-1/4) instead of 1/(1-1/2))
Line 112 "while the last uses the definition..." and Lemma 4 as well?
With the 2/3 improvement, you should get \tilde \sigma_t^2 \leq 11/3 \sigma^2 + \tilde \sigma_t^2/3.
6) Without section 3, the paper would have been a little weak, since the proof techniques are extremely straightforward once we have the estimation errors of the median of means, which are known. Thus this section is welcomed.
7) Proof of Theorem 7: "Let Delta_\epsilon = \Delta_\epsilon + \epsilon" I guess it should be "Let Delta_\epsilon = \Delta + \epsilon".
8) There are some missing words on page 7.
9) Section 4:
i)Optimal constants. For the seek improvement, one should mimic the KL optimization, at least implicitly. I am not sure this can be achieved that easily.
Self-normalized processes would play a role in reducing the magnitude of the confidence bounds, I am not sure it will help beyond that.
ii) TS: I don't see any simple way to define a "kurtosis-based" prior/posterior. Can you elaborate on this point? Otherwise this seems like an empty paragraph and I suggest you remove it.
Decision:
--------
Overall, this is an article with a rather simple (a direct application of median of means confidence bounds for unknown mean and variance) but elegant idea, and the paper is nicely executed and very clear. I give a weak accept, not a clear accept only because most of the material is already known.


Recurrent Ladder Networks
We propose a recurrent extension of the Ladder networks [22] whose structure is motivated by the inference required in hierarchical latent variable models. We demonstrate that the recurrent Ladder is able to handle a wide variety of complex learning tasks that benefit from iterative inference and temporal modeling. The architecture shows close-to-optimal results on temporal modeling of video data, competitive results on music modeling, and improved perceptual grouping based on higher order abstractions, such as stochastic textures and motion cues. We present results for fully supervised, semi-supervised, and unsupervised tasks. The results suggest that the proposed architecture and principles are powerful tools for learning a hierarchy of abstractions, learning iterative inference and handling temporal information.

In this paper, the authors introduce an extension of ladder networks for temporal data. Similarly to ladder network, the model proposed in the paper consists in an encoder-decoder network, with "lateral" connections between the corresponding layers of the encoder and the decoder. Contrary to the original model, recurrent ladder networks also have connection between different time steps (as well as skip connection from the encoder to the next time step). These recurrent connections allows to model temporal data or to perform iterative inference (where more than one step of message passing is required). The proposed model is then evaluated on different tasks. The first one is a classification task of partially occluded and moving MNIST digit, where the authors show that the proposed model performs slightly better than baselines corresponding to the proposed models with various ablation. The second task is generative modeling of various piano pieces. Finally, the last experiment is a classification task of MNIST digits with Brodatz textures, making the classification very challenging.
While the ideas presented in this paper are natural and well founded, I believe that the current version of the papers has some issues. First, the proposed model is very succinctly described (in section 2), assuming that the reader is familier with ladder networks. This is not the case for me, and I had to look at previous papers to understand what is the model. I believe that the paper should be self-contained, which I do not think is the case for the submitted version. I also believe that some statement in the paper are a bit unclear or misleading: for example, I do not think that Fig. 1c corresponds to message passing with latent variables varying in time (I think that it corresponds to iterative inference). Indeed, in the case of a temporal latent variable model, there should also be messages from t to t-1. I also found the experimental section a bit weak: most of the experiments are performed on toy datasets (with the exception of the piano dataset, where the model is not state of the art).
Overall, I am a bit ambivalent about the paper. While the ideas presented are interesting, I think that the proposed model is a bit incremental compared to previous work, and that the experiments are a bit weak.
Since I am not familiar at all with this research area, I put a low confidence score for my review.


Excess Risk Bounds for the Bayes Risk using Variational Inference in Latent Gaussian Models
Bayesian models are established as one of the main successful paradigms for complex problems in machine learning. To handle intractable inference, research in this area has developed new approximation methods that are fast and effective. However, theoretical analysis of the performance of such approximations is not well developed. The paper furthers such analysis by providing bounds on the excess risk of variational inference algorithms and related regularized loss minimization algorithms for a large class of latent variable models with Gaussian latent variables. We strengthen previous results for variational algorithms by showing that they are competitive with any point-estimate predictor. Unlike previous work, we provide bounds on the risk of the Bayesian predictor and not just the risk of the Gibbs predictor for the same approximate posterior. The bounds are applied in complex models including sparse Gaussian processes and correlated topic models. Theoretical results are complemented by identifying novel approximations to the Bayesian objective that attempt to minimize the risk directly. An empirical evaluation compares the variational and new algorithms shedding further light on their performance.

General comments:
Overall this is an interesting paper with some nice theoretical results, though the techniques are rather standard. As far as I know, the results on risk bounds for Bayes predictor are novel. It provides theoretical evidence for Bayesian treating of parameters (at least for Gaussian priors) and the generalization guarantee of variational approximations (1-layer, also 2-layer by collapsing the second), which should be of wide interest to the Bayesian community. The experiments on CTMs demonstrate the risk bounds of different variational approximations, though I find the direct loss minimization case less significant because usually it’s hard to compute the marginal log likelihoods.
The main concern that I have is the presentation of the results. It seems that the authors induce unnecessary complexity during the writing. The three kinds of risks induced in the paper ($r_{Bay}, r_{Gib}, r_{2A}$) are just negative expectations of marginal log    likelihood $\log p(y|x)$, the first term in elbo for w, and the first term in elbo for w and f (using terminology from variational inference). In Section 2.2, a 2-layer Bayesian model is introduced. I understand the authors’ intention to align with real model examples used in the paper after I finish later sections. But I still feel it’s better to derive the main results (Th.2, 6, 8) first on the 1-layer model $p(w)p(y|x,w)$ and later extend them into 2-layer settings. In fact the main theoretical result is on collapsed variational approximations, which can be seen as a 1-layer model (This is also how the proof of Corollary 3 comes).
Minor: In Section 2.2, if the 2-layer model is defined as $p(w)p(f)\prod_i p(y_i|f_i, x_i)$, should the risks be $...E_{p(f|w)} p(y|f, x)$ instead of $...E_{p(f|w,x)}p(y|f)$? I’m not sure why there is a dependence of f on x.


Learning from uncertain curves: The 2-Wasserstein metric for Gaussian processes
We introduce a novel framework for statistical analysis of populations of nondegenerate Gaussian processes (GPs), which are natural representations of uncertain curves. This allows inherent variation or uncertainty in function-valued data to be properly incorporated in the population analysis. Using the 2-Wasserstein metric we geometrize the space of GPs with L 2 mean and covariance functions over compact index spaces. We prove uniqueness of the barycenter of a population of GPs, as well as convergence of the metric and the barycenter of their finite-dimensional counterparts. This justifies practical computations. Finally, we demonstrate our framework through experimental validation on GP datasets representing brain connectivity and climate development. A MATLAB library for relevant computations will be published at https://sites.google.com/view/antonmallasto/software.

The paper extends the Wasserstein (earth-mover's) metric from
finite-dimensional Gaussians to GPs, defining a geometry on spaces of
GPs and in particular the notion of a barycenter, or 'average' GP. It
establishes that 2-Wasserstein distances, and barycenters, between GPs
can be computed as the limit of equivalent computations on finite
Gaussians. These results are applied to analysis of uncertain
white-matter tract data and annual temperature curves from multiple
stations.
The writing is serviceable and relevant background is presented
clearly. I found no errors in the results themselves, though did not
check all of them thoroughly. My main concerns are with motivation and
significance. I agree that it is in some sense natural to extend the
Wasserstein geometry on Gaussians to GPs, and it's reassuring to see
that this can be done and (as I understand the results of the paper)
basically nothing surprising happens -- this is definitely a
contribution. But there is not much motivation for why this is a
useful thing to do. The experiments in the paper are (as is standard
for theory papers) fairly contrived, and the conclusions they arrive
at (the earth is getting warmer) could likely be gained by many other
forms of analysis --- what makes the Wasserstein machinery
particularly meaningful here?
For example: from a hierarchical Bayesian perspective it would seem
reasonable to model the climate data using a latent, unobserved global
temperature curve f_global ~ GP(m_global, k_global), and then assume
that each station i observes that curve with additional local GP noise,
f_i ~ GP(f_global + mi, ki). Then the latent global temperature curve
could be analytically computed through simple Gaussian conditioning,
would be interpretable in terms of this simple generative model, and
would (correctly) become more certain as we gathered evidence from
additional stations. By contrast I don't understand how to
statistically interpret the barycenter of independently estimated GPs,
or what meaning to assign to its uncertainty (am I correct in
understanding that uncertainty would *not* decrease given additional
observations?  what are the circumstances in which this is desirable
behavior?).
Perhaps other reviewers with a more theoretical bent will find this
work exciting. But in the absence of any really surprising results or
novel theoretical tools, I would like to see evidence that the machinery
it builds is well-motivated and has real advantages for practical data
analysis -- can it reach conclusions more efficiently or uncover structure
more effectively than existing techniques? As currently written the paper
doesn't cross my personal NIPS significance bar.
misc notes:
- Fig 1: the 'naive mean' column claims to average the pointwise
standard deviations, but the stddevs shown are smaller than *any* of
the example curves: how can this be true? Also why is averaging
pointwise stddevs, rather than (co)variances or precisions, the relevant
comparison?
- I wonder if the Wasserstein metric could be useful for variational
fitting of sparse/approximate GP models? Currently this is mostly done
by minimizing KL divergence (as in Matthews et al. 2015,
https://arxiv.org/abs/1504.07027) but one could imagine minimizing the
Wasserstein distance between a target GP and a fast approximating
GP. If that turned out to yield a practical method with statistical or
computational advantages over KL divergence for fitting large-scale
GPs, that'd be a noteworthy contribution.
130-131: missing close-paren in inf_\mu\inP2 (H
152: 'whe' -> 'we'
155: 'This set is a convex'
170: why is 'Covariance' capitalized?
178: 'GDss' -> 'GDs'?


Robust Hypothesis Test for Nonlinear Effect with Gaussian Processes
This work constructs a hypothesis test for detecting whether an data-generating function h : R p → R belongs to a specific reproducing kernel Hilbert space H 0 , where the structure of H 0 is only partially known. Utilizing the theory of reproducing kernels, we reduce this hypothesis to a simple one-sided score test for a scalar parameter, develop a testing procedure that is robust against the misspecification of kernel functions, and also propose an ensemble-based estimator for the null model to guarantee test performance in small samples. To demonstrate the utility of the proposed method, we apply our test to the problem of detecting nonlinear interaction between groups of continuous features. We evaluate the finite-sample performance of our test under different data-generating functions and estimation strategies for the null model. Our results reveal interesting connections between notions in machine learning (model underfit/overfit) and those in statistical inference (i.e. Type I error/power of hypothesis test), and also highlight unexpected consequences of common model estimating strategies (e.g. estimating kernel hyperparameters using maximum likelihood estimation) on model inference.

This paper considers an interesting issue of hypothesis testing with complex black models. I will say from the outset that I am skeptical of the NHST framework, especially as it is currently applied in classical statistical settings, where I think it is clear the ongoing harm that it has done to applied science (see, e.g. the ASA's statement on p-values). Whether I should be more or less skeptical of it in nonparametric settings is an interesting questions and I'd be happy to be convinced otherwise. I guess the standard arguments aren't worth rehashing in great detail but just so you're aware of where I'm coming from, if I care about the (possibly non-linear) interaction effect then I want a kernel parameterized as:
beta1 * k1a(x1,x1') + beta2 * k2c(x2,x2') + beta12 * k1b(x1,x1')*k2b(x2,x2')
I want to fit that model using all of my expensive data, and then say something about beta12. Bayesian methods would give me a posterior credible interval for beta12---given limited data this is going to be my approach for sure and I'm going to spend time constructing priors based on previous studies and expert knowledge. But if you want to take a frequentist approach that's fine, you can just bootstrap to obtain a confidence interval for beta12, which I'm going to be happier with than the asymptotics-based approach you take for small samples.
OK, now that I've gotten that disclaimer out of the way (apologies!), taking your paper on its own terms, here are some areas that I think could be improved.
On line 29 I'm not sure why the null hypothesis is additive; shouldn't this be a multiplicative interaction between air pollutants and nutrient intake?
Be careful in Section 2: the fact that h ~ GP with covariance kernel k means that the probability that h is in the RKHS H_k is 0! This is a surprising fact (see Wahba 1990 and Lukic and Beder 2001) and I don't think it invalidates your approach (e.g. E[h] is in the RKHS), but it's worth fixing this point.
I like the GP-LMM connection and think it needs more explanation as I am not very familiar with REML. Could I, for example, posit the following model:
y = h1 + h2 + epsilon, h1 ~ GP(0,K1) and h2 ~ GP(0,K2) and then use your approach to decide between these models:
y = h1 + epsilon
y = h2 + epsilon
y = h1 + h2 + epsilon?
I think the answer is yes, and I'm wondering if it'd be clearer to explain things this way before introducing the garrote parameter delta.
Turning the test statistic into a model residual also helped me think about what was going on, so maybe it'd be good to move that earlier than Section 4. It also might be nice to show some training/testing curves (to illustrate bias/variance) in Section 4 to explain what's going on.
It wasn't obvious to me why the additive main effects domain H1 \osum H12 is orthogonal to the product domain H1 \otimes H2 (section 5).
For the experiments, if this were real data I could imagine trying all of the difference kernel parameterizations and choosing the one that the fits the best (based on marginal likelihood I guess) as the model for the null hypothesis. Of the different models you considered, which one fit the data best? And did this choice end up giving good type 1 and type 2 error?
A real data experiment would be nice.


Deep Sets
We study the problem of designing models for machine learning tasks defined on sets. In contrast to traditional approach of operating on fixed dimensional vectors, we consider objective functions defined on sets that are invariant to permutations. Such problems are widespread, ranging from estimation of population statistics [1], to anomaly detection in piezometer data of embankment dams [2], to cosmology [3,4]. Our main theorem characterizes the permutation invariant functions and provides a family of functions to which any permutation invariant objective function must belong. This family of functions has a special structure which enables us to design a deep network architecture that can operate on sets and which can be deployed on a variety of scenarios including both unsupervised and supervised learning tasks. We also derive the necessary and sufficient conditions for permutation equivariance in deep models. We demonstrate the applicability of our method on population statistic estimation, point cloud classification, set expansion, and outlier detection.

Summary of Review:
Although the techniques are not novel, the proofs are, and the paper seems well done overall.
I’m not familiar with the experiments / domains of the tasks, and there’s not enough description for me to confidently judge the experiments, but they appear fairly strong.
Also, the authors missed one key piece of related work (“Learning Multiagent Communication with Backpropagation” - Sukhbaatar et al. 2016).
Overview of paper:
This paper examines permutation-invariant/equivariant deep learning, characterizing the class of functions (on countable input spaces) and DNN layers which exhibit these properties.
There is also an extensive experimental evaluation.
The paper is lacking in clarity of presentation and this may hide some lack of conceptual clarity as well; for instance, in some of the applications given (e.g. anomaly detection), the input is properly thought of as a multiset, not a set.
The proof of Theorem 2 would not apply to multisets.
Quality:
Overall, the paper appears solid in motivation, theory, and experiments.
The proofs (which I view as the main contribution) are not particularly difficult, impressive, or involved, but they do provide valuable results.
The other contributions are a large number of experiments, which I have difficulty judging.
The take-aways of the experiments are fairly clear:
1) expanding deep learning to solve set-based tasks can (yet again) yield improvements over approaches that don’t use deep learning
2) in many cases, a naive application of deep learning is bound to fail (for instance, section 4.3, where the position of the anomaly is, presumably, random, and thus impossible for a permutation *invariant* architecture, which I presume the baseline to be), and permutation invariance / equivariance *must* be considered and properly incorporated into the model design.
Clarity:
While the overall message is clear, the details are not, and I’ll offer a number of corrections / suggestions, which I expect the authors to address or consider.
First of all, notation should be defined (for instance, this is missing in de Finetti’s theorem).
Section 2.3 should, but does not explicitly show how these results relate to the rest of the paper.
Line 136-137: you should mention some specific previous applications, and be clear, later on in the experiments section, about the novelty of each experiment.
Line 144: which results are competitive with state-of-the-art (SOTA)?  Please specify, and reference previous SOTA.
As mentioned before, the tasks are not always explained clearly enough for someone unfamiliar with them (not even in the appendix!).
In particular, I had difficulty understanding the tasks in section 4.1.3 and 4.2.
In 4.2 I ask: How is “set expansion” performed? What does “coherent” mean? Is the data labelled (e.g. with ground-truth set assignments?) What is the performance metric? What is the “Beta-Binomial model”?
detailed comments/questions:
There are many missing articles, e.g. line 54 “in transductive” —> “in the transductive”
The 2nd sentence of the abstract is not grammatical.
line 46: “not” should be removed
line 48: sets do not have an ordering; they are “invariant by definition”.  Please rephrase this part to clarify (you could talk about “ordered sets”, for instance).
Line 151: undefined notation (R(alpha), M).
Lines 161-164 repeat the caption of Figure 1; remove/replace duplicate text.
the enumeration in lines 217-273 is poorly formatted
Line 275 d) : is this the *only* difference with your method?  Max-pooling is permutation invariant, so this would still be a kind of Deep Set method, right??
line 513: the mapping must be a bijection for the conclusions to follow!
As I mentioned, set and multiset should be distinguished; a multiset is permutation invariant but may have repeated elements.
Multisets may be more appropriate for some machine learning tasks.
BTW, I suspect the result of Theorem 2 would not hold for uncountable sets.
Bayesian Sets is cited twice (35 and 49).
Originality:
I’m a bit out of my depth in terms of background on the tasks in this paper, and I’m not 100% sure the proofs are novel; I would not be particularly surprised if these results are known in, e.g. some pure mathematics communities.
My best guess is that the proofs are novel, and I’m fairly confident they’re novel to the machine learning or at least deep learning community.
Significance:
I think the proofs are significant and likely to clarify the possible approaches to these kinds of set-based tasks for many readers.
As a result (and also due to the substantial experimental work), the paper may also play a role in popularizing the application of deep nets to set-based tasks, which would be quite significant.


Dual Path Networks
In this work, we present a simple, highly efficient and modularized Dual Path Network (DPN) for image classification which presents a new topology of connection paths internally. By revealing the equivalence of the state-of-the-art Residual Network (ResNet) and Densely Convolutional Network (DenseNet) within the HORNN framework, we find that ResNet enables feature re-usage while DenseNet enables new features exploration which are both important for learning good representations. To enjoy the benefits from both path topologies, our proposed Dual Path Network shares common features while maintaining the flexibility to explore new features through dual path architectures. Extensive experiments on three benchmark datasets, ImagNet-1k, Places365 and PASCAL VOC, clearly demonstrate superior performance of the proposed DPN over state-of-the-arts. In particular, on the ImagNet-1k dataset, a shallow DPN surpasses the best ResNeXt-101(64 × 4d) with 26% smaller model size, 25% less computational cost and 8% lower memory consumption, and a deeper DPN (DPN-131) further pushes the state-of-the-art single model performance with about 2 times faster training speed. Experiments on the Places365 large-scale scene dataset, PASCAL VOC detection dataset, and PASCAL VOC segmentation dataset also demonstrate its consistently better performance than DenseNet, ResNet and the latest ResNeXt model over various applications.

The authors propose a new network architecture which is a combination of ResNets and DenseNets. They introduce a very informative theoretical formulation which can be used to formulate ResNets, DenseNets and their proposed architecture. The authors present compelling results, analysis and statistics on compelling benchmarks.
Pros:
(+) The paper is well written with theoretical and empirical results
(+) The authors provide useful analysis and statistics
(+) The impact of DPNs is shown on a variety of computer vision tasks
(+) The performance of the DPNs on the presented vision tasks is compelling
Cons:
(-) Optional results on MS COCO would make the paper even stronger
Network engineering is an important field and it is important that it is done correctly, with analysis and many in depth experiments. The impact of new architectures comes through their generalization capabilities. This paper does a good job on all of the above. Even though there is no groundbreaking novelty in the proposed micro-blocks, the authors provide extensive analysis and statistics for their proposed model. They also present results on a variety of computer vision tasks which makes their paper complete. Especially the analysis of memory usage, training speed and number of flops is very revealing and informative.
The authors present results on compelling computer vision tasks, such as ImageNet, Places365 and PASCAL VOC. These are traditionally the most difficult datasets and tasks in computer vision. It would make the paper even stronger if the authors presented results on MS COCO, which is harder than PASCAL VOC.


Analyzing Hidden Representations in End-to-End Automatic Speech Recognition Systems
Neural networks have become ubiquitous in automatic speech recognition systems. While neural networks are typically used as acoustic models in more complex systems, recent studies have explored end-to-end speech recognition systems based on neural networks, which can be trained to directly predict text from input acoustic features. Although such systems are conceptually elegant and simpler than traditional systems, it is less obvious how to interpret the trained models.
In this work, we analyze the speech representations learned by a deep end-to-end model that is based on convolutional and recurrent layers, and trained with a connectionist temporal classification (CTC) loss. We use a pre-trained model to generate frame-level features which are given to a classifier that is trained on frame classification into phones. We evaluate representations from different layers of the deep model and compare their quality for predicting phone labels. Our experiments shed light on important aspects of the end-to-end model such as layer depth, model complexity, and other design choices.

The authors conduct an analysis of CTC trained acoustic models to determine how information related to phonetic categories is preserved in CTC-based models which directly output graphemes. The work follows a long line of research that has analyzed neural network representations to determine how they model phonemic representations, although to the best of my knowledge this has not been done previously for CTC-based end-to-end architectures.
The results and analysis presented by the authors is interesting, although there are some concerns I have with the conclusions that the authors draw that I would like to clarify these points. Please see my detailed comments below.
- In analyzing the quality of features learned at each layer in the network, from the description of the experiments in the paper it would appear that the authors only feed in features corresponding to a single frame at time t, and predict the corresponding output label. In the paper, the authors conclude that (Line 159--164) "... after the 5th recurrent layer accuracy goes down again. One possible explanation to this may be that higher layers in the model are more sensitive to long distance information that is needed for the speech recognition task, whereas the local information which is needed for classifying phones is better captured in lower layers." This is a plausible explanation, although another plausible explanation that I'd like to suggest for this result is the following:
It is know from previous work, e.g., (Senior et al., 2015) that even when predicting phoneme output targets, CTC-based models with recurrent layers can significantly delay outputs from the model. In other words, the features at a given frame may not contain information required to predict the current ground-truth label, but this may be shifted. Thus, for example, it is possible that if a window of features around the frame at time t was used instead of a single frame, the conclusions vis-a-vis the quality of the recurrent layers in the model would be very different. An alternative approach, would be to follow the procedure in (Senior et al., 2015) and constrain the CTC loss function to only allow a certain amount of delay in predicting labels. I think it is important to conduct the experiments required to establish whether this is the case before drawing the conclusions in the paper.
Reference: A. Senior, H. Sak, F. de Chaumont Quitry, T. Sainath and K. Rao, "Acoustic modelling with CD-CTC-SMBR LSTM RNNS," 2015 IEEE Workshop on Automatic Speech Recognition and Understanding (ASRU), Scottsdale, AZ, 2015, pp. 604-609.
- In analyzing phoneme categories I notice that the authors use the full 61 label TIMIT phoneme set (60 after excluding h#). However, it is common practice for TIMIT to also report phoneme recognition results after mapping the 61 label set down to 48 following (Lee and Hon, 89). In particular, this maps certain allophones into the same category, e.g., many closures are mapped to the same class. It would be interesting to confirm that the same trends as identified by the authors continue to hold in the reduced set.
Reference: Lee, K-F., and H-W. Hon. "Speaker-independent phone recognition using hidden Markov models." IEEE Transactions on Acoustics, Speech, and Signal Processing 37.11 (1989): 1641-1648.
- There have been many previous works on analyzing neural network representations in the context of ASR apart from the references cited by the authors. I would suggest that the authors incorporate some of the references listed below in addition to the ones that they have already listed:
* Nagamine, Tasha, Michael L. Seltzer, and Nima Mesgarani. "On the Role of Nonlinear Transformations in Deep Neural Network Acoustic Models." INTERSPEECH. 2016.
* Nagamine, Tasha, Michael L. Seltzer, and Nima Mesgarani. "Exploring how deep neural networks form phonemic categories." Sixteenth Annual Conference of the International Speech Communication Association. 2015.
* Yu, Dong, et al. "Feature learning in deep neural networks-studies on speech recognition tasks." arXiv preprint arXiv:1301.3605 (2013).
- In the context of grapheme-based CTC for ASR, I think the authors should cite the following early work:
* F. Eyben, M. Wöllmer, B. Schuller and A. Graves, "From speech to letters - using a novel neural network architecture for grapheme based ASR," 2009 IEEE Workshop on Automatic Speech Recognition & Understanding, Merano, 2009, pp. 376-380.
- Line 250: "... and micro-averaging F1 inside each coarse sound class." What is meant by "micro-averaging" in this context?
- Minor comments and typographical errors:
* Line 135: "We train the classifier with Adam [22] with default parameters ...". Please specify what you mean by default parameters.
* In Figure 1 and Figure 2: The authors use "steps" when describing strides in the convolution. I'd recommend changing this to "strides" for consistency with the text.
* In Figure 1. d: The scale on the Y-axis is different from the other figures which makes the two figures in (c.) and (d.) not directly comparable. Could the authors please use the same Y-axis scale for all figures in Figure 1.