Linear Response Methods for Accurate Covariance Estimates from Mean Field Variational Bayes
Mean field variational Bayes (MFVB) is a popular posterior approximation method due to its fast runtime on large-scale data sets. However, a well known major failing of MFVB is that it underestimates the uncertainty of model variables (sometimes severely) and provides no information about model variable covariance. We generalize linear response methods from statistical physics to deliver accurate uncertainty estimates for model variables-both for individual variables and coherently across variables. We call our method linear response variational Bayes (LRVB). When the MFVB posterior approximation is in the exponential family, LRVB has a simple, analytic form, even for non-conjugate models. Indeed, we make no assumptions about the form of the true posterior. We demonstrate the accuracy and scalability of our method on a range of models for both simulated and real data.

I think the basic idea of the approximation presented by the authors is very interesting.However it is hard to appreciate the broad applicability of the idea from the presentations in Sections 2.1 and 2.2.  The presentation of the idea in the text differs a bit from what needs to be done to apply the method in the examples.There is a model with parameter theta, and then we're told that the variational distribution q(theta) is in the exponential family with log q(theta)=eta^T theta-A(eta).Now, for most exponential families (such as the multivariate Gaussian family) the sufficient statistic would be say t(theta) where the dimension of t(theta) is larger than theta.This is problematic for the explanation of the method presented, since the authors go on to parametrize the lower bound in terms of E(theta) rather than eta, which would require theta and eta to be of the same dimension.I think what the authors mean here is that the lower bound can be parametrized in terms of m=E(theta) and some additional parameters lambda say, and then for any fixed m the additional parameters can be set to their optimum values for the given m according to the lower bound.By this kind of "profiling" we obtain a function of m only and then I think the arguments they give go through for this new profiled lower bound.Perhaps the authors think this is an obvious technicality but it confused me for a while and I think would confuse other readers too.  I think it's also true that if one is interested in marginal inferences on a subset of the parameters the same idea can be applied to get a function from the lower bound depending only on the mean parameters for the parameters of interest.This is something like a variational Bayes version of profile likelihood I guess.I would conjecture that this kind of profiling focusing only on the parameters of interest increases the accuracy of the method compared to the method presented which deals with all the parameters at once.  A minor error I noted was that in the equation above (6), it should be -(-V^{-1}+H)^{-1} not (V^{-1}+H)^{-1}.


Orthogonal NMF through Subspace Exploration
Orthogonal Nonnegative Matrix Factorization (ONMF) aims to approximate a nonnegative matrix as the product of two k-dimensional nonnegative factors, one of which has orthonormal columns. It yields potentially useful data representations as superposition of disjoint parts, while it has been shown to work well for clustering tasks where traditional methods underperform. Existing algorithms rely mostly on heuristics, which despite their good empirical performance, lack provable performance guarantees. We present a new ONMF algorithm with provable approximation guarantees. For any constant dimension k, we obtain an additive EPTAS without any assumptions on the input. Our algorithm relies on a novel approximation to the related Nonnegative Principal Component Analysis (NNPCA) problem; given an arbitrary data matrix, NNPCA seeks k nonnegative components that jointly capture most of the variance. Our NNPCA algorithm is of independent interest and generalizes previous work that could only obtain guarantees for a single component. We evaluate our algorithms on several real and synthetic datasets and show that their performance matches or outperforms the state of the art.

The paper proposes a subspace exploration algorithm for the problems of orthogonal NMF and nonnegative PCA (which are tightly related). According to the authors, this is the first algorithm that comes with performance guarantees that are characterized by a tradeoff between the approximation error and the computational complexity. Although the complexity scales exponentially with some user controlled parameters (order of low rank approximation, and target rank), simulations indicate good performance in practice.Quality: The paper is technically sound, with proofs about the main theorems, discussion about their shortcomings (exponential complexity), and simulations to establish state of the art results in practice. Clarity: The paper is mostly clearly written. I think one clarifications that is important but is missing is the following: Algorithm 3 LocalOptW runs with complexity O(2^k*k*m). Naively, for Algorithm 1, algorithm 3 has to be run for every choice of C. But the number of choices for C is also exponentially large (the authors should state explicitly how large it is). This would make the whole algorithm even more computationally hard. However the authors argue (footnote 2) that when the two algorithms are combined (as in the application shown in this paper), Alg 3 can be simplified into a O(k*m) procedure. While intuitive, the authors should make explicitly clear how this is done, and perhaps re-write algs 1 and 3 to demonstrate that. Originality: To the best of my knowledge the approach is original.Significance: The paper establishes both theoretical bounds (for the first time), and state of the art results through simulations. I'd like to see a comparison (or a discussion) of the various different methods, also with respect to the run-time of the various algorithms. The complexity bound O((1/\epsilon)^(1/\epsilon)) can grow rapidly, so it'd be nice to see that the gained performance does not come at a terrible computational cost. Since ONMF and NNPCA are largely equivalent, why does the proposed method perform better than EM for NNPCA (table 1) but worse than EM for ONMF (table 2) for the same dataset MFEAT PIX? Is there something that I miss here?Typo: last paragraph of the results section, Table 1 should be Table 2 (I think)


Associative Memory via a Sparse Recovery Model
An associative memory is a structure learned from a dataset M of vectors (signals) in a way such that, given a noisy version of one of the vectors as input, the nearest valid vector from M (nearest neighbor) is provided as output, preferably via a fast iterative algorithm. Traditionally, binary (or q-ary) Hopfield neural networks are used to model the above structure. In this paper, for the first time, we propose a model of associative memory based on sparse recovery of signals. Our basic premise is simple. For a dataset, we learn a set of linear constraints that every vector in the dataset must satisfy. Provided these linear constraints possess some special properties, it is possible to cast the task of finding nearest neighbor as a sparse recovery problem. Assuming generic random models for the dataset, we show that it is possible to store super-polynomial or exponential number of n-length vectors in a neural network of size O(n). Furthermore, given a noisy version of one of the stored vectors corrupted in near-linear number of coordinates, the vector can be correctly recalled using a neurally feasible algorithm.

Associative memory via a Sparse Recovery Model -------------------------------------------------- This paper builds an associative memory using message passing in a random graph setting.For an appropriately chosen set of constraints one can create an associative memory that seems to perform well using only local computations.I think the task is valid and the solution is interesting.I would have preferred more experimental work, so I am recommending a weak accept.More technical comments: ----------------------- Minor quibble - on line 46 - please refer to 1.22^n as exp(0.2*n).It's just more consistent with the rest of your notation in the rest of the paperOn line 95, you mention that your model is robust against adversarial error patterns but then don't mention it again.Can you please go into more detail about this claim or drop it entirely?Regarding equation 20, you casually mention that it is a neurally feasible algorithm since it involves only matrix vector multiplications.However matrix B appears twice in the equation, which seems to imply that there is a non-local dependency in your algorithm. (i.e. if one were to change a tweak a single parameter in the B matrix then one would have to modify your neural implementation in two, non local places).This makes it hard to see how such an algorithm could be learnt.If you do have any ideas n how this could be learnt then it would be great to add them, as it would really strengthen the paper.For your experimental section I would like to see more evidence showing that scaling behaviour that you predict does indeed show up in practice as you scale the network out.(i.e. iterative soft thresholding doesn't change the theoretical analysis.)It would also be great to see information theoretic performance measures.e.g. how does the mutual information between the initial cue and true pattern compare with the mutual information between the recalled pattern and the true pattern?Please consider also citing: Sterne (2012) - Efficient and Robust Associative Memory from a Generalized Bloom Filter which also does associative memory using a random graph with message passing.If you think the paper is very different then I won't push the point, but it seemed relevant to me.


Pointer Networks
We introduce a new neural architecture to learn the conditional probability of an output sequence with elements that are discrete tokens corresponding to positions in an input sequence. Such problems cannot be trivially addressed by existent approaches such as sequence-to-sequence [1] and Neural Turing Machines [2], because the number of target classes in each step of the output depends on the length of the input, which is variable. Problems such as sorting variable sized sequences, and various combinatorial optimization problems belong to this class. Our model solves the problem of variable size output dictionaries using a recently proposed mechanism of neural attention. It differs from the previous attention attempts in that, instead of using attention to blend hidden units of an encoder to a context vector at each decoder step, it uses attention as a pointer to select a member of the input sequence as the output. We call this architecture a Pointer Net (Ptr-Net). We show Ptr-Nets can be used to learn approximate solutions to three challenging geometric problems -finding planar convex hulls, computing Delaunay triangulations, and the planar Travelling Salesman Problem -using training examples alone. Ptr-Nets not only improve over sequence-to-sequence with input attention, but also allow us to generalize to variable size output dictionaries. We show that the learnt models generalize beyond the maximum lengths they were trained on. We hope our results on these tasks will encourage a broader exploration of neural learning for discrete problems.

This paper addresses the question of using neural networks to solve computational geometry problems, such as Traveling Salesman Problem (TSP), finding the convex hull in the plane, or finding Delaunay triangulations. To that end, the authors propose a slight variant of recent content-based attention mechanisms allowing them to predict a sequence of positions (or indices) of the input sequence, and thus train on the aforementioned tasks. Numerical results show that their model learns approximate solutions and is able to generalize beyond the sizes used during training.The main contribution of this paper is the problem set-up, casting some fundamental computational geometry problems as particular instances of "sequence-to-sequence" tasks, similarly as other recent efforts (such as "Learning to Execute" by Zaremba & Sutskever), and asking whether data-driven algorithms can compete with their analytical counterparts. Besides the practical interest (which might be relevant for the TSP, since it is NP hard and therefore we typically rely on approximate algorithms), this is a scientifically interesting question.This work might serve as motivation and benchmark for future efforts in that direction. The paper is clearly written (although some aspects are discussed in too broad terms cf below comments), the set-up is well presented and the three problems are described with enough detail. As I mentioned before, the paper is original in its set-up, less so in its proposed architecture. As pointed out by the authors themselves, the so-called Pointer Networks are a minor variant of the attention model of [5], so that it can be used in the setting where the output of the network are the attention positions themselves rather than some exterior targets.However, despite these positive aspects, I would like to point out some comments, in the hope to see them addressed in the rebuttal:--> Although it is true that some recent architectures are unable to adapt to variable output dictionary size, my impression is that other works do permit this setup, such as "Inferring Algorithmic Patterns with stack-augmented recurrent nets", by A.Joulin and T.Mikolov, or the Memory networks by J.Weston. If that is the case, it would be interesting to compare the performance of these methods and to report the results.--> Training procedure. A convex hull is presented here as a sequence of input points. However, there is a natural equivalence class in the output space, since the starting point and the direction are arbitrary. After the first two points of the output are known, the conditional probabilities of the next point do not suffer from this ambiguity, but couldn't the model be improved by resolving the ambiguity? In Section 3.2, where the outputs define an even larger equivalence class, the authors choose a lexicographic ordering (thus choosing one representative of the equivalence class). Why not do the same thing for the convex hull? --> I found the analysis of the results to be a bit superficial, I would have liked to see more introspection into what is happening. For example, in Section 4.2, the authors stress the fact that their model generalizes to larger sequence lengths. My impression is that, while true, this claim should be studied more carefully. In this task, there are two essential cardinal quantities, the length n of the input sequence, and the size k of the convex hull. In the setup of points distributed iid in the square, k is typically of the order of o(4 log n) (see for example "On the expected complexity of random convex hulls" ), and it is unclear which quantity is more relevant to measure the generalization of the model, n or k. The latter typically grows at logarithmic rate. --> Lines 403-404 mentions one of the most interesting numerical results of the paper, namely that the network seems to "regularize" the A1 algorithm. Why? Even if it is a conjecture, this seems to be one of the essential messages of the paper, replacing a priori operations by data-driven modules. --> Lastly, in lines 406-410, the authors compare the generalization performance of the network between the convex hull problem and the TSP. They explain the worse generalization of the TSP case from the complexity of the underlying algorithm (NP-complete vs n log(n)). While this is a valid point, I am wondering if there is another even more important aspect, namely the effective size of the output. In the case of the convex hull, as mentioned earlier, the typical size of the set is o(4 log n), and thus the number of bits to encode the output solution is of the order of log ( binomial (n, 4log n) ) = o( (log n)^2 ), whereas for the TSP, the output is essentially any permutation of the input, which requires of the order of log (n !) = n log(n) bits. Therefore, one could argue that in the convex hull case, the network needs to learn over a space of outputs which is of significantly smaller dimension than in the TSP. The fact is that there exsit o(n^2) approximate algorithms (A2) that do well -- but learning those operations over a large output space is challenging.--> A fundamental tool to fight curse of dimensionality and NP-hardness is to rely on dynamic programming or multiscale tools. The paper proposes a very small variant of a readily available model, which has the advantage of being simple, but does not address this property. However, I am wondering if the interest of this line of work is precisely the class of intractable problems, in which being able to learn this multiscale structure seems a necessary step (similar in spirit to the fast multipole transform). --> Even if it is not in the spirit of what is presented here, a natural question is why not try to learn the TSP by optimizing the length of the path (assuming one could backpropagate a continuous relaxation), or perhaps a fine-tuning step similar to a Viterbi decoding.


Planar Ultrametrics for Image Segmentation
We study the problem of hierarchical clustering on planar graphs. We formulate this in terms of finding the closest ultrametric to a specified set of distances and solve it using an LP relaxation that leverages minimum cost perfect matching as a subroutine to efficiently explore the space of planar partitions. We apply our algorithm to the problem of hierarchical image segmentation.

The paper generalizes the LP-relaxation approach to correlation clustering (multicut) to produce hierarchical clustering/segmentation. This can also be understood as fitting an ultrametric into the input distance data (which is called, sligtly confusingly, ultrametric rounding). The additional constraint is that at a higher level of the hierarchy, edges can be only removed from the multicuts and not added.The main contribution is a formulation of a suitable LP relaxation for this problem. The paper considers only planar graphs, which enables to employ column generation (detecting violated cutting planes) by planar max-cut / perfect matching (using Blossom 5). Besides that, the LP uses a number of tricks, using experience gained in past research on correlation clustering [6,16]. The LP is solved by CPLEX with Blossom5 plugged in.The experiments are done on the Berkeley segmentation dataset, using superpixels as the graph vertices. The appendix/supplement gives a thorough and detailed comparison with one other algorithm (UCM, reference [2] in the supplement), which is a version of algglomerative clustering.The experiments show that the method works well, which is not self-evident given several non-self-evident features in the LP formulation. On the practical side, the experiments suggest that the method is marginally better than UCM (slight refinements of contours that do not much contribute to evaluation statistics but can be visible) but notably slower - see e.g. Figure 3b in the main paper.Clarity of the text is not great but acceptable. One objection is that it often refers to non-trivial results from previous works, which makes it hard to read for a reader not completely familiar with the topic.I have one theoretical question. You write (paragraph starting at line 095) that the relaxation CC of MC is the convex hull (ie, the integral hull) of MC. That means, the relaxation of (1) alone is exact (ie, the optimal value of linear optimization over MC and CC are equal). But at the same time you seem to suggest (line 097) that the relaxation of (4) is proper (ie, not tight). That should mean that adding the hierarchy constraint X^l\ge X^{l+1} spoils this exactness - ie, linear optimization over \Omega_L is not the same as that over \bar{\Omega}_L. If so, this is theoretically interesting. Or did I miss something?Minor comment: the term "ultrametric rounding" is confusing because it clashes with "rounding schemes" in combinatorial optimization (see, eg, https://en.wikipedia.org/wiki/Randomized_rounding). Note that the paper [1] uses the term in this sense, while your paper uses it for "fitting an ultrametric". Is the term used in enough other works, preferably in theoretical computer science? If not, please do not promote wrong terminology and consider changing the title.


M -Statistic for Kernel Change-Point Detection
Detecting the emergence of an abrupt change-point is a classic problem in statistics and machine learning. Kernel-based nonparametric statistics have been proposed for this task which make fewer assumptions on the distributions than traditional parametric approach. However, none of the existing kernel statistics has provided a computationally efficient way to characterize the extremal behavior of the statistic. Such characterization is crucial for setting the detection threshold, to control the significance level in the offline case as well as the average run length in the online case. In this paper we propose two related computationally efficient M -statistics for kernel-based change-point detection when the amount of background data is large. A novel theoretical result of the paper is the characterization of the tail probability of these statistics using a new technique based on change-ofmeasure. Such characterization provides us accurate detection thresholds for both offline and online cases in computationally efficient manner, without the need to resort to the more expensive simulations such as bootstrapping. We show that our methods perform well in both synthetic and real world data.

This paper proposes two M-statistics, for the offline and online settings, for change-point detection that are based on the kernel MMD B-test statistic for two-sample testing. The B-test statistic is a recently developed alternative to the MMD that is more efficient; it involves taking an average of the MMD over a partitioning of the data into N blocks. Since the blocks are independent, the B-statistic is asymptotically normal so it is simple to obtain thresholds without resorting to bootstrapping. This procedure is modified for the case of change point detection by averaging the MMDs computed from each of N blocks occurring before a potential change point with the block of data of the same size as each N occurring after the potential change point. To determine whether a change-point has occurred somewhere in a block of data, the above procedure is carried out for each potential change point location and a Z-statistic is obtained for each. The M-statistic proposed by the authors is a maximum over the standardized Z-statistics. The authors derive an expression for the variance of the Z-statistics (needed to standardize it) and for the significance threshold of the M-statistic (since the Z-statistics are not averages of independent blocks of data, they are not asymptotically normal like in the case of the B-test) and give a theoretical performance analysis which is also verified numerically. Finally, they apply their method to several real data sets, where it seems to perform slightly better than a baseline method in terms of AUC. The work is original and significant. Competing methods require bootstrapping, which is much less efficient with larger amounts of data. Furthermore, the theoretical work involved in deriving the variance of the Z-statistic and thresholds for the M-statistic may prove to be useful elsewhere. The theoretical work is rigorous and seems reasonable. I did not check the proofs in the appendix.   The presentation is generally good, but could probably be improved. I needed to read through section 3 several times before I understood the procedure. The fact that the data are being formed into blocks at two different levels (for each N and each B) make things difficult to understand when described casually in the body of the section. Figure 1 was somewhat helpful but I was still confused. If the procedure were described more formally, e.g. pseudocode, it might be easier to understand. In my opinion, some of the space given to details in sections 4 and 5 could be better used explaining the test procedure better in section 3 (and those details moved to the appendix). Additionally, it would be useful to have more of an explanation of the baseline procedures in section 6 and incorporate some of the results from the appendix. There are some minor typographical / editing errors in the paper; it should be proofread a few times. Line 111 begins with something that is not a sentence.


Multi-class SVMs: From Tighter Data-Dependent Generalization Bounds to Novel Algorithms
This paper studies the generalization performance of multi-class classification algorithms, for which we obtain-for the first time-a data-dependent generalization error bound with a logarithmic dependence on the class size, substantially improving the state-of-the-art linear dependence in the existing data-dependent generalization analysis. The theoretical analysis motivates us to introduce a new multi-class classification machine based on p -norm regularization, where the parameter p controls the complexity of the corresponding bounds. We derive an efficient optimization algorithm based on Fenchel duality theory. Benchmarks on several real-world datasets show that the proposed algorithm can achieve significant accuracy gains over the state of the art.

The paper describes a family of multi-class SVMs parameterised by a single parameter p. The family contains the Crammer & Singer SVM as a special case.Corresponding generalization bounds are derived, which for certain parameter choices have a logarithmic dependency on the number of classes. The latter has - to my knowledge - not been shown for any multi-class classifier.The corresponding learning machine is derived including its Fenchel dual problem. There is an empirical evaluation, which shows that the approach works. However, the experiments are the weak spot of the paper.Only three test problems are considered. It is not clear which kernel function is used, therefore I assume a linear kernel. The Crammer & Singer SVM serves as the baseline, the parameter p is optimised on a fine grid (including the value p=2, which gives the Crammer & Singer SVM). The results for the best p value are reported (optimised for each data set independently), which are slightly better than the Crammer & Singer SVM results.That the Crammer & Singer SVM results are achieved does not come as a surprise, as p=2 is tested for the new machines, i.e., the new machines include the Crammer & Singer SVM. The grid for optimising p is very fine. That is, the best of many values of p (i.e., many different machines) are compared to the Crammer & Singer variant. That one always find one machine for each data set is not surprising, actually I think that the improvements are so small is rather disappointing. What if the model selection for the Crammer & Singer variant is refined such that in the end as many machines are tested as in the case when also p is optimised? For example, using a finer grid for regularisation parameter and kernel bandwidth parameter (in case of a Gaussian kernel SVM). Better statistics about the robustness w.r.t. p should be reported.Testing more data sets with a linear as well as a non-linear kernel is mandatory.Computation times for solving the optimisation problems are not given. The general problem of the new machines is more difficult to solve than the Crammer & Singer QP, and the scaling of the optimisation time w.r.t. the number of training data points is important to see. Timing experiments are a must.Long training times (which I assume) and the necessity to adapt an additional hyperparameter reduce the practical value of the proposed approach.Anyway, the theoretical result is strong.  Minor comments: mercer -> Mercer lines 394,395: The description of the grid for p does not seem to be consistent. 1.2 and 1.25 differ by 0.5 - how is this related to the step sizes 0.5 and 0.1?


The Brain Uses Reliability of Stimulus Information when Making Perceptual Decisions
In simple perceptual decisions the brain has to identify a stimulus based on noisy sensory samples from the stimulus. Basic statistical considerations state that the reliability of the stimulus information, i.e., the amount of noise in the samples, should be taken into account when the decision is made. However, for perceptual decision making experiments it has been questioned whether the brain indeed uses the reliability for making decisions when confronted with unpredictable changes in stimulus reliability. We here show that even the basic drift diffusion model, which has frequently been used to explain experimental findings in perceptual decision making, implicitly relies on estimates of stimulus reliability. We then show that only those variants of the drift diffusion model which allow stimulusspecific reliabilities are consistent with neurophysiological findings. Our analysis suggests that the brain estimates the reliability of the stimulus on a short time scale of at most a few hundred milliseconds.

The authors show the drift diffusion model (DDM) of perceptual decision making implicitly assumes that the brain makes online estimates of stimulus reliability. By formulating a probabilistic equivalent, the authors compare versions of the DDM with and without trial-dependent reliability estimates, and show that behavioral and neurophysiological data are consistent with DDMs that do estimate reliability.In the introduction, it seems to me that the authors conflate two points: first, that the brain uses estimates of stimulus reliability when making decision; and second, that stimulus reliability affects tuning curves multiplicatively. Although the two are related in the PPC framework cited by the authors, in general the former relates to decoding, the latter to encoding. Also in the introduction, the authors propose that "the multiplicative effect on tuning curves results from adaptation of internal estimates of measurement reliability". However, throughout the paper there is no further explanation of what they mean by adaptation, and the above idea is not further elaborated, except perhaps in the last paragraph of discussion. Section 2, first paragraph: since the authors highlighted multiplicative effects of tuning curves, it is worth pointing out that the exact definition of coherence actually matters and could determine how reliability (or coherence) changes tuning curves. For instance, in the experiments cited, coherent dots move in a fixed direction and the other dots directions are sampled from the uniform distribution; an alternative choice could be to draw all dots from eg a circular Gaussian peaked at the chosen direction, and vary reliability by changing the width of the distribution. Importantly, this is in fact the generative model of the stimulus considered in eq. 3. This latter choice would lead to changes in tuning width, not just multiplicative scaling, when reliability changes. Fig 1, the use of rose histograms is misleading: the area of the circular sector is quadratic (not linear) in the radius (i.e the count). Eq 2 requires some more detail about \rho, the "population" firing rate. To be optimal, the "population rate" should be obtained by weighting each neuron's response by a weight that depends on its tuning curve derivative and on the covariance matrix of the population (see eg Ma et al 2006).Eq 4 is not precise; It should be x_t conditioned on the true stimulus being left (or right). Similarly, in eq. 5, the l.h.s. should be similarly conditioned. Eq 6: the term \delta y defined in this equation is Gaussian distributed only if conditioned on a particular (left or right) true value of the stimulus. Otherwise, across different true stimuli, it is a mixture of gaussians.


Policy Gradient for Coherent Risk Measures
Several authors have recently developed risk-sensitive policy gradient methods that augment the standard expected cost minimization problem with a measure of variability in cost. These studies have focused on specific risk-measures, such as the variance or conditional value at risk (CVaR). In this work, we extend the policy gradient method to the whole class of coherent risk measures, which is widely accepted in finance and operations research, among other fields. We consider both static and time-consistent dynamic risk measures. For static risk measures, our approach is in the spirit of policy gradient algorithms and combines a standard sampling approach with convex programming. For dynamic risk measures, our approach is actor-critic style and involves explicit approximation of value function. Most importantly, our contribution presents a unified approach to risk-sensitive reinforcement learning that generalizes and extends previous results.

The paper proposes a new formula for the class of static coherent risks from which policy gradients for many common risk measures can be derived analytically. It also presents a general sampling-based algorithm for cases where no analytical solution is available and shows consistency guarantees of the policy gradient estimates. Eventually, the paper provides expressions for policy gradients of Markov coherent dynamic risk measures.Significance and quality:The unified derivation of policy gradients for coherent risk measures is intriguing and a valuable and novel contribution to the risk-sensitive decision making literature. Nevertheless, analytical solutions of the Lagrangian saddle-points are only available for some risk measures.The proposed sample-based approximation scheme could allow us to apply the derived results for general coherent risk measures by avoiding to derive analytical forms of the saddle points. However, the paper does not provide any empirical evidence that the proposed estimation scheme is viable in practice. An experimental comparison between the using the sample-based policy gradients and the analytical ones in different problems, e.g. for CVaR would be extremely helpful.   Originality: To the best of my knowledge, the derived policy gradients are - in this general form - are novel.Clarity: The paper is generally well written, however, it appears to me that the clarity of the main paper is hindered considerably by number of different issues attempted to be addressed in the eight pages. Especially the dynamic risk policy gradients in Section 5 are presented on a very high level and the lack of any algorithm details and experimental evaluations do not allow the reader to judge whether the derived policy gradients yield approaches that are viable in practice. It seems to me this content is rather suited for a longer journal publication. An alternative that I would see as beneficial is to split this paper and rather focus on static risk measures which allows to make the paper more standalone and provide more details and a more comprehensive empirical evaluation beyond the very basic toy example in the current submission.


Algorithms with Logarithmic or Sublinear Regret for Constrained Contextual Bandits
We study contextual bandits with budget and time constraints, referred to as constrained contextual bandits. The time and budget constraints significantly complicate the exploration and exploitation tradeoff because they introduce complex coupling among contexts over time. To gain insight, we first study unit-cost systems with known context distribution. When the expected rewards are known, we develop an approximation of the oracle, referred to Adaptive-Linear-Programming (ALP), which achieves near-optimality and only requires the ordering of expected rewards. With these highly desirable features, we then combine ALP with the upper-confidence-bound (UCB) method in the general case where the expected rewards are unknown a priori. We show that the proposed UCB-ALP algorithm achieves logarithmic regret except for certain boundary cases. Further, we design algorithms and obtain similar regret bounds for more general systems with unknown context distribution and heterogeneous costs. To the best of our knowledge, this is the first work that shows how to achieve logarithmic regret in constrained contextual bandits. Moreover, this work also sheds light on the study of computationally efficient algorithms for general constrained contextual bandits.

Review after rebuttal : - I agree that it is not exactly the same, but it seems close enough to me : if the horizon is large enough, the coupling of the context vanishes (and given that, I don't understand why . So I am not sure why each occurrence of a context cannot be treated as a separate instance of a bandit problem). - To me, this paper would be much better with a clean (the supplementary material's bound has to be reworked) upper bound in term of arm gaps for the problem dependent case, and ideally matching lower bounds in problem dependent and independent cases. ----------------------The authors consider the problem of a budgeted and time constrained contextual bandit. The main finding of their paper is that if the number of context and actions is finite, and if the time constraint T is sufficiently large (with respect to the problem at hand) then they prove that a regret of order \log(T) is achievable, which is a new result for contextual bandits.The results seem thorough, and the writing is precise. The results are interesting but not very surprising. Indeed, since the number of contexts J and arms K are finite, this bandit problem can be seen as a a time constrained and budgeted bandit problem with JK arms, which is also action constrained since at each time, not all JK arms are accessible (only the K arms corresponding to the context). Since the number of arms is finite, for T large enough, the regret is \log(T) as in action constrained bandit problems. So I am not sure how innovative this paper is - maybe the authors can clarify this in their rebuttal.I have a few questions regarding this paper : - In Theorem 2, the regret inn case 2) is O(\sqrt(T) + ...). But is it really O(\sqrt(T) + ...)? Shouldn't it be O(\sqrt(KT) + ...) at least? - Still in Theorem 2, the regret is either O(KJ \log(T)) or O(\sqrt(T) + KJ\log(T)) depending on the configuration of the arms. In general, problem dependent bounds are expressed in a more refined way with the sum of inverse of arm gaps. Isn't it possible to do something similar here? I understand that you have KJ instead of a certain sum of gaps because your proof is based on events characterizing the correct orders of the UCB, and not concentration bounds on their gaps. But is it an optimal idea? - It would be interesting to have lower bounds for this problem, ideally problem dependent and problem independent. Trivial lower bounds are the ones of classical bandit which correspond to cost equals 0 and context which is always the same. Can it be refined to take into account the complexity of the context?


A Dual-Augmented Block Minimization Framework for Learning with Limited Memory
In past few years, several techniques have been proposed for training of linear Support Vector Machine (SVM) in limited-memory setting, where a dual blockcoordinate descent (dual-BCD) method was used to balance cost spent on I/O and computation. In this paper, we consider the more general setting of regularized Empirical Risk Minimization (ERM) when data cannot fit into memory. In particular, we generalize the existing block minimization framework based on strong duality and Augmented Lagrangian technique to achieve global convergence for general convex ERM. The block minimization framework is flexible in the sense that, given a solver working under sufficient memory, one can integrate it with the framework to obtain a solver globally convergent under limited-memory condition. We conduct experiments on L1-regularized classification and regression problems to corroborate our convergence theory and compare the proposed framework to algorithms adopted from online and distributed settings, which shows superiority of the proposed approach on data of size ten times larger than the memory capacity.

The main result is the linear convergence rate in terms of outer rounds of the method, assuming exact solvers of the (still large) ERM subproblems which fit in memory.The presentation of the paper could be improved significantly, but I think the contributions are still valuable, practical and interesting. Several of my comments below are suggestions for further improvements. UPDATE: Unfortunately the experiments lack a comparison to Quasi-Newton methods and d-GLMNET (i.e. the serialized variants of these). this issue was not answered in the author feedback, which questions parts of the value of the experiments. 1) Exact solution of the subproblems is assumed (See also Section 4.3.1), which is disappointing in view of practical applications. Even Theorem 2 only provides approximate solutions for the inner solvers. On the positive side, the experiments already work with approximate inner solvers.2) Choice of \eta_t in theory and experiments: In the paper, please make more clear that the algorithm depends on this step-size parameter. In experiments, how was this parameter selected? How does the performance depend on the parameter?3) Basic consensus-type ADMM variants which are well-studied for this type of problem. Their discussion (i.e. the summary) should be moved from the appendix to the main paper, and made more precise. Currently a good amount of space in the main paper is used by redundant formulas (e.g. useless dual, if working on the augmented primal, many repetitions from Section 3 in Section 4). It would be important to more clearly explain the difference in the updates proposed here as to the ADMM updates currently in the Appendix (Algorithm 3 and 4). I.e. write the actual ADMM updates and L_k for the case here, and compare to (15).4) Experiments:The paper unfortunately lacks an experimental comparison with other state of the art distributed l1-solvers (i.e. their serialized counterparts), such as d-GLMNET [A], Shotgun [B] and most importantly orthant-wise variants of Quasi-Newton methods [C] which are standard for l1-problems.The experiments are currently presented in a way that is far from reproducible, and we'd hope this will be improved. For example, the stopping criteria for the subproblem solvers, the \eta_k parameters, dependence on the regularization parameters etc is not given. Also, timing plots alone might not be very informative, but it might be important to discuss the IO load and number of passes through the data. In particular for online methods such as MD/SGD, they might suffer unfairly from inefficient random access, but could also be run in a memory-aware fashion. The tradeoff of IO vs computation could also be investigated, i.e. how much does one benefit from more local computation before swapping in a new data block.5) The discussion around lines 422 suggests that in contrast to MD, the approach here has linear convergence. This however is very oversimplifying, as the linear convergence is in number of outer rounds, each round consisting in an exact solution of a problem in millions of datapoints (same structure as the original problem). In contrast, the convergence rates of the online methods is in terms of individual points processed. The constants in the linear rate theory should be discussed, and could potentially be bounded in some relevant cases, e.g. for Lasso.6) The level of generality and the function classes assumed for the main optimization problems should be stated more precisly.7) *Dual* nature of the algorithm: Why do you call the algorithm 'dual-BCD' (motivated by [31]), if it is here applied to (7), which is the primal problem, with augmentation? This also relates to 'dual' in the title of the paper, as well as the title of Section 3, which in this view is confusing to start with the dual (4) if in the end the interest is (7). The reader will always be confused since all your proposed 'dual' methods are not solving (4) but (7). Also in line 247, calling it just BCD (or primal BCD or saddle-point BCD) would make it easier for the reader to understand which methods work on the dual, and which on the primal.8) Presentation The paper in some places seems to have been written in a rush, and the presentation and related work discussions could be greatly improved. In particular, the paper would benefit from a more concise discussion of related proximal gradient methods and their rates, as there is existing literature already even for the case of approximate proximal steps. 9) Why don't you generalize your result by allowing arbitrary solvers for the subproblem, why restricting to BCD (Theorem 2 etc), especially as this gets easily confused with outer BCD? In this respect, Theorem 3 is crucial for your contribution, while Theorem 2 is not.10) Minor: In Section 3, the authors extend the block-minimization framework of [31,2] from linear SVM to general ERM problems. However, this was already done by [D,E] for the ERM dual, though not including the l1 primal case. The transformation of those algorithms from the distributed setting to the 'limited memory' case here is straightforward, same as for ADMM and related methods.11) About easily adapting local solvers around line 280: Make precise the comment only covers first-order solvers, not solvers which potentially rely on more problem structure.12) Clarify how \mu_Bk is a variable in (7), i.e. if alpha_Bk is also an optimization variable. Same later in (15) etc.  minor - w^*() in line 3.2 of algorithm 2 is not clearly defined. -the readability of the whole paper and in particular the algorithms could easily be improved. 064: descnet 065,321: subsumes -> includes 176: define the function F, \bar F (objective in which problem?) 286: _in is not defined and properly related with S or Theorem 2. appendix: 581: appendix of lemma? 589: prposed. 2-norm messed up 630: define assumptions on \mu in Theorem 5 695: w and z are vectors in (38), not numbers, so need norms 784: note that 785: the analysis of *** additional references:[A] I Trofimov and A Genkin. Distributed Coordinate Descent for L1-regularized Logistic Regression. arXiv.org, 2014.[B] J K Bradley, A Kyrola, D Bickson, and C Guestrin. Parallel Coordinate Descent for L1-Regularized Loss Minimization. ICML, 2011.[C] G Andrew and J Gao. Scalable training of L1-regularized log-linear models. ICML, 2007.[D] T Yang. Trading Computation for Communication: Distributed Stochastic Dual Coordinate Ascent. NIPS, 2013.[E] M Jaggi et al. Communication-Efficient Distributed Dual Coordinate Ascent. NIPS, 2014.


A Bayesian Framework for Modeling Confidence in Perceptual Decision Making
The degree of confidence in one's choice or decision is a critical aspect of perceptual decision making. Attempts to quantify a decision maker's confidence by measuring accuracy in a task have yielded limited success because confidence and accuracy are typically not equal. In this paper, we introduce a Bayesian framework to model confidence in perceptual decision making. We show that this model, based on partially observable Markov decision processes (POMDPs), is able to predict confidence of a decision maker based only on the data available to the experimenter. We test our model on two experiments on confidence-based decision making involving the well-known random dots motion discrimination task.
In both experiments, we show that our model's predictions closely match experimental data. Additionally, our model is also consistent with other phenomena such as the hard-easy effect in perceptual decision making.

The authors' model confidence data from two experiments (conducted by others and previously published in the scientific literature) using a POMDP. In both experiments, subjects saw a random-dot kinematogram on each trial and made a binary choice about the dominant motion direction. The first experiment used monkeys as subjects and stimuli had a fixed duration. The second experiment used people as subjects and stimuli continued until a subject made a response. The paper reports that the POMDP model does a good job of fitting the experimental data, both the accuracy data and the confidence data.I should start by stating that I know a bit about perceptual decision making and a (small) bit about POMDPs but I am unfamiliar with the application of POMDPs to perceptual decision making. The paper reports that this is the first time that a POMDP has been used to model confidence data in a perceptual decision making task. I'll take the authors' word on this.Taken as a whole, I have a positive opinion of this manuscript. I very much like the problem studied here (modeling confidence data in perceptual decision making) and I like the idea of applying a POMDP to this data. The fit to the experimental data is qualitatively good (though quantitatively mediocre).A significant shortcoming of this paper is the lack of a literature review section providing context for the work provided here. Other researchers have previously applied POMDPs to perceptual decision making. Did they only model accuracy data? Did they model reaction time data too? How does the POMDP studied here relate to the POMDPs presented in these earlier papers? Is it largely the same or is it substantially different? If the latter , how (and why) is it different? If the current paper is the first to apply POMDPs to confidence data in a perceptual decision making task, how did researchers model confidence data previously? Without answers to these questions, there is no way to evaluate the novelty and significance of the contribution made by the current paper.


LASSO with Non-linear Measurements is Equivalent to One With Linear Measurements
Consider estimating an unknown, but structured (e.g. sparse, low-rank, etc.), signal x 0 ∈ R n from a vector y ∈ R m of measurements of the form y i = g i (a i T x 0 ), where the a i 's are the rows of a known measurement matrix A, and, g(·) is a (potentially unknown) nonlinear and random link-function. Such measurement functions could arise in applications where the measurement device has nonlinearities and uncertainties. It could also arise by design, e.g., g i (x) = sign(x + z i ), corresponds to noisy 1-bit quantized measurements. Motivated by the classical work of Brillinger, and more recent work of Plan and Vershynin, we estimate x 0 via solving the Generalized-LASSO, i.e.,x := arg min x y − Ax 0 2 + λf (x) for some regularization parameter λ > 0 and some (typically non-smooth) convex regularizer f (·) that promotes the structure of x 0 , e.g. 1 -norm, nuclear-norm, etc. While this approach seems to naively ignore the nonlinear function g(·), both Brillinger (in the non-constrained case) and Plan and Vershynin have shown that, when the entries of A are iid standard normal, this is a good estimator of x 0 up to a constant of proportionality µ, which only depends on g(·). In this work, we considerably strengthen these results by obtaining explicit expressions for x−µx 0 2 , for the regularized Generalized-LASSO, that are asymptotically precise when m and n grow large. A main result is that the estimation performance of the Generalized LASSO with non-linear measurements is asymptotically the same as one whose measurements are linear y i = µa i T x 0 + σz i , with µ = Eγg(γ) and σ 2 = E(g(γ) − µγ) 2 , and, γ standard normal. To the best of our knowledge, the derived expressions on the estimation performance are the first-known precise results in this context. One interesting consequence of our result is that the optimal quantizer of the measurements that minimizes the estimation error of the Generalized LASSO is the celebrated Lloyd-Max quantizer.

Summary: This paper considers the Generalized Lasso with non-linear measurements and Gaussian design. A known heuristic often used is that non-linear observations may be treated as noisy linear observations. Capitalizing on the work of Plan and Vershynin [17], the authors here extend the results of [17], and by moving to an asymptotic regime, they provide novel precise explicit expressions of the error estimation. The accuracy of their predictions is confirmed with many special cases (including the Lasso, the group Lasso and the nuclear norm). The application to the design of optimal quantizers in the context of q-Bit CS and its relation to the Lloyd-Max quantizer is very insightful.             Quality: This is a nice piece of work.            Clarity: The paper is quite well-written, though the treatment is somewhat unbalanced. The lack of space however prevents including all the necessary details.                  Originality: The paper builds upon the work of Plan and Vershynin [17] and expands in several aspects that are novel. For instance, the explicit expressions of the performance estimation error are new.            Comments:     1) Page 4, line 193: the regularized and penalized versions are equivalent with appropriate correspondence.     2) Caption of Figure 1: 'n=768' shouldbe '$n$=768'.     3) Page 5, line 247: a typo at 'When ...'.     4) Page 5, line 294: what the authors call the proximal function is (very well) known in convex analysis as the Moreau envelope of index \tau. The notation used there is awkward as prox is generally used for the proximal point/operator but not for the Moreau envelope. I suggest changing it.     5) Page 5, Assumption 1: I think it would instructive to give this assumption a more formal interpretation using results from variational analysis.     6) The authors should avoid calling the generalized Lasso an algorithm in the text. It is certainly a (convex) program (or decoder to borrow the terminology widely used in CS), but not an algorithm.     7) Experiments: the authors show the results when applying the generalized Lasso to the non-linear measurements, and how the theorems predict the estimation error.


From random walks to distances on unweighted graphs
Large unweighted directed graphs are commonly used to capture relations between entities. A fundamental problem in the analysis of such networks is to properly define the similarity or dissimilarity between any two vertices. Despite the significance of this problem, statistical characterization of the proposed metrics has been limited. We introduce and develop a class of techniques for analyzing random walks on graphs using stochastic calculus. Using these techniques we generalize results on the degeneracy of hitting times and analyze a metric based on the Laplace transformed hitting time (LTHT). The metric serves as a natural, provably well-behaved alternative to the expected hitting time. We establish a general correspondence between hitting times of the Brownian motion and analogous hitting times on the graph. We show that the LTHT is consistent with respect to the underlying metric of a geometric graph, preserves clustering tendency, and remains robust against random addition of non-geometric edges. Tests on simulated and real-world data show that the LTHT matches theoretical predictions and outperforms alternatives.

Abstract: Large unweighted directed graphs are commonly used to capture relations be- tween entities. A fundamental problem in the analysis of such networks is to properly define the similarity or dissimilarity between any two vertices. Despite the significance of this problem, statistical characterization of the proposed met- rics has been limited.What particular problem do the authors have in mind? The same argument is true for weighted, undirected graphs but they seem to be excluded. If one would know what problem the authors are talking about this would become clear. In the paper I could not find any connection to some 'real world' problem but only a mathematical consideration of the problem.How about betweenness centrality, as used to identify bottleneck genes in gene networks. http://journals.plos.org/ploscompbiol/article?id=10.1371/journal.pcbi.0030059 Is that a problem the author has in mind? Can one compare both methods? Compare with any other method?Without a clear connection to a real world problem I consider the paper out of scope for NIPS. However, I think the paper could make a nice contribution for a mathematics conference/journal.The paper concludes without a summary/discussion.


On Top-k Selection in Multi-Armed Bandits and Hidden Bipartite Graphs
This paper discusses how to efficiently choose from n unknown distributions the k ones whose means are the greatest by a certain metric, up to a small relative error. We study the topic under two standard settings-multi-armed bandits and hidden bipartite graphs-which differ in the nature of the input distributions. In the former setting, each distribution can be sampled (in the i.i.d. manner) an arbitrary number of times, whereas in the latter, each distribution is defined on a population of a finite size m (and hence, is fully revealed after m samples). For both settings, we prove lower bounds on the total number of samples needed, and propose optimal algorithms whose sample complexities match those lower bounds.

Summary - - - - - - -This paper studies the problem of finding the top k (out of n) arms (the k arms with the highest mean) in multi-armed bandits fixed confidence setting. More specifically, the goal is to derive an algorithm that with the smallest number of pulls returns a set of k arms such that with high probability (w.p. \geq 1-\delta)1) for each 1 \leq i \leq k, the mean of the i'th arm in the set is larger than (1-\epsilon) times the mean of the i'th arm (overall).  2) the average mean of the arms returned by the algorithm is larger than (1-\epsilon) times the average mean of the top k arms. The above settings are different and more general than those have been considered in the multi-armed bandits fixed confidence pure exploration literature (e.g., the work by Kalyanakrishnan et al., 2012 and the one by Gabillon et al, 2012), in the sense that the algorithms in those papers guarantee that every arm they return is (\epsilon,k)-optimal (the mean of every arm returned is not smaller than the mean of the k'th best arm minus \epsilon, with high probability).  For each of the above two problems, the authors present an algorithm and prove an upper-bound and a matching lower-bound for it. The authors motivate these two settings with the problem of crowdsourcing. The authors also show how their algorithm for the first problem should be changed to solve the same problem with additive instead of multiplicative guarantee (the mean of the i'th arm plus \epsilon instead of (1-\epsilon) times the mean of the i'th arm). Then the authors focus on the problem of exploring hidden bipartite graphs and state two problems, similar to those stated above. These two graph problems can be easily reduced to the two problems above, and thus, the algorithms for those problems can be easily used here. The only contribution in this part of the paper is the derivation of a lower-bound for each of these two graph problems.   Comments - - - - - - - -- The paper is well-written and is easy to follow. - To the best of my knowledge, the formulation considered in this paper is novel. The objective functions studied here (in Problems 1 and 2), although close, are different than those studied in the literature of pure exploration multi-armed bandits with fixed confidence. They are different in two senses: 1) the mean of each returned arm should be close to the mean of a specific top k arm (instead of being close to the mean of the k'th arm), and 2) the guarantee is multiplicative instead of additive ((1-\epsilon) times the mean of the i'th arm instead of the mean of the i'th arm plus \epsilon). The first difference could play an important role in the case where there exists many arms whose mean is in [\theta_k(B) - \epsilon, \theta_k(B)]. In this case, the existing (\epsilon,k)-optimal algorithms may return k arms, none of them among the top k, while this is not the case for the algorithm proposed in this paper. The authors have a discussion (on Pages 2 and 3) on the significance of multiplicative guarantee and mention its importance when the optimal solution is very small, which is apparently the case in many hidden graph problems. This is a good reason to consider multiplicative guarantee in addition to additive. They also discuss how we can define \epsilon'=\epsilon\theta_k(B), and use algorithms for additive guarantee to have multiplicative guarantee.  - In addition to the objective function, there is a very important difference between the top-k algorithms in the literature (e.g., the work by Kalyanakrishnan et al., 2012 and the one by Gabillon et al, 2012) and the one here, which is in the bound when the complexity appears in their bounds instead of 1 / \epsilon^2*\theta_k(B). This means that the bounds of the existing methods can be much tighter than the one for the algorithm in this paper. This is the main reason that those algorithms are more complex than the racing type algorithms presented in this paper. My main criticism about this work is the lack of discussion about this issue. As I mentioned above I admit that the formulation considered in the paper is novel, but the authors have considered the simplest approach (the racing type algorithms) and of course their bound is of the form 1 / \epsilon^2\theta_k(B). I think it would be possible to make the complexity appear in the bound for the problems considered in this paper, of course (probably) not by the type of algorithms considered here. It would be great if the authors make this difference clear and discuss the difficulties of having algorithms with the complexity in their bounds for the problems studied in this paper. Why didn't the authors derived algorithms of that form? - Why the proposed algorithms use the empirical mean based on the uniform allocation at that specific iteration instead of the empirical mean based on all the available samples from that arm? Is this just because of simplifying the proof and theory or has other reasons? Please clarify.   Minor Comments - - - - - - - - - - - -  - Line 102: "instead an additive" should be "instead of an additive". - Why the guess for the value of the mean of the k'th arm (\mu) starts at \mu=1 in ME-AS and QE-AS algorithms? This seems to be useless since \hat{\theta}^{US}(a_k) will never be greater than 2*\mu=2. I guess it should start at 1/2.


GP Kernels for Cross-Spectrum Analysis
Multi-output Gaussian processes provide a convenient framework for multi-task problems. An illustrative and motivating example of a multi-task problem is multi-region electrophysiological time-series data, where experimentalists are interested in both power and phase coherence between channels. Recently, Wilson and Adams (2013) proposed the spectral mixture (SM) kernel to model the spectral density of a single task in a Gaussian process framework. In this paper, we develop a novel covariance kernel for multiple outputs, called the cross-spectral mixture (CSM) kernel. This new, flexible kernel represents both the power and phase relationship between multiple observation channels. We demonstrate the expressive capabilities of the CSM kernel through implementation of a Bayesian hidden Markov model, where the emission distribution is a multi-output Gaussian process with a CSM covariance kernel. Results are presented for measured multi-region electrophysiological data.

The authors propose a flexible and interpretable kernel (the CSM kernel), building on spectral mixture kernels, for learning relationships between multiple tasks.The starting point is to use Gaussian processes with 1 component spectral mixture kernels as the basis functions in a linear model of coregionalisation (SM-LMC). However, SM-LMC does not contain information about the phases between channels.Thus the authors propose the cross spectral mixture kernel, which mixes phase shifted versions of spectral mixture kernels across channels.The resulting kernel is interpretable and flexible. The authors note that if they have an evenly spaced temporal grid, this gives rise to covariance matrix structure which can be exploited for fast computations.In particular, they approximate their covariance matrices with circulant matrices, and use discrete Fourier transforms to speed up inference and learning.They implement the CSM kernel as part of a Bayesian HMM, and apply the model to a neuroscience application. The paper proposes a useful technically sound method, with a compelling application.Detailed comments:- A generalized version of the "SM-LMC" was discussed in chapter 4.4, p. 137 of Wilson [2014], "Covariance kernels for fast automatic pattern discovery and extrapolation with Gaussian processes", PhD thesis.In particular, it was proposed to model $\bm{y}(x) = W(x) \bm{f}(x)$, where $\bm{y}(x)$ are the responses, $\bm{f}(x)$ is a vector of GPs each with 1 component SM kernels, and W(x) is an input dependent matrix of mixing weights.Setting W(x) = W, one recovers the SM-LMC.    Quality:The paper is technically sound, and the application is very compelling.I would expect, however, the computational complexity to be O(C^3 N log N) rather than O(C^3 N).An alternative to the proposed approach would be to exploit the Toeplitz structure of your matrices and perform linear conjugate gradients, which could be exact to within numerical precision, but somewhat slower than your approach.Evaluating the log determinant would require O(N^2) computations and O(N) memory. By following an inference strategy similar to "Gaussian Process Regression Networks", ICML, you may be able to further reduce your complexity to O(C N).In general, I was a little unsure of how much the scalability of the proposed approach was necessary for this application.It would help to clarify this in a final version. ClarityThe paper was a pleasure to read.  OriginalityThere is valuable novelty in decoupling the phase shifts between different tasks, as proposed in the CSM kernel, which usefully builds on the SM kernel.   SignificanceI can easily imagine others finding this methodology widely applicable.I really enjoyed the neuroscience application, and only wish the paper went into this application more deeply, with more interpretation of results.It's very exciting that we can use these kernel learning approaches to gain new significant insights into our data, in addition to improved predictive performance.


Sample Complexity Bounds for Iterative Stochastic Policy Optimization
This paper is concerned with robustness analysis of decision making under uncertainty. We consider a class of iterative stochastic policy optimization problems and analyze the resulting expected performance for each newly updated policy at each iteration. In particular, we employ concentration-of-measure inequalities to compute future expected cost and probability of constraint violation using empirical runs. A novel inequality bound is derived that accounts for the possibly unbounded change-of-measure likelihood ratio resulting from iterative policy adaptation. The bound serves as a high-confidence certificate for providing future performance or safety guarantees. The approach is illustrated with a simple robot control scenario and initial steps towards applications to challenging aerial vehicle navigation problems are presented.

Paper considers iterative stochastic policy optimization, and independent of how the policy is updated, the authors provide a bound on the expected cost resulting from the execution of a new stochastic policy. using observed samples from a previous policy. The bound is illustrated in an example of aerial vehicle navigation.Quality, Results suggest that original policy could be improved to achieve only a 10% predicted collision rate (down from 50%) - while that shows good improvement, it would be helpful to know how the original policy was chosen, as a 50% collision probability seems very high. I.e. in this case, could the authors provide evidence that a good selection of hand picked control parameters cannot do better than 50% coll prob to start? How much does the choice of alpha in Line 245 impact the tightness of the bound? While closed form, that does not appear to be the best choice.Clarity, Section 2.1 seems quite separate from the rest of the paper, and given the level of specificity given in the discrete time dynamics, the concepts are not well connected to what follows. Line 105 talks about "an equivalent stochastic optimization" - are these problems really equivalent? If so, equivalent in what sense, and if not, why discuss the discrete form?Also, the stochastic problemon Lines 107 and 110 appear similar to chance constrained control (an MPC) - thus the authors should discuss those similarities. Numerous papers are available - the authors could start here (though many newer results exist):@article{schwarm1999chance, title={Chance-constrained model predictive control},author={Schwarm, Alexander T and Nikolaou, Michael},journal={AIChE Journal}, volume={45}, number={8}, pages={1743--1752},year={1999} }The figure text is far too small to be readable Legends of figs 2(b), 2(c), 3(b), 3(c) have 3 lines, but the blue one is not observable  The discussion of the results at the end of section 4 could be greatly improved - there is very little discussion of the plots in the following pages, and what is there doesn't shed much more light on the results. The graphs suggest that something interesting is going on, but exactly what is hard to know.  Originality Authors point out that obtaining bounds for the cost is challenging because of the unbounded change-of-measure likelihood ratio, making approaches based on hoeffdings inequality impractical. They propose an alternative from Catoni [25], but that leads to statements like "The key step is to use the the [sic] moment truncation map \psi proposed by Catoni [25]..." [Line 232]. It is good to build on prior work, but this statement somewhat reduces the apparent originality of this work as it is indicates that Catoni's contribution was significant.SignificanceThe result is significant in that the bounds provided will enable the development of algorithms that yield policies that are robust in the sense that the resulting cost will not exceed an apriori known cost (at least with high probability).Minor points:[7] seems like an odd choice to reference for robust MPC - there better papers to consider, e.g. http://homes.esat.kuleuven.be/~maapc/static/files/CACSD/survey-robust-mpc.pdf


Robust Gaussian Graphical Modeling with the Trimmed Graphical Lasso
Gaussian Graphical Models (GGMs) are popular tools for studying network structures. However, many modern applications such as gene network discovery and social interactions analysis often involve high-dimensional noisy data with outliers or heavier tails than the Gaussian distribution. In this paper, we propose the Trimmed Graphical Lasso for robust estimation of sparse GGMs. Our method guards against outliers by an implicit trimming mechanism akin to the popular Least Trimmed Squares method used for linear regression. We provide a rigorous statistical analysis of our estimator in the high-dimensional setting. In contrast, existing approaches for robust sparse GGMs estimation lack statistical guarantees. Our theoretical results are complemented by experiments on simulated and real gene expression data which further demonstrate the value of our approach.

The paper presents a method for robust inference of sparse GGM models. Theoretical guarantees on estimated models are given even when these estimates correspond to merely local optima of the relevant objective function.The paper is clearly presented and organised logically: after introducing the problem to be solved the objective function (3) is presented. A reasonable approach to optimisation is presented - Algorithm 1. Since the objective is only bi-convex not convex no attempt is made to find a global minimum. Instead theoretical guarantees are given on *any* local minimum subject to certain conditions. Some work is done in Section 3 to provide the reader with some feeling for when these conditions will be met. Experiments with both synthetic and real data are provided.(3) is a reasonable way of dealing with outliers/noise and it is clearly good to have the theoretical guarantees. However, I found it hard to be convinced that much had been gained - for practical problems - over existing technqiues. (The authors deserve praise for comparing against a good selection of alternatives.) The results in Fig 1 show that the proposed method is indeed "competitive" but not superior by a large margin. On the other hand doing somewhat better than alternatives *and* having some theoretical guarantees is a contribution.Other points...Please mention that \tau_{1}, \tau_{2} and \tau are all defined in the supplementary material - I searched hard for their definitions in the main paper!I did not understand the sentence at line 236. What does it mean to choose a parameter "from [a] quantity"?51 paid on -> paid to 123 if i-th sample -> if the i-th sample 125 only small number -> only a small number 242 how easy we can satisfy -> how easily we can satisfy 277 Note that -> Note that the 278 Corollary 2 recover -> Corollary 2 reveals an 285 by arbitrary optimization -> by an arbitrary optimization286 multiple local optimum -> multiple local optima 291 from other -> from another365 Fix j subscript. Also is -> if 463 gaussian -> Gaussian


Spectral Norm Regularization of Orthonormal Representations for Graph Transduction
Recent literature [1] suggests that embedding a graph on an unit sphere leads to better generalization for graph transduction. However, the choice of optimal embedding and an efficient algorithm to compute the same remains open. In this paper, we show that orthonormal representations, a class of unit-sphere graph embeddings are PAC learnable. Existing PAC-based analysis do not apply as the VC dimension of the function class is infinite. We propose an alternative PAC-based bound, which do not depend on the VC dimension of the underlying function class, but is related to the famous Lovász ϑ function. The main contribution of the paper is SPORE, a SPectral regularized ORthonormal Embedding for graph transduction, derived from the PAC bound. SPORE is posed as a non-smooth convex function over an elliptope. These problems are usually solved as semi-definite programs (SDPs) with time complexity O(n 6 ). We present, Infeasible Inexact proximal (IIP): an Inexact proximal method which performs subgradient procedure on an approximate projection, not necessarily feasible. IIP is more scalable than SDP, has an O(
) convergence, and is generally applicable whenever a suitable approximate projection is available. We use IIP to compute SPORE where the approximate projection step is computed by FISTA, an accelerated gradient descent procedure. We show that the method has a convergence rate of O(
). The proposed algorithm easily scales to 1000's of vertices, while the standard SDP computation does not scale beyond few hundred vertices. Furthermore, the analysis presented here easily extends to the multiple graph setting.

The paper starts by introducing the background and showing the contributions. The authors then use the Zhang and Ando result showing that SSL reduces to an equivalent kernel-based supervised learning problem for the rest of the paper. In section 2 and appendix, they provide an error bound showing that minimising the spectral norm of the graph kernel matrix is a good way to improve generalisation. In section 3, they add a spectral norm regularization term to the Zhang and Ando formulation, and provide links between the error convergence and the Lovasz number of the data graph. Section 4 proposes a proximal solver, where the specificity is that it deals with the projection step approximately to handle the constraint of a cone-polytope intersection. Section 5 proposes an MKL extension, and section 6 concludes with experimental results on several datasets.            Overall, the paper is interesting and contains a lot of material (maybe even a bit dense for a single paper), with several important contributions. The link between large-scale topological properties of the data graph (Lovasz number), which reflect its organisation in some space or manifold, and the convergence of the embedding estimation (eq. 5) is particularly enlightening and bridges several fields.            In section 1 after equation 3 I would perhaps make explicit that the correspondance between eqs. (3) and (1) means that a semi-supervised learning problem becomes a supervised learning problem, so that the learning is on labelled samples, otherwise it is a bit surprising to see indices over S only (with no \bar{S}) in the sums in equation (3), whereas in eq. 1 the regulariser is also on non-labeled samples while the loss is on labelled data.      In section 2, The notation of equation 5 is confusing - why is there an equal sign just before \omega_C ?            In section 3 around line 162, maybe mention how the embedding (U) is recovered from K ? There are several (practical) choices for the factorisation.            In section 4,Algorithm 1 step 8-9 - ensuring that K_r is PSD typically quite expensive (for general dense square symmetric matrices), and here an eigendecomposition is done every time (supplementary D.3.2). Is this where the algorithm takes most time? Would it be feasible to just optimise on a factorisation of K (say, Cholesky with K=LL^T), projection to the feasible set would be to a matrix with all-zeros in upper triangular, which would by construction ensure that K is PSD ?            Zhang and Ando is in NIPS 2005, not 2006


Online Learning with Gaussian Payoffs and Side Observations
We consider a sequential learning problem with Gaussian payoffs and side observations: after selecting an action i, the learner receives information about the payoff of every action j in the form of Gaussian observations whose mean is the same as the mean payoff, but the variance depends on the pair (i, j) (and may be infinite). The setup allows a more refined information transfer from one action to another than previous partial monitoring setups, including the recently introduced graph-structured feedback case. For the first time in the literature, we provide non-asymptotic problem-dependent lower bounds on the regret of any algorithm, which recover existing asymptotic problem-dependent lower bounds and finitetime minimax lower bounds available in the literature. We also provide algorithms that achieve the problem-dependent lower bound (up to some universal constant factor) or the minimax lower bounds (up to logarithmic factors).

The authors consider a sequential decision problem in stochastic environment with Gaussian payoffs and Gaussian side observations. In each time step a learner chooses an action from a fixed finite set of actions, receives a random reward associated with the action, and observes a feedback from other actions. All the observation are Gaussian random variables with fixed expectations and variance depending on a chosen action. The setting includes several previously studied feedback models: full information, bandit, and partial monitoring with feedback graphs (previously studied in adversarial environment).The setting of the paper is very interesting and naturally extends previous settings, which provides tools to capture problems with complicated feedback structure.Pros of the paper:The main contribution of the paper are lower bounds on the regret for any algorithm. The authors provide non-asymptotic problem-dependent lower bound, which recovers asymptotic problem-dependent lower bound, and finite-time minimax lower bound.The authors proposed two algorithms and their finite-time analysis. The first algorithm achieves upper bound which matches asymptotic problem-dependent lower bound (up to constants) and the second algorithm achieves upper bound which matches minimax lower bound (up to logarithmic factors).Cons of the paper:Even though the setting provides rich feedback structures, the upper bounds for the algorithms are analysed only for a special case of uniform variance, where the variance terms are equal to either sigma or infinity (feedback graphs). In particular, the proofs of Theorem 6 andTheorem 8, provided in appendix B and C(proved using standard tools) are only considering this special case and it is unclear if the approach would extend to the general case which seems more interesting. However, my main objection is that the paper lacks intuition behind the proofs (especially for the lower bound). A good example is Section 3.2.1.: There is no reason given why are the quantities defined as such. What does the set of the reasonable items mean? Where are the definitions for m_i^+ and m_i^- coming from? The whole Section 3.2.1. is extremely hard to parse and features almost no explanations or relations to the studied problem.With regards to the lower bound, it is also unclear whether this is indeed the difficult case, since the upper-bound for the nonuniform case is not given.Furthermore, there is a significant amount of mistakes, including wrong or missing indices. Examples: assumption (iii) in section 2, simplex using set C_N^T as upper index insection 3.1), missing terms (e.g. definition of C_T in section 2.1)). Combined with missing descriptions of the ideas, this makes it even harder toto follow the paper.Conclusion:Even though the results in the paper are substantial, the paper needs significant changes to eliminate mistakes, provide intuition behind (mostly the lower bound) proofs, improve clarity and readability.Therefore in my opinion, the paper is not ready to be published in the current form.


Smooth Interactive Submodular Set Cover
Interactive submodular set cover is an interactive variant of submodular set cover over a hypothesis class of submodular functions, where the goal is to satisfy all sufficiently plausible submodular functions to a target threshold using as few (cost-weighted) actions as possible. It models settings where there is uncertainty regarding which submodular function to optimize. In this paper, we propose a new extension, which we call smooth interactive submodular set cover, that allows the target threshold to vary depending on the plausibility of each hypothesis. We present the first algorithm for this more general setting with theoretical guarantees on optimality. We further show how to extend our approach to deal with realvalued functions, which yields new theoretical results for real-valued submodular set cover for both the interactive and non-interactive settings.

*Summary*This paper considers the interactive submodular set cover problem, where one is given a set of submodular functions, and needs to optimize an unknown true function using a minimal amount of actions. The paper considers two extensions of previous work: (1) varying "saturation" conditions (i.e., threshold) for different hypothesis functions in the hypothesis class; and (2) settings when the hypothesis submodular functions are real-valued. The paper provides a unified greedy framework to (approximately) solve both problems. In general the paper is well organized and the theory seems sound, even though some parts could have been made clearer. My major criticism is the empirical part of the paper, namely there is no empirical support why one should use the proposed algorithm. please see my detailed comments below.*Quality*The paper provides worst case cost bound for the smooth interactive set cover problem, by designing a submodular meta-objective objective for various scenarios (of the threshold function). The key result of the paper the approximation guarantee for the multiple threshold version (and the authors show that one can reduce the convex threshold curve version can be approximately reduced to the previous case). The theory and proofs non-trivially generalize the results of Guillory and Bilmes, 2010.The authors gives an example of a real-world applications, but show no empirical results. In the simulation results, there are no competitive approaches (even random selection) as well, thus it's hard to tell the empirical effectiveness of the algorithm.The bound depends on the number of thresholds $N$. When N=1 it reduces to the Noisy ISSC setting. It would be helpful to see empirically how different N's influence the performance of the algorithm. *Clarity*The paper mentions active learning a couple of times in the paper. However, along the line of submodular maximization, active learning is more of "exploring" tests that can mostly reduces size of the version space, and does not favors the actions that are "exploitative". It may cause some confusion to readers.There are a list of typos / points that are unclear, as summarized below.Ln 73, in the cost bound, define $a$. Ln 208-210, if alpha_n > alpha_j, then by Ln 140, it should hold that kapa_n < kapa_j. Hence the kapa_n and kapa_j in equation (2) should be flipped.Ln 230, from the figure, shouldn't the condition only enforces that either of the two threshold must be satisfied, according to (A) (Ln 228-229)?In section 4.5, there seem to be a couple of typos / unclear explanations to the experiment: Ln 327, each cluster was randomly assigned 15 clusters of interest: do you mean 15 hypotheses of interest? Ln 334 states that "at epsilon = 15, the algorithm is allowed to terminate immediate immediately". Why is the cost still non-negative when epsilon >= 15?Ln 429, threshold for to*Originality and Significance*The work builds upon and extends the work of Guillory and Bilmes, 2010. As I mentioned earlier, even though it relies heavily on the previously work, the generalization in the proofs is novel.Since the construction of the algorithm (specifically, the surrogate meta objective) is non-trivial, I would like to see some empirical evidences showing that why one should use the proposed approach.


A Convergent Gradient Descent Algorithm for Rank Minimization and Semidefinite Programming from Random Linear Measurements
We propose a simple, scalable, and fast gradient descent algorithm to optimize a nonconvex objective for the rank minimization problem and a closely related family of semidefinite programs. With O(r 3 κ 2 n log n) random measurements of a positive semidefinite n×n matrix of rank r and condition number κ, our method is guaranteed to converge linearly to the global optimum.

This paper proposes a gradient descent type algorithm for rank minimization in the space of symmetric positive semidefinite matrices with linear equality constraints. It was motivated by the recent work by Candes et al [5]. The basic algorithm as in [5] is based on the Wirtinger flow algorithm. Convergence analysis and sample complexity analysis are also provided for the proposed algorithm. The paper is well-written with consistent notations and a high level outline of the convergence analysis. These make this paper more accessible to readers. In addition, with a nice initialization strategy, it converges to the global optimum under some assumptions.This is a good paper. I was pleased to review this paper.My general comments and suggestion are the following.  L177 : Authors claim that the method generalizes Wirtinger's flow algorithm which I'm not sure is right because the proposed method is defined with only real values (NOT complex); z and ai's. Please clarify that.Please briefly explain about a "phase retrieval" for readers who are not familiar with the previous work by Candes et al [5].Assumption on the true rank:L183 : The paper assumes that the true rank is given prior to the algorithm. It might be a quite strong assumption in some sense and the nuclear norm method does not assume this. Discuss the effect of misspecification of rank with the proposed method.L253 : As authors suggested, if one wants to search for the proper rank by running the algorithm multiple times, is there a systematic way of choosing r after running the algorithm say n number of times and looking at the value f(Z) after each r?Eventually, the problem solved by the method is reduced to an optimization for a fixed (low) rank solution. In literature, it was addressed by multiple methods especially using manifold frameworks, which are not discussed in the paper. You may want to refer relevant works (R1-R3) for readers.R1. Journee, Michel, et al. "Low-rank optimization on the cone of positive semidefinite matrices." SIAM Journal on Optimization 20.5 (2010): 2327-2351. R2. Vandereycken, Bart, and Stefan Vandewalle. "A Riemannian optimization approach for computing low-rank solutions of Lyapunov equations." SIAM Journal on Matrix Analysis and Applications 31.5 (2010): 2553-2579. R3. Vandereycken, Bart. "Low-rank matrix completion by Riemannian optimization."SIAM Journal on Optimization 23.2 (2013): 1214-1236.


Learning with Group Invariant Features: A Kernel Perspective
We analyze in this paper a random feature map based on a theory of invariance (I-theory) introduced in [1]. More specifically, a group invariant signal signature is obtained through cumulative distributions of group-transformed random projections. Our analysis bridges invariant feature learning with kernel methods, as we show that this feature map defines an expected Haar-integration kernel that is invariant to the specified group action. We show how this non-linear random feature map approximates this group invariant kernel uniformly on a set of N points. Moreover, we show that it defines a function space that is dense in the equivalent Invariant Reproducing Kernel Hilbert Space. Finally, we quantify error rates of the convergence of the empirical risk minimization, as well as the reduction in the sample complexity of a learning algorithm using such an invariant representation for signal classification, in a classical supervised learning setting.

Haar-group invariant kernels have the form:K(x,z) = \int_G \int_G k(gx, g'z) dm(g) dm(g'), (1)where G is a compact unitary group. Therefore, we say these kernels are group invariant since k(x,z) = k(gx,g'z) for any pair of transformations (g,g') belonging to the group G. For example, if we deal with images, an useful group G could a collection of translations and rotations.Since the double integral in (1) is expensive to compute, the authors propose to use sampling to compute a finite dimensional, randomized feature map that approximates the kernel K. In particular, the feature map is constructed by sampling random elements g from G using the measure m(g), sampling "templates" t from a reference distribution (which is set to be Gaussian in this paper), and then constructing the discretized CDF of the random variable $\langle gt,x \rangle$, for an input x.The construction is accompanied with concentration inequalities that depict how fast does this randomized feature map converge to the true Haar-group-invariant kernels, and an adaptation of the risk bounds from [Rahimi and Recht, 2007] for the introduced group-invariant feature map.Question: how does the number of templates affect the quality of approximation? It seems like all asymptotic guarantees are wrt n (the number of bins in the CDF discretization) and not wrt m (the number of templates, as played with in the experiments). What is the learning rate between K_s and K?The numerical simulations seem preliminary, as the method is demonstrated only on two small-scale datasets. Furthermore, there is no comparison to other related methods (such as regular random features where the number of them is increased at no cost thanks to, for example, a Fastfood approximation). Also, the plots are difficult to interpret: perhaps a logarithmic scale on the Y-axis would help.Question: how would you learn G if you are not sure about the nature of your data? This approach seems to invoke the human to be more in the loop, drifting one step away from automatic machine learning.The technical quality of the paper is good, and it is fairly well written. The significance of the paper is undermined by the minimal experimental section, which is conducted on small-scale digit datasets (this Section would be greatly strengthened by incorporating competing methods such as large random feature ensembles, histogram random features, convolutional neural networks... And extending the simulations to larger datasets such as natural images ).


On some provably correct cases of variational inference for topic models
Variational inference is an efficient, popular heuristic used in the context of latent variable models. We provide the first analysis of instances where variational inference algorithms converge to the global optimum, in the setting of topic models. Our initializations are natural, one of them being used in LDA-c, the most popular implementation of variational inference. In addition to providing intuition into why this heuristic might work in practice, the multiplicative, rather than additive nature of the variational inference updates forces us to use non-standard proof arguments, which we believe might be of general theoretical interest.

The authors prove that variational inference in LDA converges to the ground truth model, in polynomial time, for two different case studies with different underlying assumptions about the structure of the data. In this analysis, the authors employ "thresholded" EM updates which estimate the per-topic word distribution based on the subset of documents where a given document dominates. The proofs, which are provided in a 35-page supplement, require assumptions about the number of words in a document that are uniquely associated with each topic, the number of topics per document, and the number documents in which a given word exclusively identifies a topic.I am not enough of a specialist to evaluate the provided proofs in detail, so I will restrict myself to relatively high level comments. Empirically speaking, variational inference can and does get stuck in local maxima. This suggests that at least some of the assumptions that the authors make will not (always) hold for real-world datasets. As such my intuition is that the authors may have identified a set of conditions under which variational inference in LDA is provably "easy". This in itself is a potentially useful contribution. Assuming the proofs are sound, I would argue for acceptance.## Comments on content- I'm a bit confused about the role of the  parameters, about which the authors appear to make several conflicting statements    > The  Dirichlet parameters do not have a closed form expression and are   > updated via gradient descent.         Do the authors in fact perform emprical Bayes / type II max likelihood   estimation? The statement below suggests otherwise    > In particular, in the E-step, in the large document limit, the first   > term in the update equation for  has a vanishing contribution    The authors further on state that    > These assumptions are a less smooth version of properties of the   > Dirichlet prior. Namely, it's a folklore result that Dirichlet draws   > are sparse with high probability, for a certain reasonable range of   > parameters.    Dirichlet draws are indeed sparse when  << 1.0, but not necessarily when    => 1.0. This is not so much a "folklore result" as a well known   property of the Dirichlet distribution. Indeed, the lemma referenced in   the supplementary material assumes _i = C_i / K^c with c > 1.    The authors assume  can be ignored in EM updates, but at the same time   make assumptions about sparsity. It would appear they are implicitly   making assumptions about the  true generative process of the data, even   when the same values for  are not necessarily used during inference. Can   they comment on this?- What is meant by "if i does not belong to document d" in Algorithm 1?- The exposition in section 4.2 as a whole is somewhat confusing. I would recommend that the authors simply write down their proposed updates, then explain how these updates relate to the ones in [Lee and Seung 2000], and then argue what guarantees apply to these updates.## Comments on style- It would benefit the readability of this paper if the authors would put equations on separate lines (and number them) -- at the very least in those cases where equations require a summation sign. This also eliminates awkward back references like "the first term in the update equation for ".- Similarly, explicitly writing out the generative model, even if well known, never hurts. It avoids vague phrases like "sampled according to a prior distribution ".- There is some confusion of notation between X and w, which are both used to refer to the words in the document set.- There is also some confusion as to the indexing of variables, which in some cases do and in others do not include the document index d.- This is a matter of personal preference, but I would use `discrete` rather than `multinomial` when describing the generative model of LDA.- The words "latent Dirichlet allocation" do not appear in the manuscript. Why did the authors consistently write "topic model" to refer to LDA?## Typos- 131: min -> \min