Variational Information Maximisation for Intrinsically Motivated Reinforcement Learning
The mutual information is a core statistical quantity that has applications in all areas of machine learning, whether this is in training of density models over multiple data modalities, in maximising the efficiency of noisy transmission channels, or when learning behaviour policies for exploration by artificial agents. Most learning algorithms that involve optimisation of the mutual information rely on the Blahut-Arimoto algorithm -an enumerative algorithm with exponential complexity that is not suitable for modern machine learning applications. This paper provides a new approach for scalable optimisation of the mutual information by merging techniques from variational inference and deep learning. We develop our approach by focusing on the problem of intrinsically-motivated learning, where the mutual information forms the definition of a well-known internal drive known as empowerment. Using a variational lower bound on the mutual information, combined with convolutional networks for handling visual input streams, we develop a stochastic optimisation algorithm that allows for scalable information maximisation and empowerment-based reasoning directly from pixels to actions.

The paper concerns a stochastic variational approach towards mutual information maximisation with applications to reinforcement learning. The authors present a treatment of MI following a bound taken form earlier work by Barber&Agakov and using it stochastically by mini-batch descent. In order to estimate the MI they introduce a novelty and use neural networks to predict parameters for factors in state transition models as shown in Equation 6. This replaces the need for an explicit generative model of the data. The authors use this algorithm in the context of empowerment, which is a measure that can be used to connect MI with reinforcement learning. they demonstrate very simple empowerment based policies using their MI approximation which yield experimentally good results while running much more efficiently than previous algorithms used to estimate the MI.The paper is written clearly and appears to be of good quality throughout. The organization is slightly perplexing, as frequently the purpose of the paper is mixed up between the efficient estimation of MI and reinforcement learning and Iwould urge the authors to clarify what their precise goals are: the paper is not sufficiently focused on reinforcement learning to be evaluated as such and it also fails to clearly show broad applicability for stochastic variational MI in non-reinforcement learning settings.That said, the results appear encouraging and efficient flexible methods for modeling actions in environments using neural networks are currently relevant directions for research.Comparisons with Auto-Encoding Variational Bayes would be interesting, as the approximation of factors using neural networks in a variational setting is reminiscent of this paper. I would also be interested in seeing how exactlythe performance of the model depends on the length of the action states given that the variational stochastic approximation to the gaussian factors does not use any kind of control variates to help learning of these factors and the fact that a factorization over many variational distributions will become worse and worse as the number of the factors grows without modeling interactions of components.


BACKSHIFT: Learning causal cyclic graphs from unknown shift interventions
We propose a simple method to learn linear causal cyclic models in the presence of latent variables. The method relies on equilibrium data of the model recorded under a specific kind of interventions ("shift interventions"). The location and strength of these interventions do not have to be known and can be estimated from the data. Our method, called BACKSHIFT, only uses second moments of the data and performs simple joint matrix diagonalization, applied to differences between covariance matrices. We give a sufficient and necessary condition for identifiability of the system, which is fulfilled almost surely under some quite general assumptions if and only if there are at least three distinct experimental settings, one of which can be pure observational data. We demonstrate the performance on some simulated data and applications in flow cytometry and financial time series.

%%%%%%%%%%%%%%%%%%%%%%%%%%% Update after authors response: I thank the authors for taking the time to clearly and honestly respond to my (our) questions. I am even more convinced that the work is worth presenting at NIPS. Probably my lack of understanding (or me being too lazy to read the appendices) but would really like to see the step-by-step explanation on the link between Assumption C and the ability of the algorithm to identify situations where the permutation (1 step only? Other remark: possible since you only have few nodes as well right otherwise the permutation space simply can't be explored?) which makes CP < 1.%%%%%%%%%%%%%%%%%%%%%%%%%%% The manuscript introduces linear causal dependencies in a complex system as measured by x's. The originality is that the location of the intervened variables is unknown. The authors present assumptions (section 2.3) under which they derive theoretical results in terms of model identifiability (section 3), in both an ideal setting where the sample size is infinite (algorithm presented in section 2.4) and when it isn't (algorithm presented in section 2.5). A good aprt of the paper is devoted to much appreciated numerical experiments on both simulated (ground truth known) and real data networks.I quite appreciated reading this paper and found the problem well motivated and nicely tackled. The only nuance is that it is not a huge breakthrough, rather a nice and useful increment from existing theory.Comments to the authors:- l36, def of "equilibrium data" is never clear.- l44, "solutions to (1)": mind that (1) is only a model. I guess you mean solutions of an optimisation problem related to Eqn (1) by a score function; this is detailed in sections 2.4 and 2.5 I know so merely a language issue here.- l61: "surgical" used here and defined later.- l62: iterations? Haven't talked about an algorithm yet- l64: interactionS (typo)- l84: unknown location of interventions; OK why not but in fields like modern Molecular Biology, this is fully determined, e.g. gene KO. Or on the contrary there would be way too many interventions at the same time (e.g. genetical genomics data sets where tiny interventions are performed at each node at the same time, see Mooij et al NIPS 2015 or Vignes et al. PLoS ONE 2011)- l102-103 is intriguing, could you clarify?- section 2.3: the 3 assumptions are in fact 3 (coherent?) sets of assumptions? Sentences in each item are not always equivalent. Moreover I-B invertible roughly forbids considering high-dimensional or even moderate dimension scenarii. So you don't really compare to many approaches you quote? With nice implementations like Kalisch and Buhlmann JMLR 2007 or their pcalg R package. In the same vein, see also Rau et al. BMC Syst. Biol. 2013.- l177: can you expand why Eqn (7) motivates the estimator in Eqn (10)? Rational for the loss defined by L as well?- l204, can you quickly explain how FFDiag [23] ( I mean how it works, reads like the difficult part in your computational work) allows you to solve problem (11)?- l207: (a) permuting -> looks like a permutations of potentially many edges? Can be harsh at the end or do I understand incorrectly? For permutation effect on columns of an adjacency matrix, see Champion et al. http://arxiv.org/abs/1507.02018- l212, variant of LAP -> any guarantee of an acceptable solution at this step?- l268: producing the code is VERY appreciated!


Expressing an Image Stream with a Sequence of Natural Sentences
We propose an approach for retrieving a sequence of natural sentences for an image stream. Since general users often take a series of pictures on their special moments, it would better take into consideration of the whole image stream to produce natural language descriptions. While almost all previous studies have dealt with the relation between a single image and a single natural sentence, our work extends both input and output dimension to a sequence of images and a sequence of sentences. To this end, we design a multimodal architecture called coherent recurrent convolutional network (CRCN), which consists of convolutional neural networks, bidirectional recurrent neural networks, and an entity-based local coherence model. Our approach directly learns from vast user-generated resource of blog posts as text-image parallel training data. We demonstrate that our approach outperforms other state-of-the-art candidate methods, using both quantitative measures (e.g. BLEU and top-K recall) and user studies via Amazon Mechanical Turk.

The paper attacks the problem of describing a sequence of images from blog-posts with a sequence of consistent sentences. For this the paper proposes to first retrieve the K=5 most similar images and associated sentences from the training set for each query image. The main contribution of the paper lies in defining a way to select the most relevant sentences for the query image sequence, providing a coherent description. For this sentences are first embedded in a vector and then the sequence of sentences is modeled with a bidirectional LSTM. The output of the bi-directional LSTM is first fed through a relu and fully connected layer and then scored with a compatibility score between image and sentence. Additionally a local coherence model [1] is included to enforce the compatibility between sentences.   Strength / positive aspects: - The paper proposes a novel and effective architecture to retrieve coherent sentences for an image sequence. - The paper provides an extensive quantitative, qualitative, and human evaluation, showing the superiority of their approach against several baselines, not using coherence. They also provide an ablation experiment, removing the coherence model [1]. - The authors promise to release source code and dataset. Weaknesses / Questions / Unclarities: 1. Line 94: The paper claims that there is "no mechanism for the coherence between sentences" in [5]. Although not the contribution of [5], [5] predicts an intermediate semantic representation of videos, which is coherent across sentences by modeling the topic of the multi-sentence description. 2. Correctness/clarity: Figure 2b does not seem to correspond to the description 3.3/Equation (2). While Figure 2b implies that the fully connected layer are connected to all sentences, this it not the case in Eq2, which implies that the parameters are shared across sentences, but only connected to the vector representing a single sentence. 3. A better metric to automatically evaluate the generated sentences is Meteor (http://www.cs.cmu.edu/~alavie/METEOR/) instead of BLEU, especially if there is only a single reference sentence. 4. Why two linear functions in Eq2 (W_{f2}, W_{f1}) are applied behind each other? Given that two linear functions are again a linear function the benefit is unclear. An ablation study showing the benefit of these functions would be interesting. 5. Why the same parameters of the fully connected layers are used for the BRNN output (o_t) and the local coherence model q (Equation 2)? 6. Is the paragraph vector [16] fine-tuned or kept fixed? === post rebuttal === After reading the rebuttal I recommend the paper for acceptance. The authors successfully addressed issues with the formulation, evaluation, and related work.Please make the promised changes to the final and also clarify the following point in the final. 6. Is the paragraph vector [16] fine-tuned or kept fixed?


Embedding Inference for Structured Multilabel Prediction
A key bottleneck in structured output prediction is the need for inference during training and testing, usually requiring some form of dynamic programming. Rather than using approximate inference or tailoring a specialized inference method for a particular structure-standard responses to the scaling challengewe propose to embed prediction constraints directly into the learned representation. By eliminating the need for explicit inference a more scalable approach to structured output prediction can be achieved, particularly at test time. We demonstrate the idea for multi-label prediction under subsumption and mutual exclusion constraints, where a relationship to maximum margin structured output prediction can be established. Experiments demonstrate that the benefits of structured output training can still be realized even after inference has been eliminated.

This paper deals with a particular task of structured prediction: binary variables, score function consisting of unary potentials only, structure consists in the feasibility constraints (logical and mutual exclusion). The paper attempts to get rid of the costly inference at the test and train stages by posing special constraints on the model parameters such that trivial inference procedure automatically satisfies the feasibility constraints. The proposed approach is compatible with multiple structured losses, e.g. conditional log-likelihood (2) and structured large margin (3) are mentioned. To the best of my knowledge the idea is original and seems interesting. The paper is relatively clear although sometimes it is hard to follow the technical details (I have several major comments/questions about clarity/correctness, see below). The idea of embedding inference might be quite influential (lead to structured prediction methods without inference) even given that the direct applications of the paper are quite narrow.Major: 1)Results of sections 3.2 and 3.3 (key results of the paper) are based on Lemma 2, which has no proof. Lemma 2 consists of claims (9) and (10). I agree that (9) is analogous to lemma 1 and therefore clear. Equation (10), however, appears for the first time and in my opinion seems non-trivial. If it does not hold (under the assumptions of lemma 2) than section 3.2-3.3 are incorrect.2) One of the contributions of the paper (mentioned in the abstract) is "equivalence to maximum margin structured output prediction". I do not see how this follows from the results of the paper. In particular, proposition 6 claims that is the score function satisfies constraints (5) and (6) than the large-margin objective can be computed without performing inference (r.h.s. of (22)). But general claim of equivalence in my opinion should sound something like this: "A minimizer of objective (3) (can be simplified by (22) under all the assumptions) under model (8) with theta_PAQ being positive semi-definite and constraints (11), (12), (16) (imposing (5) and (6)) at the same time minimizes objective (3) under model (8) with theta_PAQ being positive semi-definite (constraints are satisfied by inference)". I'd really like the authors to clarify what is meant by "equivalence". 3) In the experiments it is not specified how the hyper-parameters of the methods were chosen. In particular, choice of the weight of the regularizer r(theta) could significantly influence the results. Minor: Line 153. I do not understand why s_k(x) \geq 0 is a substantial requirement. Lines 159 and 319. Notation "argmax_{k \in {0,1} } f(k)= 1" is a bit confusing i.e. it is not clear what happens if f(1) = f(0). I suggest writing "1 \in argmax_{k \in {0,1} }". Line 295. Notation "t_{ik}" (two indices) is not introduced. Being slightly more careful would help the reader.*Comments after the response Thanks for for prooving Lemma 2 and clarifying Proposition 6.Please, incorporate these comments into the paper. In particular, phrase from the abstract "We demonstrate this idea for multi-label prediction under subsumption and mutual exclusion constraints, where equivalence to maximum margin structured output prediction is established." sounds like an overclaim.


Copula variational inference
We develop a general variational inference method that preserves dependency among the latent variables. Our method uses copulas to augment the families of distributions used in mean-field and structured approximations. Copulas model the dependency that is not captured by the original variational distribution, and thus the augmented variational family guarantees better approximations to the posterior. With stochastic optimization, inference on the augmented distribution is scalable. Furthermore, our strategy is generic: it can be applied to any inference procedure that currently uses the mean-field or structured approach. Copula variational inference has many advantages: it reduces bias; it is less sensitive to local optima; it is less sensitive to hyperparameters; and it helps characterize and interpret the dependency among the latent variables.

The typical mean-field approximation made in variational inference results in a well-known range of issues, from poor approximation of the distribution as a whole to poor covariance estimates in particular. This is a pressing problem for variational inference, and the authors aim to address it by introducing copulas into the variational approximating distribution. The authors' experimental results suggest that this may be a promising direction for improving on variational approximations. However, some details of the method are not covered as much as desired, and the experiments seem somewhat limited.Major points(1) In order to run their method, the authors need to choose a vine and a copula. While the authors spend a little time discussing that they choose these according to an algorithm, it's not clear what the approximative significance of the vine in particular is. Why are some vines better than others? Why are some copulas better than others? Are all possible vines considered? Is that problematic in many-parameter models? How are the possible copulas chosen? Is there any reason to think these choices of copulas are enough? E.g., on the tree point,p. 4: line 163--168: Where does the first tree come from? Can it be any tree on the model variables? Does it matter?p. 5, line 243: What does it mean to learn the tree structure? What is there to learn?(2) The authors seem to use one model per performance metric: a Gaussian mixture model for variance modeling and a latent state space model for predictive likelihood. (Why not compare both models on both metrics?) This isn't necessarily bad; it could be a starting point for future exploration. But then the authors seem to make somewhat broad conclusions based on one model each. It's also unclear why the authors plot just one particular set of parameters from the Gaussian mixture model in Figure 3. Aren't there are other mixture model parameters (like the component means in Eq. 18)? For that matter, what are the hyperparameters in the each generative model (the ones used in the experiment)?p. 8, line 393--396: It seems from Fig 3 that CVI and LRVB both estimate variances equally well on this problem. LRVB focuses on covariances. Presumably the point of CVI is to get an entire posterior that is better. Table 1 gets at this a little, but it would be nice to see more comparisons on, say, estimates of other posterior statistics than just covariances. Or on other models. And why not compare to structured mean-field or other recent methods that get beyond mean-field for predictive likelihood?p. 8, line 418: "CVI always outperforms LRVB". This statement seems a bit broad to be warranted from the experiments presented. The authors might reasonably say CVI outperforms LRVB in predictive likelihood (based on Table 1)---but does it outperform across other models besides just the one mixture model presented? Or do the authors really mean to say it outperforms LRVB in other ways? It is unclear how the two compare in covariance estimation (besides looking roughly the same in Figure 3).Minor pointsp. 1, line 35--37: This sentence is a little difficult to parse on the first pass. It sounds like the authors are saying it's a generalization but also a special case. Presumably they are just saying that their algorithm iterates between mean-field steps and copula steps. Does "structured factorizations" here refer to [19]? If so, it would help the reader to be explicit.p. 2, line 83: It seems a little unusual of a notational choice to use z for all latent variables; it seems to be usually reserved for indicatorsp. 2, line 84: KL is technically not a distance since it is not symmetricp. 3, line 143: "the bivariate Gaussian distribution with zero mean and Pearson correlation coefficient $\rho$" <-- Does this distribution have arbitrary strictly positive variance parameters in both directions? (or all the variances one since the authors refer to "the bivariate Gaussian"?)p. 5, line 247--248: What does this mean; e.g., what are these "future iterations"? Are the authors saying that the choice of tree and copula family is stable across multiple independent runs of the algorithm? If so, what changes from run to run that this choice might be different? If not, what are the iterations? What is the significance of the lack of change described?p. 7, line 357: "reduce the variance" <-- The authors might want to use more careful language here to clarify what variance is being reduced since the word "variance" is being used in a few different ways around this point in the text.p. 8, Table 1 seems to be missing running time. It's in the text, but it's a little hard to parse the comparison in paragraph form. Is there any reason not to put it in the table?p. 8, line 389: "consistent estimates" <-- This seems to be a poor use of the word "consistent" in the statistical sense (there's no limit as the number of data points becomes infinite). Perhaps "accurate" is the desired word.p. 8, line 412: This paragraph is a little hard to read. It sounds like CVI took hours, but then the authors say it's better in runtime. Perhaps this could be clarified.------------------------------ After Author Feedback ------------------------------The author feedback seemed on point; I look forward to reading more detail about the vine methodology and a broader range of experiments in the final version.


Are You Talking to a Machine? Dataset and Methods for Multilingual Image Question Answering
In this paper, we present the mQA model, which is able to answer questions about the content of an image. The answer can be a sentence, a phrase or a single word. Our model contains four components: a Long Short-Term Memory (LSTM) to extract the question representation, a Convolutional Neural Network (CNN) to extract the visual representation, an LSTM for storing the linguistic context in an answer, and a fusing component to combine the information from the first three components and generate the answer. We construct a Freestyle Multilingual Image Question Answering (FM-IQA) dataset to train and evaluate our mQA model. It contains over 150,000 images and 310,000 freestyle Chinese question-answer pairs and their English translations. The quality of the generated answers of our mQA model on this dataset is evaluated by human judges through a Turing Test. Specifically, we mix the answers provided by humans and our model. The human judges need to distinguish our model from the human. They will also provide a score (i.e. 0, 1, 2, the larger the better) indicating the quality of the answer. We propose strategies to monitor the quality of this evaluation process. The experiments show that in 64.7% of cases, the human judges cannot distinguish our model from humans. The average score is 1.454 (1.918 for human). The details of this work, including the FM-IQA dataset, can be found on the project page: http://idl.baidu.com/FM-IQA.html.
function. To lower down the risk of overfitting, we allow the weight sharing of the word embedding layer between the LSTMs in the first and third components. We also adopt the transposed weight sharing scheme as proposed in [25], which allows the weight sharing between word embedding layer and the fully connected Softmax layer.
To train our method, we construct a large-scale Freestyle Multilingual Image Question Answering dataset 1 (FM-IQA, see details in Section 4) based on the MS COCO dataset [21]. The current version of the dataset contains 158,392 images with 316,193 Chinese question-answer pairs and their corresponding English translations. 2 To diversify the annotations, the annotators are allowed to raise any question related to the content of the image. We propose strategies to monitor the quality of the annotations. This dataset contains a wide range of AI related questions, such as action recognition (e.g., "Is the man trying to buy vegetables?"), object recognition (e.g., "What is there in yellow?"), positions and interactions among objects in the image (e.g. "Where is the kitty?") and reasoning based on commonsense and visual content (e.g. "Why does the bus park here?", see last column of Figure 3).
Because of the variability of the freestyle question-answer pairs, it is hard to accurately evaluate the method with automatic metrics. We conduct a Visual Turing Test [38] using human judges. Specifically, we mix the question-answer pairs generated by our model with the same set of questionanswer pairs labeled by annotators. The human judges need to determine whether the answer is given by a model or a human. In addition, we also ask them to give a score of 0 (i.e. wrong), 1 (i.e. partially correct), or 2 (i.e. correct). The results show that our mQA model passes 64.7% of this test (treated as answers of a human) and the average score is 1.454. In the discussion, we analyze the failure cases of our model and show that combined with the m-RNN [24] model, our model can automatically ask a question about an image and answer that question.

The paper was fairly clear overall. As stated above, my main concern is that the VTT evaluation might produce very different results from one set of evaluators to another, as is a danger for any such manual evaluation. The scale of resources needed to reproduce it makes it somewhat inaccessible to academic groups. A more detailed experimental protocol, including instructions given to examiners, should be provided.Aside from that, there are a few few questions/comments for the authors to consider:* The first part of the title is a little weird. The main/best contribution of the paper is the model. Something more standard might be better.* Section 3.1 explains the model hyperparameters and architecture, but the selection mechanism was not specified? Why did you use 400 memory cells? Why were the weight matrices of LSTM(Q) and LSTM(A) not shared in the first components?* 'where "+" denotes element-wise addition" <- as opposed to what??* Are the top layers of the ConvNet fixed after separate training, or only the convolutional layers?* Section 3.2 <- in neural language modelling, a common strategy is to (counter-intuitively, perhaps) decouple input and output embeddings, as representations for the purpose of predicting the next symbol may need to encode different information as those modelling the context of the prediction. If tying these matrices (which is also done in many models) provided better validation errors, this should be clearly stated. Otherwise, I'd recommend dropping some of the hand-wavvy fluffiness from this section.* Section 3.3 <- The initial learning rate is 1 and we decrease it by a factor of 10 for every epoch of the data" <- this sounds massively hacky. How was this procedure selected? Surely not against test results, right?* "We stop training when the loss does not decrease within three epochs" <- also hacky. How was this procedure selected? Why not use loss on a validation set and some form of patience/early stopping?* For VTT in section 5.1, what is variance in annotator scoring of models? Is there some form of inter-annotator agreement you could report?Overall, the idea for the model is not bad, but resembles many LSTM/RNN based QA or sequence to sequence models. My main concern with this paper is that many aspects of the model design and training procedure seem ill-motivated, and make the reported results somewhat hard to judge. The Visual Turing Test is also a problematic means of evaluation, as discussed above. I'd like to see these issues substantially addressed before recommending acceptance.


Spectral Representations for Convolutional Neural Networks
Discrete Fourier transforms provide a significant speedup in the computation of convolutions in deep learning. In this work, we demonstrate that, beyond its advantages for efficient computation, the spectral domain also provides a powerful representation in which to model and train convolutional neural networks (CNNs). We employ spectral representations to introduce a number of innovations to CNN design. First, we propose spectral pooling, which performs dimensionality reduction by truncating the representation in the frequency domain. This approach preserves considerably more information per parameter than other pooling strategies and enables flexibility in the choice of pooling output dimensionality. This representation also enables a new form of stochastic regularization by randomized modification of resolution. We show that these methods achieve competitive results on classification and approximation tasks, without using any dropout or max-pooling. Finally, we demonstrate the effectiveness of complex-coefficient spectral parameterization of convolutional filters. While this leaves the underlying model unchanged, it results in a representation that greatly facilitates optimization. We observe on a variety of popular CNN configurations that this leads to significantly faster convergence during training.

Comments/questions about spectral pooling: - L212: "First [spectral pooling] representation maintains more information for the same number of degrees of freedom."-- I disagree with this statement.The amount of information in an image or its DFT is the same, after all it is just a linear transformation of the data.More information is present because the output of spectral pooling is larger than typical max/avg-pooling. In fact, avg-pooling is very similar to spectral pooling: it's a moving average filter (a different low-pass filter) followed by downsampling.The same amount of info could be kept using max/avg-pooling by using overlapping windows (also related to the claim on L250 that max-pooling has a large reduction of dimensionality). - maxpooling is translation invariant, while spectral pooling is not translation invariant (translation is encoded in the phase of the DFT).Supposedly translation invariance helps in recognition tasks.Does spectral pooling get similar results to max-pooling on object recognition tasks (e.g., ImageNet tested in Fig 4a)? - given the similarity to avg-pooling, it would be nice to see a comparison on CIFAR.Comments/questions about spectral parameterization: - for spectral parameterization, it would be interesting to see whether the re-parameterization allows better learning for smaller training set sizes. - L301 - Does the reparameterization affect the regularization of the weights?I suppose not, since it is based on a unitary linear transform.It is worth mentioning this. - L319 - unitary transform is also a condition for the training to be the same. Otherwise regularization of the weights is affected. - L424-425 - The experiments show that fewer iterations are needed for the spectral representation.But this ignores the time-cost for calculating the DFT and inverse DFT.What is the speed-up in terms of actual clock time?


3D Object Proposals for Accurate Object Class Detection
The goal of this paper is to generate high-quality 3D object proposals in the context of autonomous driving. Our method exploits stereo imagery to place proposals in the form of 3D bounding boxes. We formulate the problem as minimizing an energy function encoding object size priors, ground plane as well as several depth informed features that reason about free space, point cloud densities and distance to the ground. Our experiments show significant performance gains over existing RGB and RGB-D object proposal methods on the challenging KITTI benchmark. Combined with convolutional neural net (CNN) scoring, our approach outperforms all existing results on all three KITTI object classes.

-- Summary --This paper addresses the problem of generating 3D object proposals given a stereo image pair from an autonomous driving vehicle. The paper proposes a set of features for a 3D cuboid over a point cloud and ground plane derived from the stereo image pair. The features include point cloud density, free space, object height prior, and object height relative to its surroundings. Note that the features are dependant on knowledge of the object class (other "objectness" proposal methods are agnostic to the object class). A structural SVM is trained to predict the "objectness" of the 3D cuboid proposal. Bounding box and orientation regression is performed for the final object detection.The approach is evaluated on the KITTI benchmark, and the approach outperforms competing baselines for oracle recall and object detection. -- Comments --Positives: Generating good object proposals has shown to be an important step for object detection.While most prior work has focused on 2D still images, this paper focuses on 3D data, which has arguably received less attention. The approach outperforms baselines in this space (including approaches informed by 3D), which also translates to improved performance for object detection.The paper is reasonably clear to get the ideas of the approach and cites well prior art in this space.Negatives: The main ideas behind the paper may be perceived as incremental from a machine learning perspective.I think the boost in performance would be appreciated more by the 3D vision community.Detailed comments:+ Section 3.4: I'm not clear why a structured SVM is used here. Section 3.1 introduced a feature vector, and it seems that only a linear classifier is needed.I'd like to see how performance of a linear SVM with hard negative mining would perform.+ Prior work in generating object proposals are object class agnostic.I'd like to see performance where object class information is not used (e.g. remove object height from feature vector).  + Line 107: It would be good to clarify that the input is a calibrated stereo image pair since the approach relies on metric information. Also perhaps mention the dependency on ground plane estimation.+ Paragraph starting Line 142: What is the parameterization of t? The indices for t and \theta are not needed.Finally, it may be good to mention that it is assumed the boxes live on the ground and/or its base is parallel to the ground plane.+ Equation (1): It may be good to make explicit that each \phi depends on x.+ Equation (4): Why not "max" over the heights?+ Line 194: I think this potential assumes objects are well segmented in 3D (i.e. no overlap or intersection).Is this a valid assumption?+ Line 246: Please provide more details on the ground plane estimation, as it is not reproducible as described.+ Line 255: NMS => non maximum suppression+ Section 3.4: What is the setting of C?Also, is "y" supposed to be bold?


Differentially Private Learning of Structured Discrete Distributions
We investigate the problem of learning an unknown probability distribution over a discrete population from random samples. Our goal is to design efficient algorithms that simultaneously achieve low error in total variation norm while guaranteeing Differential Privacy to the individuals of the population. We describe a general approach that yields near sample-optimal and computationally efficient differentially private estimators for a wide range of well-studied and natural distribution families. Our theoretical results show that for a wide variety of structured distributions there exist private estimation algorithms that are nearly as efficient-both in terms of sample size and running time-as their non-private counterparts. We complement our theoretical guarantees with an experimental evaluation. Our experiments illustrate the speed and accuracy of our private estimators on both synthetic mixture models and a large public data set.

This paper studies the problem of learning discrete probability distributions over large finite sets under privacy constraints. The contributions are split into three parts: a theoretical study with formal accuracy, privacy, and computational guarantees; a fast empirical algorithm with privacy guarantees; and, a set of experiments on synthetic data and a real dataset.Broadly speaking, the main idea of the paper is to leverage previous work on algorithms for learning adaptive one-dimensional histograms over large sets and combine them with noise-generating procedures to obtain private algorithms. This is an important theoretical problem, and the authors manage to show in Section 3 that privacy can be achieved at a very reasonable price in terms of statistical accuracy. The techniques involved in Theorems 1 and 2 are not especially novel: they rely on the Laplace mechanism and a simple reduction. Theorem 3 is more interesting, but two important details were missing from the original submission. These were provided in the rebuttal and, though they are relatively simple, my suggestion is the authors incorporate them in the supplementary material in further versions of the paper.The practical impact of the contributions presented in the paper might be limited. The authors mention several families of univariate discrete distributions which can be agnostically learned using their techniques. These are quite broad, and seem obviously related to the cases where a data analyst would benefit from a histogram representation of the target distribution. However, the one-dimensional assumptions seems a rather strong assumption for many practical applications. If one had access to samples from a high-dimensional discrete distribution, it does not seem possible that the algorithms presented in the paper can be useful in extracting relevant information about the geometry of such distribution, since different projections to a one-dimensional setting would probably yield quite different histograms. It would be nice if the authors commented on these limitations and whether there are obvious extensions of their techniques for such cases, or on the other hand one would need completely different methods.*** Detalied Comments and Typos ***- In general, I think the material in Section 3 would benefit from some restructuring where algorithms and analyses are decoupled; after all, statements of the form "there exists an algorithm for doing that and this" are less satisfying that statements about a clearly specified algorithm.- According to Definition 1, an (alpha,beta)-agnostic learning algorithm produces a "hypothesis distribution h". However, the output of the algorithm analyzed in Theorem 1 might not be a distribution due to the random perturbation. Therefore, this algorithm is not strictly an (alpha,beta)-agnostic learning algorithm in the sense of Definition 1. This mismatch should be discussed and/or fixed.- Citations for the results mentioned in Lines 60-62 are missing.- When Theorems 4 and 5 are cited in Line 183, please say they can be found in the supplementary material. Though most theory conferences accept long papers describing their essential contributions in the first pages, NIPS still has a clear division between main paper and supplementary material, and the former is supposed to be self-contained.- Lemma 1 should say that the algorithm in the thesis is also private. Furthermore, the relevance of requiring in the hypothesis of the lemma that the first algorithm has a particular sample complexity is not entirely clear.- Non-increasing -> Non-decreasing (Line 240)- \hat{P}_n -> \hat{p}_n (Line 240)- distance Combining (Line 215)- curve For example (Line 251)


Probabilistic Curve Learning: Coulomb Repulsion and the Electrostatic Gaussian Process
Learning of low dimensional structure in multidimensional data is a canonical problem in machine learning. One common approach is to suppose that the observed data are close to a lower-dimensional smooth manifold. There are a rich variety of manifold learning methods available, which allow mapping of data points to the manifold. However, there is a clear lack of probabilistic methods that allow learning of the manifold along with the generative distribution of the observed data. The best attempt is the Gaussian process latent variable model (GP-LVM), but identifiability issues lead to poor performance. We solve these issues by proposing a novel Coulomb repulsive process (Corp) for locations of points on the manifold, inspired by physical models of electrostatic interactions among particles. Combining this process with a GP prior for the mapping function yields a novel electrostatic GP (electroGP) process. Focusing on the simple case of a one-dimensional manifold, we develop efficient inference algorithms, and illustrate substantially improved performance in a variety of experiments including filling in missing frames in video.

This paper proposes a new method to learn a mapping function from a d-dimensional data point, y, to one-dimensional embedding, x. The paper extends the standard GP to a hierarchical model by incorporating what they call Coulomb repulsive process (Corp). It first generates samples of the one-dimensional x and then generates observable d-dim data points through a GP, whose covariance is constrained to be diagonal. Intuitively, Corp is used to ensure reasonable inter-point distance in the embedding space through the repulsive force in [0,1] to make the distance of 0.5 is the least stressful state. The equi-distance prior in [0,1] looks like a natural choice when the original distribution for y forms a compact single continuous manifold in the original space. To find the embeddings, they propose an alternating algorithm, where GP parameters and the embeddings are solved alternatingly. To the best of my knowledge, the use of Corp in the context of GP-LVM is new. The proposed approach looks interesting enough to accept the paper. I personally think that the paper can be improved by describing theoretical aspects of the model more carefully. It looks obvious to me that the model still has major limitations such as the strong dependency on the initial guess (x_0). I feel that the authors are overselling the model by making too strong statements on the practical utility of the model. For example, in practice, we all know that careful pre-process is required to make "your favorite distance-preserving manifold" work. What if the distribution looks multi-modal? What if x_0 misses to capture the major features of the original distribution? Your claim in Sec. 5 is scientifically sound? I know recently more and more reviewers tend to demand extensive experimental comparison. However, for papers like this, the most important thing is to build a common understanding with the audience on the major features of the model and wide applicability of the model. I hope the camera-ready version gets more understandable and sophisticated by addressing it. Minor comments. - Do not mix between bold and non-bold notation to represent vectors. Use boldface for e.g. \mu in Introduction. - Introduction of Corp in Sec.2 can be improved. This is basically to introduce the repulsive force. Don't make it overly mathematical. - If you are to use r=1, you don't have to show r=20. You can show illustrative figures instead. - Dependency on x_0 should be discussed in depth. If you cannot, do not overclaim the practical utility. - Also, describe practical limitations in addition to practical advantages. - Sec. 5 sounds like a frivolous sales talk. This is a good paper as is. You don't have to worry about the "more experiment!" criticism. I'd like you to make more sophisticated academic comments.   - Redraw Fig.2. Visibility is extremely bad.


Efficient Non-greedy Optimization of Decision Trees
Decision trees and randomized forests are widely used in computer vision and machine learning. Standard algorithms for decision tree induction optimize the split functions one node at a time according to some splitting criteria. This greedy procedure often leads to suboptimal trees. In this paper, we present an algorithm for optimizing the split functions at all levels of the tree jointly with the leaf parameters, based on a global objective. We show that the problem of finding optimal linear-combination (oblique) splits for decision trees is related to structured prediction with latent variables, and we formulate a convex-concave upper bound on the tree's empirical loss. Computing the gradient of the proposed surrogate objective with respect to each training exemplar is O(d 2 ), where d is the tree depth, and thus training deep trees is feasible. The use of stochastic gradient descent for optimization enables effective training with large datasets. Experiments on several classification benchmarks demonstrate that the resulting non-greedy decision trees outperform greedy decision tree baselines.

The submission proposes an algorithm for global optimization of decision trees based on a reformulation of the problem as (latent) structured prediction. The latent structured variables are sign vectors encoding the decisions made at each internal node in the tree.A non-differentiable and non-convex loss is proposed, which is then approximated by a differentiable (but still non-convex) upper bound.Minimizing this loss via SGD entails solving a sequence of loss-augmented inference problems, for which efficient algorithms are given. Experiments are shown comparing this method to a standard greedy training method.The key contributions of the work are casting the global DT optimization as structured prediction and devising efficient algorithms to optimize the objective.The way this is done is novel and interesting, and raises interesting questions.I have some concerns, however, about some practical considerations-especially concerning the motivation for the method and how significant the performance gains over naive methods are.Considered purely as a method to globally optimize DTs, the method is intriguing in several ways.The first advantage gained by casting the optimization as structured prediction is being able to perform SGD by solving a sequence of loss-augmented optimization problems that leverage the special structure of the problem.However, I think the most interesting result of this approach is that the gradient updates are sparse-if I'm not mistaken, each gradient step involves changing only one node's weight vector.The submission presents this mainly as a computational advantage, but I believe that this property raises more intriguing possibilities that are not really acknowledged in the paper.One could imagine optimizing over an infinitely deep tree by initially setting all weights to zero, and then gradually growing the tree by making more weights nonzero, thus letting the model's complexity grow naturally to fit the data.This raises interesting questions: for example, would such a method prove to be equivalent to, or a variation of, any known greedy strategy?This may be the case, if it turns out that it is always preferable to introduce a new node rather than revise an old node's weights.Or, with appropriate regularization, would such a method instead converge to a tree with a limited number of nodes (as Fig. 2 might suggest)?I think this issue cuts to the core of what makes the proposed method worthwhile compared to naive DTs, and deserves perhaps more careful analysis than is currently offered by the paper.For instance, a critical issue left ambiguous is this: what happens if some node's weight is equal to zero?The discussion on page 3 implies that we take the right branch, which seems like a dubious choice.It seems like it would be preferable to output a decision at this node; later, we might decide to make this weight nonzero, corresponding to splitting the node.Precisely defining what happens here is important, because it has a direct analog to pruning strategies for greedy training.I have one significant practical concern regarding the motivation for the work.Namely, I would say that plain DTs are useful mainly to the extent that they are interpretable; if the interpretability requirement is dropped, then it is usually better to use other methods, which generally offer much better performance.As stated, I would expect the method to produce non-interpretable results, since the L2 regularization would produce dense weight vectors.Although switching to L1 regularization might help, it is possible that performance would drop if we were forced to regularize to the point where exactly 1-sparse splits were produced.So, this would definitely need to be tested.My other main concern is that the experiments don't go into enough detail regarding other baselines.No details are given as to which pruning strategies were employed for the naive approach.Since a range of regularization parameters were tried for the proposed method, I would expect the naive method to be similarly tried with a few different pruning strategies and/or parameters.I think it would also be fair to try some other trivial strategies for global optimization, such as training a greedy method with random subsets of the training data, and choosing the one that minimizes the desired loss on the full training or validation set.Regarding clarity, the paper is reasonably well-written. My only comment is that the parts describing inference could be a bit clearer.Section 6.2, for example, could be summarized by saying that the objective is first maximized for all g corresponding to a given leaf, which is then maximized over all leaves.Section 6.4 could also be a little more explicit.In summary, I think this is a very clever approach that raises some very interesting questions.However, it is unclear whether this is a practically useful method at this point, both due to the aforementioned interpretability issue (why use plain DTs in the first place, if they are not interpretable?) and due to a lack of details in the experiments.PS: although the title mentions forests, the paper does not address this case.POST-REBUTTAL COMMENTSI still think that the case of zero weights is a critical case that deserves further analysis, for the reasons I brought up in my review.Reading the rebuttal, it sounds like the weights are probably initialized randomly, which is why this issue doesn't seem to come up in practice.However, I have a feeling that what would really make this method interesting would be the case where the weights are initially zero and are gradually increased.I encourage the authors to consider this direction.


Anytime Influence Bounds and the Explosive Behavior of Continuous-Time Diffusion Networks
The paper studies transition phenomena in information cascades observed along a diffusion process over some graph. We introduce the Laplace Hazard matrix and show that its spectral radius fully characterizes the dynamics of the contagion both in terms of influence and of explosion time. Using this concept, we prove tight non-asymptotic bounds for the influence of a set of nodes, and we also provide an in-depth analysis of the critical time after which the contagion becomes super-critical. Our contributions include formal definitions and tight lower bounds of critical explosion time. We illustrate the relevance of our theoretical results through several examples of information cascades used in epidemiology and viral marketing models. Finally, we provide a series of numerical experiments for various types of networks which confirm the tightness of the theoretical bounds.

- It would to point out that p_ij(t) are actually subprobabilities, since \int_{0}^{\infty} p_{ij}(t) < 1.- Daneshmand et al. [ICML '14] also use the naming "hazard matrix" to denote a different quantity on a set of observed cascades. It would be useful to point out this, highlighting the differences.- In section 4.1., the authors do not explicitly state that \alpha_{ij} in [0, 1). I think this is necessary for p_{ij}(t) to be a (sub)probability.- The writing of the first paragraph of the introduction could be heavily improved.- The authors refer to [3] as early work on influence maximization. [3] does not tackle the influence maximization problem, but a different one.- The proof of Lemma 3 could be described more explicitly. In particular, explicitly relate f(x), g(x) and \mu(x) to the actual functions in the problem setting. This would help a reader to understand the definition of "right positive correlation", which is never formally written.- In Eq. (11) in the proof of Lemma 4, the authors should explicitly write what is a and x in ln(1 - ax) >= ln (1-a)x to help the reader.- In Eq. 12, should f_i(s) on the right side not be f_j(s)?- There is some very recent work that aims to establish the theoretical foundations of influence models. In particular, there are sample complexity results on the recovery of influence models:Daneshmand et al., ICML 2014 Pouget-Abadie et al., ICML 2015Although different to this work, it would help the reader understanding that there is already an on-going interest on the theoretical analysis of such models.


A class of network models recoverable by spectral clustering
Finding communities in networks is a problem that remains difficult, in spite of the amount of attention it has recently received. The Stochastic Block-Model (SBM) is a generative model for graphs with "communities" for which, because of its simplicity, the theoretical understanding has advanced fast in recent years. In particular, there have been various results showing that simple versions of spectral clustering using the Normalized Laplacian of the graph can recover the communities almost perfectly with high probability. Here we show that essentially the same algorithm used for the SBM and for its extension called Degree-Corrected SBM, works on a wider class of Block-Models, which we call Preference Frame Models, with essentially the same guarantees. Moreover, the parametrization we introduce clearly exhibits the free parameters needed to specify this class of models, and results in bounds that expose with more clarity the parameters that control the recovery error in this model class.

The authors present an analysis that establishes the applicability of the classical spectral clustering algorithm on a wide variety of generalized block stochastic models (various communities, sizes, transition probabilities, etc). Their results consider the weak recovery regime, where "most of the nodes" are clustered correctly; certainly a more realistic setup.The ideas presented seem novel, however I am not an expert in the area. The paper is well-written, however it is extremely heavy in notation, making the technical result inaccessible. One example where the technical discussion is confusing: It is not clear how parameter kappa should be thought of. For example, what is a maximum degree bound with a reasonable kappa, where the probability bounds are meaningful?Also, there are many relevant and important recent results on the subject, and a comparison against them will be very useful - Results by Mossel, Neeman, Sly et al. http://arxiv.org/abs/1407.1591 http://arxiv.org/abs/1404.6325 http://arxiv.org/abs/1311.4115 - Results by Abbe, Bandeira, Sandon et al. http://arxiv.org/abs/1506.03729 http://arxiv.org/abs/1503.00609 http://arxiv.org/abs/1405.3267 - Results by Lelarge, Massoulie et al. http://arxiv.org/abs/1506.08621 http://arxiv.org/abs/1406.6897 Finally, a minor comment: the authors mention at some point that: "there are re strong reasons to believe that the KPFM is the most general class for which community recovery by direct spectral clustering is possible." Why is this true? Overall, it seems that the contribution could be a valuable addition to the literature of community detection on SBMs, however it is not clear how it compares against the state of the art spectral methods for extended SBMs as noted above. Some typos: - [19,18,9].In particular -> space missing- overfill in line 090- "degree model"[8]-> space missing


Efficient Learning by Directed Acyclic Graph For Resource Constrained Prediction
We study the problem of reducing test-time acquisition costs in classification systems. Our goal is to learn decision rules that adaptively select sensors for each example as necessary to make a confident prediction. We model our system as a directed acyclic graph (DAG) where internal nodes correspond to sensor subsets and decision functions at each node choose whether to acquire a new sensor or classify using the available measurements. This problem can be posed as an empirical risk minimization over training data. Rather than jointly optimizing such a highly coupled and non-convex problem over all decision nodes, we propose an efficient algorithm motivated by dynamic programming. We learn node policies in the DAG by reducing the global objective to a series of cost sensitive learning problems. Our approach is computationally efficient and has proven guarantees of convergence to the optimal system for a fixed architecture. In addition, we present an extension to map other budgeted learning problems with large number of sensors to our DAG architecture and demonstrate empirical performance exceeding state-of-the-art algorithms for data composed of both few and many sensors.

Resource constraint prediction has to select the smallest/ cheapest set of sensors in order to make accurate decisions. A new approach to this problem is proposed in this paper where the authors basically define a deterministic Markov decision process and learn its policy applying cost sensitive learning in every state of the MDP. Given a new testing example to classify, the policy has to decidewhether a new sensor should be queried (if yes, which sensor), or whether the prediction should be made with the current set of sensors measured.In line 44, the authors mention the "expected budget constraint". Please be clear about that because your paper is not budgeted learning; budgeted learning is more challenging, please see, e.g. papers of Russell Greiner (University of Alberta) on this topic.The DAG used in this paper defines an MDP. Perhaps a closer connection with MDPs could be made, e.g., you could mention that this is a special MDP that can be solved efficiently using your method. Also, it would be good to explain what are the key techniques/assumptions that allow solving this MDP efficiently. Putting your method in a broader MDP context would be very useful.Line 122, a quotation mark is missing.I have some problems understanding equation (5). If (s_j,s_k) is and edge, how can I read from the first case in the equation (5) that this represents the cost of acquiring a new sensor? Please explain. Shouldn't there be $t\in s_k \ s_j$ ?One of my key concerns is R that is used in Lemma 2.1. In [2], the authors use importance-weighted binary classification. This is basically classification that considers misclassification cost. What I am missing in Lemma 2.1 and its proof is a clear link between misclassification cost in [2] and the cost of features (or sets of features) in Lemma 2.1. Please explain. I have another problem with lemma 2.1. If node j leads to a leaf then the cost of sensors is zero because there is no need to pay for any sensors; why then, the cost of an edge j->k is considered in the proof of lemma 2.1.Vectors w are not defined in eq. 8. Well, (2) in Alg. 1 says "construct the weight vector w of edge costs per action". Please explain how to do that.It looks that the key element in the analysis in section 2.3 is to optimise at each stage over all examples in the training set. It sounded weird to me initially, but I believe that that's correct. My second main concern are Lemma 3.1 and theorem 3.2. As long as sub-modularity in Lemma 3.1 is correct I believe, I am not sure why we should not consider sub-modularity in the theorem 3.2 as well. The risk in the theorem is monotonically decreasing indeed, but we should really talk about diminishing returns which don't exist because adding a new feature f_1 may not change anything until we add another features f_2 when those two features interact. So, there are not diminishing returns and sub-modularity does not apply. Additionally, in the proof of theorem 3.2 the authors consider adding a new subset (line 34 in the appendix), but doing that will eventually violate the constraint in e.q. (1) in the appendix. I believe that this is another reason why this problem is not sub-modular, and the greedy approach can be very bad.Also, "the reward" in lemma 3.1 shouldbe explained.


Testing Closeness With Unequal Sized Samples
We consider the problem of testing whether two unequal-sized samples were drawn from identical distributions, versus distributions that differ significantly. Specifically, given a target error parameter ε > 0, m 1 independent draws from an unknown distribution p with discrete support, and m 2 draws from an unknown distribution q of discrete support, we describe a test for distinguishing the case that p = q from the case that ||p − q|| 1 ≥ ε. If p and q are supported on at most n elements, then our test is successful with high probability provided m 1 ≥ n 2/3 /ε 4/3 and m 2 = Ω max{
We show that this tradeoff is information theoretically optimal throughout this range in the dependencies on all parameters, n, m 1 , and ε, to constant factors for worst-case distributions. As a consequence, we obtain an algorithm for estimating the mixing time of a Markov chain on n states up to a log n factor that usesÕ(n 3/2 τ mix ) queries to a "next node" oracle. The core of our testing algorithm is a relatively simple statistic that seems to perform well in practice, both on synthetic and on natural language data. We believe that this statistic might prove to be a useful primitive within larger machine learning and natural language processing systems.

The paper considers the problem of testing if two samples are from identical or very different distributions. This is an interesting question since in many practical scenarios, one of the data sources might be more precious/hard to obtain than the other. This paper characterizes the precise tradeoff between the sample lengths required to solve this problem, and also nail down the dependence on the accuracy, epsilon up to constant factors. From my understanding, the actual upper bounds (Achievability) bounds of [3] are actually nearly optimal (up to logarithmic terms), and this paper removes all spurious log-factors and reduces the exponent of epsilon to the optimal value. Even if the upper bound is a small improvement in terms of the number of samples, it still requires new insights and better testers, something the authors have managed to achieve. The lower bounds of the paper are similar to those in [8], [28] and matches the upper bounds shown up to constant factors.  Here are some specific comments on the paper:In Theorem 1, the authors require eps \ge n^{-1/12}. Is this because there is an additional term with higher powers ofepsilon, which disappears when this happens. If yes, it would be nice if the main result contains the entire complexity, and put the conditional result as a corollary. 98: uniform distribution minimizes the number of domain elements appearing more than once. This is slightly incorrect. Please rephrase as it maximizes the number of elements appearing once.108-112: I got a little confused with using n for domain size and then for # of samples when describing [1,2]. 217: should m1 be something else. I am not sure if it is n^{1-\gamma}. 270-278: the authors can give some further intuition and requirement of the additional step needed.321: Pois (O(1/\sqrt n)) ... should it be Pois(O(\sqrt n)).I am not fully sure what conclusions to draw from the experiments. I might be missing something elementary here. The paper is well written, and is an enjoyable read.


Learning Large-Scale Poisson DAG Models based on OverDispersion Scoring
In this paper, we address the question of identifiability and learning algorithms for large-scale Poisson Directed Acyclic Graphical (DAG) models. We define general Poisson DAG models as models where each node is a Poisson random variable with rate parameter depending on the values of the parents in the underlying DAG. First, we prove that Poisson DAG models are identifiable from observational data, and present a polynomial-time algorithm that learns the Poisson DAG model under suitable regularity conditions. The main idea behind our algorithm is based on overdispersion, in that variables that are conditionally Poisson are overdispersed relative to variables that are marginally Poisson. Our algorithms exploits overdispersion along with methods for learning sparse Poisson undirected graphical models for faster computation. We provide both theoretical guarantees and simulation results for both small and large-scale DAGs.

The paper considers the inference of the structure of the DAG underlying a Poisson directed graphical model. The proposed strategy relies on the fact that nodes with parents are conditionally Poisson, whereas nodes without parents are marginally Poisson. The paper aims at distinguishing the former from the later based on an over-dispersion criterion.    I do like the principle of the approach but the construction is only partly convincing. My main concerns are the following:   1 - I am not completely clear about the proof if identifiability. The proof of Thm 3.1. starts with "assume that the true causal ordering \pi^* is {1, 2, ...p}". To me, in a general oriented graphical model, there may be several true causal orderings. As the proof relies on induction, this should be clarified.   2 - The exploration of the DAG space is indeed a critical task and the authors simply suggest to use GLMlasso at each step to reduce the search space. The objective function of the GLMlasso is not necessarily consistent with the overdispersion criterion, so the global procedure is a bit heuristic.   3 - In the simulation study, I did not understand why all regression coefficients $\theta_{jk}$ are chosen to be negative (between -1 and -.7). This is allthemore surprising as the authors states that Yang requires this negativity constraint, while they do not. Does this choice make the problem easier?   4 - As for the performances, indeed they are good will large $n$, even for fairly large $p$, by I have been very surprised by the bad performances for $p = 10$ and $n = 500$ which seems to be a rather comfortable case. Do the authors have an idea why they fail in such a, apparently, easy situation?        Minor remarks:   1 - In the proof of Thm 3.1 should "Pa(j)/{1, 2, ..., m}" be "Pa(j)\{1, 2, ..., m}"?   2 - The threshold $c_0$ is likely to have some influence on the results, which is not explored in the simulation study.


On the Convergence of Stochastic Gradient MCMC Algorithms with High-Order Integrators
Recent advances in Bayesian learning with large-scale data have witnessed emergence of stochastic gradient MCMC algorithms (SG-MCMC), such as stochastic gradient Langevin dynamics (SGLD), stochastic gradient Hamiltonian MCMC (SGHMC), and the stochastic gradient thermostat. While finite-time convergence properties of the SGLD with a 1st-order Euler integrator have recently been studied, corresponding theory for general SG-MCMCs has not been explored. In this paper we consider general SG-MCMCs with high-order integrators, and develop theory to analyze finite-time convergence properties and their asymptotic invariant measures. Our theoretical results show faster convergence rates and more accurate invariant measures for SG-MCMCs with higher-order integrators. For example, with the proposed efficient 2nd-order symmetric splitting integrator, the mean square error (MSE) of the posterior average for the SGHMC achieves an optimal convergence rate of L −4/5 at L iterations, compared to L −2/3 for the SGHMC and SGLD with 1st-order Euler integrators. Furthermore, convergence results of decreasing-step-size SG-MCMCs are also developed, with the same convergence rates as their fixed-step-size counterparts for a specific decreasing sequence. Experiments on both synthetic and real datasets verify our theory, and show advantages of the proposed method in two large-scale real applications.

The paper presents a convergence analysis for stochastic gradient Langevin algorithms. The results are very encouraging, since they show that it is possible to improve the rate of convergence by using high-order integrators. Indeed, previous convergence analysis for the SGLD algorithm have been rather discouraging, since the best possible rate in MSE has been shown to be L^{-2/3} (compared to the standard L^{-1} Monte Carlo rate). Guided by the new results, the authors also propose a 2nd order slitting integrator which attains a rate of L^{-4/5}, i.e. much closer to the standard Monte Carlo rate than SGLD (but still not quite there).I admit to not having checked the proofs in the supplementary material in detail, but the theory looks solid. I have a few questions and suggestions for improvement:1. You state in the introduction that you carry out the analysis for general SG-MCMC algorithms, but I do not understand what you mean by this. Is not the analysis in fact rather tailored to numerically integrated stochastic versions of the second order Langevin system? In particular, the way you define $\tilde L_l$ and $\Delta V_l$ seems to be specific to the 2nd order Langevin system. For instance, in the SG-HMC algorithm by Chen et al. 2014 [1] they use different scaling matrices for the drift and the noise terms, and it thus seems to me that this algorithm does not quite fit into the framework that you consider. Does your analysis apply to this algorithm? Likewise, the SG-NHT algorithm by Ding et al. 2014 [2] is based on a different, extended, system and it is not completely obvious that this fits into your framework. I think that you should be more specific about the precise requirements that you have on the diffusion (1) for your results to hold.2. In the proposed integrator, systems B and O_l together define an Ornstein-Uhlenbeck process for p (\theta is kept fixed). Why do you propose to split this into B and O_l, rather than integrating exactly the OU process? (This would also lead to a symmetric integrator, ACA, where C=B+O_l.)3. I found Section 5 "large-scale machine learning applications" too brief to provide any valuable insights. I would suggest that you focus on one of the two applications in the main paper, and move the second application completely to the supplementary material.4. Can you comment on the possibility of using other (i.e., not splitting-based) high-order integrators for SG-HMC and SGLD? Are there any standard integrators that could be used to obtain a L^{-4/5} rate for SGLD?*** Minor points/typos:5. Line 121. Are you missing a minus sign in the definition of V_l (since \grad_\theta U comes with a negative sign in the definition of F)?6. Line 261. The condition should be $\sum_{l=1}^L h_l^2 / S_L^2 \rightarrow 0$, no?7. What is L for the fixed step-size results presented in Figure 2? Is it still 10^6 (as in Figure 1) or do you use a smaller value here?8. Line 397: Should be $L = 100$K and not $L = 100K$ (or even better, $L = 10^5$).


Stochastic Online Greedy Learning with Semi-bandit Feedbacks
The greedy algorithm is extensively studied in the field of combinatorial optimization for decades. In this paper, we address the online learning problem when the input to the greedy algorithm is stochastic with unknown parameters that have to be learned over time. We first propose the greedy regret and -quasi greedy regret as learning metrics comparing with the performance of offline greedy algorithm. We then propose two online greedy learning algorithms with semi-bandit feedbacks, which use multi-armed bandit and pure exploration bandit policies at each level of greedy learning, one for each of the regret metrics respectively. Both algorithms achieve O(log T ) problem-dependent regret bound (T being the time horizon) for a general class of combinatorial structures and reward functions that allow greedy solutions. We further show that the bound is tight in T and other problem instance parameters.

Strengths:1) Technical and well executed. Upper bounds and a matching lower bound.2) Pleasure to read.3) A general class of problems which is of interest in large-scale learning.Weaknesses:1) Positioning with respect to the prior work is insufficient and needs to be improved.2) Not particular novel and surprising, given the vast amount of work on related subjects in the past few years.Major comments:This paper cites most papers from the past few years on related subjects. However, it falls short in discussing details in which this work differs from the past work, such as the rates of regret bounds and the computational complexity of the proposed methods. This needs to be improved. This is my point of view. Matroids and polymatroids are generally regarded as the largest classes of problems where the greedy algorithm is guaranteed to be optimal. [23] and:   Kveton, Branislav, Wen, Zheng, Ashkan, Azin, and Valko, Michal. Learning to act greedily: Polymatroid semi-bandits. CoRR, abs/1405.7752, 2014.which I further refer to as [33], proposed optimal learning algorithms for these problems. The main difference in this work is that the optimized function is not modular and that the feasible set is more general. In this setting, the greedy algorithm is not guaranteed to be optimal, or even to be a good approximation. This should be highlighted. Nevertheless, this work allows for solving new and interesting learning problems, such as learning to maximize a submodular function subject to a cardinality constraint in the semi-bandit setting.In the notation of this paper, [23] and [33] prove O(n (1 / \Delta) \log n) upper bounds on the regret of their learning algorithms. Let's compare this to Theorem 3.1. Clearly W = O(n) and \Delta_{max} = O(m). Therefore, the upper bound in Theorem 3.1 is O(m^2 n (1 / \Delta^2) log n), a factor of m^2 (1 / \Delta) larger than the result of [23] on more specific problems. Perhaps this is not surprising, given how general the proposed framework is, but it should be mentioned that OG is less statistically efficient than OMM [23] on many practical problems, such as learning a minimum spanning tree. The authors should also discuss the space complexity of OG. As far as I can tell, it is linear in |F|, which is n \choose m when the feasible set is a uniform matroid of rank m. Therefore, OG is not space efficient on many practical problems. When the gaps are small, even lazy initialization (line 269) could lead to exploring all n \choose m arms. In comparison, the space complexity of OMM [23] on a uniform matroid of rank m is O(n).Finally, note that [24] prove a O(m n (1 / \Delta') \log n) upper bound on the regret of CombUCB1 in stochastic combinatorial semi-bandits, where \Delta' is arguably larger than \Delta in this paper. Therefore, the upper bound of [24] is a factor of m (1 / \Delta) smaller than that in Theorem 3.1. This upper bound holds for any linear objective on any feasible set, and even for approximation oracles. Therefore, CombUCB1 is more statistically efficient than OG in this setting. The above also indicates that the extra factors in Theorem 3.1 may be due to the non-linearity of the optimized objective. This is worth mentioning.Based on the above, the novelty of this work is mostly in learning suboptimal greedy policies for non-linear objectives. The proposed learning algorithms are not guaranteed to be space efficient. Therefore, the impact of this work on practice may be limited. To make it more significant, I suggest the authors include a good motivating example, which showcases the importance of learning good suboptimal greedy policies, which can be learned efficiently by either OG or LUCB. Experimental results on this problem would be nice and convince me that the class of problems in this paper is important and worth studying. The current motivating example is bad in the sense that this paper does not solve this particular problem.My last major comment is that the lower bound in Section 5 is not very clear. In particular, I am not sure how the authors conclude that:   \Delta + 0.5 = \Delta_{max} + o(1).It can be also argued that the case where \epsilon < \Delta is not very interesting.Minor comments:Line 128: It may be less ambiguous to write ordered sets as vectors and then abuse the set notation when needed. For instance, \sigma = (S_0, S_1, \dots, S_k) instead of \sigma = \{S_0, S_1, \dots, S_k\}.Line 191: "online algorithm that are" should read as "online algorithms that are".Line 192: "objective is to minimum" should read as "objective is to minimize".


The Consistency of Common Neighbors for Link Prediction in Stochastic Blockmodels
Link prediction and clustering are key problems for network-structured data. While spectral clustering has strong theoretical guarantees under the popular stochastic blockmodel formulation of networks, it can be expensive for large graphs. On the other hand, the heuristic of predicting links to nodes that share the most common neighbors with the query node is much fast, and works very well in practice. We show theoretically that the common neighbors heuristic can extract clusters with high probability when the graph is dense enough, and can do so even in sparser graphs with the addition of a "cleaning" step. Empirical results on simulated and real-world data support our conclusions.

Spectral clustering is a standard tool for identifying link prediction. However, it is expensive for large graphs. A popular heuristic is to consider common neighbors, which is a fast and performs well in practise. This paper proves the consistency of the common neighbor heuristic for different classes of graphs. The authors also propose a cleaning step that improves the quality for sparse graphs. The approach is studied empirically on simulated and real world data.This problem is very relevant and the theoretical analysis seems to be mathematically sound. However, the practical relevance remains unclear, since the experiments fail to explore all strengths and weaknesses: - The proposed algorithm depends on a number of unknown parameters. The authors fail to explain the significance of the link probability of the clusters and the cluster sizes. It is not clear how the algorithm performs if these parameters are not optimally chosen.- Depending on if the graph is semi-sparse or not, the authors propose to conduct a cleaning step and Figure 1 shows that in that scenario it improves recall and precision. Given that it is hard to check the sparsity in practise, it is also important to show how the cleaning step affects the quality when it is not appropriate. - Equal cluster sizes (\pi = n/K) seems to be a strong assumption. You mentioned that this assumption could be easily relaxed. Can you give an idea of how that would influence your analysis? I assume that relaxing this assumption introduces more parameters that are unknown and we need to make this assumption anyhow. I would expect also an experiment showing the quality if the cluster sizes are skewed.Section 3 and 4 are very confusing and a clear structure is missing; they introducing and mixing different concepts where it is unclear how they relate together and for what they are needed. E.g., - the formular (line 182-187) are detached from the text - how do the sections "current approaches", "common neighbors", "dependence", "dependence",... relate to each other?Minor: - Line 97 seems to be a copy&paste error (see Line 126) - Line 163: "eigenvectors of the graph" -> "eigenvectors of the graph Laplacian matrix" - please write out w.h.p (Line 23) and WLOG (Line 222) - Line 214: "when average degree" -> "when the average degree"  After rebuttal: ---------------- Thank you for addressing my concerns about the algorithm's parameters, when the cleaning step is needed, and the number of clusters. However, I find Section 3 and 4 still confusing. I adapted my score accordingly and would like to encourage the authors to clarify the structure and elaborate these issues in the paper.


Unlocking neural population non-stationarity using a hierarchical dynamics model
Neural population activity often exhibits rich variability. This variability can arise from single-neuron stochasticity, neural dynamics on short time-scales, as well as from modulations of neural firing properties on long time-scales, often referred to as neural non-stationarity. To better understand the nature of co-variability in neural circuits and their impact on cortical information processing, we introduce a hierarchical dynamics model that is able to capture both slow inter-trial modulations in firing rates as well as neural population dynamics. We derive a Bayesian Laplace propagation algorithm for joint inference of parameters and population states. On neural population recordings from primary visual cortex, we demonstrate that our model provides a better account of the structure of neural firing than stationary dynamics models.

This paper extends Poisson linear dynamical systems (PLDS) to account for the non-stationarity in neural spike trains.Their method (NPLDS) uses a hierarchical framework to find the latent variables for each trial, and also scale those latent variables multiplicatively for each trial.The latent variables are found with a linear dynamical system, and the inter-trial modulators are enforced to be smooth across trials with a Gaussian process.To fit the model, the authors devised the "Bayesian Laplacian" propagation and used an iterative procedure, which may be of interest to those outside the neuroscience field.The results are shown to be more predictive than the previous PLDS method, which suggests the added complexity helps performance.The results are still preliminary in some sense, as it is unclear how this method can be used to further understand population activity, but this method shows promise for other modelers.Quality:I found all aspects of the paper to have high quality (in terms of the work).I trust the results found, although I cannot confirm without testing the algorithm. Clarity:All text was well motivated and clear to follow.Please see below for small typos, etc. Originality:While the originality of the idea is not new, the method implementation is, and I believe of interest to other modelers. Significance:This paper is a nice step up from methods from the last NIPS 2014 conference, in which inter-trial variability was not considered.I believe this paper will be of interest to the modeling community, as the method is an interesting one to fit, and is a natural accept to NIPS.However, I question if this method will be of interest to the neuroscience community (specifically, how much the method will actually be used).The authors (and collaborators) will need to demonstrate the method is useful and returns interpretable results, as the method is too complicated to warrant the risk of spending weeks/months analyzing data with it.One interesting experiment would be to vary the level of anesthesia across trials, and see if this method's latents finds interpretable fluctuations.There is also no mention that the code will be released to the community; while this is not necessary for publication in NIPS, this would be in the authors' best interests.Personally, I would not use this method for two reasons: 1) the method has too many levels of complexity with various assumptions, which makes it tough to interpret the latents and 2) I would hesitate to play around with this model (i.e., spending a day throwing my data at it), as there are too many "knobs" to play with.Thus, I hope this method can make it into a journal paper with scientific results, and I would be much more excited about it! Notes while reading:- the intro was beautifully written; the logic was clear and well-founded 083 - instead of p neurons, perhaps you could use n to match other papers (subjective) 085 - let r = N, to follow other papers (subjective) 090 - I believe this equation can be written more concisely 093 - "d controls the mean firing rate of each cell" you need to make it clear this is across all trials (since later you say the mean firing rate changes across trials, and it's confusing 094 - "latent factors" do you mean x_t here, h(i), or both? 100 - give an example of u_t here.I also question the use of u_t...what if you would like to work with natural images, etc?If you are using cos, sin, and a binary variable, do you assume all your neurons are "simple"? Figure 1: you need to say the same stimulus is shown for each trial; do you ever reference figure 1 in the text?also I would suggesting changing "recording" to "trial"...recording is not used in the text 124 - "mean firing rates vary across trials" I tripped here, because I remembered d should be a fixed mean firing rate; see if you can clarify this more by explicitly saying h(i) 205 - "where used the vectorized" typo 214 - "obtaining the the approximate" typo 255 - it would be nice to define z earlier before referencing Figure 2, it took some time to figure out Fig 2 E...can you also think of the x-axis in terms of the number of trials? 298 - "units not used in the original study" --- why were they included here? 299 - say 4 s trials 300 - I find one problem with this model is that by using an LDS, you incur a "stimulus" signal which the experimenter must define.This limits the ability of the method to be applied to all types of data (e.g., natural images). 323 - you show RMSE here, why not use log-likelihood? is that approximate or tough to compute? Figure 4 - label both axes for every panel


The Pseudo-Dimension of Near-Optimal Auctions
This paper develops a general approach, rooted in statistical learning theory, to learning an approximately revenue-maximizing auction from data. We introduce t-level auctions to interpolate between simple auctions, such as welfare maximization with reserve prices, and optimal auctions, thereby balancing the competing demands of expressivity and simplicity. We prove that such auctions have small representation error, in the sense that for every product distribution F over bidders' valuations, there exists a t-level auction with small t and expected revenue close to optimal. We show that the set of t-level auctions has modest pseudodimension (for polynomial t) and therefore leads to small learning error. One consequence of our results is that, in arbitrary single-parameter settings, one can learn a mechanism with expected revenue arbitrarily close to optimal from a polynomial number of samples.

This paper addresses the problem of learning reserve prices that approximately maximize revenue, using sample draws from an unknown distribution over bidder valuations. The authors introduce t-level auctions, in which (roughly speaking) each bidder's bid space is effectively discretized into levels, and the bidder whose bid falls on the highest level wins and pays the lowest value that falls on its lowest level required to win. The authors bound the number of samples needed to find an approximately revenue-maximizing auction from all auctions in a set C (e.g., from the set of 10-level auctions). They bound the difference in revenue between the revenue-maximizing t-level auction and the optimal auction. Results are presented for single-item auctions but are generalized to matroid settings and single-parameter settings.This paper is well-written and does a good job of comparing/contrasting the problem, approach, and results to other relevant papers in the literature. The work is novel and makes connections across multiple disciples (auction theory and learning theory). The paper is of very strong technical quality, with some of the most interesting technical results extending into the appendix. One issue that might be of practical importance: after a particular t-level auction is implemented, there is likely information lost about the subsequent bidder valuations. While truth-telling is a dominant strategy, so is reporting the lowest value that falls on the same level as the bidder's true value. If the bidder is concerned about revealing information to the auctioneer (e.g., it thinks the bidder may try to re-learn the optimal reserves in a subsequent period, in the case that F is non-stationary), it seems reasonable to think that the bidder would choose this "lowest value on interval" strategy, since it gets the same payment and allocation as truthfully reporting but reveals less information to the auctioneer. If the auctioneer wants to re-learn optimal reserves in the future or otherwise understand how bidders will respond to a change in the mechanism, losing some precision in observed bidder valuations could be problematic.As a more fundamental side-note: I understand the theoretical importance in any case, so this point does not really impact my score, but I am a little skeptical of the value of revenue-maximizing auctions in practice, at least in some well-studied real-world domains, because they don't capture the fact that bidders are often making decisions over time, and extracting as much revenue as possible in the current auction doesn't necessarily coincide with maximizing long-term revenue. I would be curious to hear any normative statement from the authors (in either direction) about the practical importance of optimal auctions in real-world domains, or how to otherwise reconcile this tension in optimizing short- versus long-term revenue. If there is space and a good argument to be made, adding a comment to the paper would be valuable.Minor question: in a footnote it is mentioned that results hold for other tie-breaking rules. Which ones? It may be worth calling out whether results hold when ties are broken uniformly at random (my initial thought is that results would not hold, but the authors probably have a definitive answer).


Online Learning with Adversarial Delays
We study the performance of standard online learning algorithms when the feedback is delayed by an adversary. We show that online-gradient-descent [1] and follow-the-perturbed-leader [2] achieve regret O( √ D) in the delayed setting, where D is the sum of delays of each round's feedback. This bound collapses to an optimal O( √ T ) bound in the usual setting of no delays (where D = T ). Our main contribution is to show that standard algorithms for online learning already have simple regret bounds in the most general setting of delayed feedback, making adjustments to the analysis and not to the algorithms themselves. Our results help affirm and clarify the success of recent algorithms in optimization and machine learning that operate in a delayed feedback model.

The authors slightly modify two well known online algorithms, OGD and FTPL, to address the problem of adversarial delays. In this problem, it is assumed that adversary might not reveal the loss functions at end of every round. The modifications to the algorithms are most natural and I appreciate the simplicity. Since the crux of the paper lies in the analysis of the modified algorithms, I need to be sure about the correctness of the proofs before accepting the paper. While it is of course possible that I have misunderstood a couple of things, I want to be convinced that it is my mistake rather than mistake in the proofs before agreeing that the paper can be accepted to a conference of such prestige.Originality- Unless someone has already published the ideas before, I cannot question the originality of the work. Significance- The authors give a few examples in the beginning as to why adversarial feedback might be delayed. I have not encountered this problem before; so I am assuming that this problem is of interest to the online learning community. Clarity- The introduction and the two (modified) algorithms are certainly clear and simple to understand. However, I have a few issues regarding the presentation of the proofs. Normally, it would not matter, but considering the paper is all about the analysis, and the proofs are in the main paper, I think the presentation is important.1. Line 033- Why is the game fundamentally unfair? Unfair to whom? Learner or adversary? If the adversary is adaptive, I would say that the adversary is very powerful. Moreover, considering the regret is w.r.t to the best single decision learner could have taken in hindsight, I do not understand the unfairness issue. 2. It is confusing how the proof is going from line 151 to 156. While the inequality in line 156 is correct ( I checked it- simply write ||x_{t+1}- y||^2= ||x_t - \sum_{s \in F_t} \nabla f_s|x_s - y||^2 and open up the expression on the right followed by simple algebra), I find what is written in the paper confusing. First, the authors fix s as max F_t, but then the expression is unrolled over $s$. Is s fixed or variable? 3. Please check the expression for x_{t,s} when it is introduced. I think in the gradient term, it should be \nabla f_r|x_r, not \nabla f_s|x_r.4. I found it very difficult to understand the proof of Lemma 3.2. Can it be presented in a better way? It is simply the end part of Theorem 1.1 and Lemma 3.2 in Kalai-Vempala FTPL paper. In their paper, it is so simple to understand (the argument of what fraction of two hypercubes overlap followed by the fact that in the fraction they dont overlap, the maximum difference in decision*cost factor is bounded by L_1 norm of decision and L_{infty} norm of cost, which is precisely what is being done in Lemma 3.2 of this paper). I dont see the need to recast Kalai-Vempala's argument in a more complicated way. 5. There are number of notational mistakes in Lemma 3.3. Please check.Quality- The main questions I have are in the proof of Theorem 1. 1. Why does the inequality in Line 192 hold? Specifically, why is x_{t,s} closer to x'_{s+1} in Euclidean space than to x_{s+1}? I simply could not understand how this came about? As far as as I can see, x_{t,s} is an arbitrarily defined point. Is x_{t,s} inside or outside the convex set K? Either way, why does the inequality hold? 2. I simply could not wrap my head around the argument between Line 200 to line 219. I tried to reason independently but could not. While of course it does not mean the proof is wrong, I would like to talk to other reviewers before being convinced about it.


Nearly-Optimal Private LASSO *
We present a nearly optimal differentially private version of the well known LASSO estimator. Our algorithm provides privacy protection with respect to each training example. The excess risk of our algorithm, compared to the non-private version, is O(1/n 2/3 ), assuming all the input data has bounded ∞ norm. This is the first differentially private algorithm that achieves such a bound without the polynomial dependence on p under no additional assumptions on the design matrix. In addition, we show that this error bound is nearly optimal amongst all differentially private algorithms.

This paper introduces differentially private Frank-Wolfe algorithms, which allows to conduct LASSO with differential privacy constraints. The proposal method is designed by taking account of the fact that the constraints induced by LASSO forms a polytope. As a result, the excess empirical risk of the proposal learning algorithm is bounded above by nearly O(1/n^{2/3}) under l1-Lipschitz continuous assumption of loss functions. This bound is better than O(polylog(n)/n^(1/3)) by [31]. The bound on the excess empirical risk is higher than O(polylog(n)/n) by [32,37]. However considering that this is derived without strong assumptions (e.g., restricted strong convexity and mutual incoherence as required in [32,37]), this can be a new contribution. Furthermore, the authors show that the bound of the excess empirical risk is near optimal under epsilon<=0.1 and delta=o(1/n^2). The authors also derive a differentially private Frank-Wolfe algorithm that can work with any convex constraint region.The derivation of the near-optimality of the proposed algorithm is an original contribution while the derivations of the differential privacy and the excess empirical risk seems to be applications of existing results.In terms of privacy preservation, assumptions on the parameter, epsilon<=0.1 and delta=o(1/n^2), are adequate. On the other hand, in terms of utility, derivation of the constant term of the excess risk is needed. The noise added by the mechanism can occupy the domain of the prediction yin realistic situations if the constant term of the bound on the excess empirical risk is not small.Say, suppose the constant term of the bound of the excess empirical risk is 1 (i.e., the excess empirical risk <= log(np/delta)/(n epsilon)^(2/3) ). Suppose privacy parameters are set to epsilon=0.1delta=1/n^2, and p=2. Then,we have excess empirical risk <= 0.5 if n<=4000 by Corollary 2.7. Since the domain of the prediction y is [-1,1], the half of the domain can be covered with not a small probability. This can be improved with a larger size of samples, but n=4000 is not a small sample size in practice. Consequently, the assumptions on privacy parameters, epsilon=0.1 and delta=1/n^2, might not be practical in some situations.