How to Hedge an Option Against an Adversary: Black-Scholes Pricing is Minimax Optimal
We consider a popular problem in finance, option pricing, through the lens of an online learning game between Nature and an Investor. In the Black-Scholes option pricing model from 1973, the Investor can continuously hedge the risk of an option by trading the underlying asset, assuming that the asset's price fluctuates according to Geometric Brownian Motion (GBM). We consider a worst-case model, in which Nature chooses a sequence of price fluctuations under a cumulative quadratic volatility constraint, and the Investor can make a sequence of hedging decisions. Our main result is to show that the value of our proposed game, which is the "regret" of hedging strategy, converges to the Black-Scholes option price. We use significantly weaker assumptions than previous work-for instance, we allow large jumps in the asset price-and show that the Black-Scholes hedging strategy is near-optimal for the Investor even in this non-stochastic framework.

This paper considers the problem of option pricing in
      finance. Using a minimax approach, the authors construct a game between
      Nature and an Investor, prove that the game value of this game converges
      to the classic Black-Scholes option price, and give an explicit hedging
      strategy that achieves this value. Clarity: This is a very
      math-heavy paper. Unfortunately, I am not very knowledgeable in the area
      of stochastic calculus, so I am unable to verify the correctness of the
      proofs in the paper and in the 14-page supplementary material. The authors
      do provide a reference to a standard book in stochastic calculus, but
      unfortunately I do not have the time to familiarize myself with the
      material. This is not a critisism of the paper. In fact, the authors
      explain the results in words very well and outline proof steps clearly for
      anyone who may look to build off these results further in the future.
      Quality: As mentioned above, I could not verify the correctness of
      the results, but rigorous proofs are provided for all of the theorems. The
      authors clearly state the strengths of their results (explicit strategy,
      weaker assumptions required) over previous work. Originality: The
      Nature vs. Investor game approach is not new, which the authors make clear
      in the introduction. This paper is basically an extension of Abernethy et
      al. [1] in two ways. Firstly, the assumptions required for the main result
      are strictly weaker than the assumptions used by Abernethy et al.
      Secondly, the authors provide an explicit hedging strategy for achieving
      the game value, whereas such a strategy was absent from [1]. While the
      mechanism may not be new, the analysis appears to be sufficiently
      original. Significance: Given the statement above, I would
      describe the contributions here as incremental as the results are not
      extremely game-changing compared to previous work. However, the analysis
      appears quite difficult and novel, and the problem does appear to be
      addressed in a better way than previous research. On the other
      hand, I am somewhat concerned about this paper's relevance to the NIPS
      community. After a quick scan through the technical areas listed in the
      call for papers, I had a hard time placing this paper in any of the given
      categories (although there is a "but not limited to" clause). Given the
      paper is purely analytical with no empirical results, perhaps this paper
      is more appropriate for a mathematics / ecommerce journal or conference.
      On that note, I was wondering if there are any other lessons to be learned
      here for a practitioner (Investor) beyond the fact that we now have more
      evidence that Black-Scholes hedging is robust. Typo: - line
      227: "to to" RESPONSE TO AUTHOR FEEDBACK: While option
      pricing does not seem to fit the typical NIPS audience, I did fail to make
      the connection with stochastic optimization and sequential decision making
      algorithms. These are certainly appropriate areas for
      NIPS.


Discovering Hidden Variables in Noisy-Or Networks using Quartet Tests
We give a polynomial-time algorithm for provably learning the structure and parameters of bipartite noisy-or Bayesian networks of binary variables where the top layer is completely hidden. Unsupervised learning of these models is a form of discrete factor analysis, enabling the discovery of hidden variables and their causal relationships with observed data. We obtain an efficient learning algorithm for a family of Bayesian networks that we call quartet-learnable. For each latent variable, the existence of a singly-coupled quartet allows us to uniquely identify and learn all parameters involving that latent variable. We give a proof of the polynomial sample complexity of our learning algorithm, and experimentally compare it to variational EM.

The paper discusses parameter estimation and two
      versions of quartet-based structure learning algorithms for bipartite
      noisy-or networks of binary variables. The presented ideas are based on a
      recent paper [Halpern & Sontag, 2013], extending to a more general
      class of networks. The performance of the proposed algorithm is
      illustrated on synthetic examples. The exposition of paper is generally
      logical but the writing needs a lot of improvement. The paper
      contains potentially interesting algorithms but it is at times hard to
      tell which ideas are new and which ones are borrowed. A summary of the
      main contributions and their significance would be useful. What are the
      real applications of the algorithm, for which existing algorithms would
      not be applicable? * Main concerns: -- Experiments: The
      authors list a number of practical applications but the performed
      experiments are rather limited. -- Exposition: There are too many
      missing definitions, missing essential information (e.g., the algorithm by
      Halpern & Sontag ), inconsistent notation, and wrong pointers to other
      parts of the text. If this paper is accepted, I hope the authors take the
      task of improving the text seriously. * Additional remarks: --
      Rank test: How practical is the epsilon-rank-testability assumption? In
      what applications would it be satisfied (with large epsilon)? Can epsilon
      be estimated from the data? -- Definition 2: Please explain what you
      mean by "the third eigenvalue of ... is at least epsilon" if the
      eigenvalues are not necessary nonnegative real numbers. -- Coherence
      test: For each quartet, 4 sets of parameters have to be estimated. State
      the computational cost of this estimation. "If the parameters learned do
      not agree" should be further explained. -- The recent tensor
      decomposition method [Halpern & Sontag] seems to be an important part
      of the proposed algorithms and should at least briefly be explained.
      -- The definition of noisy-or networks could be more thorough. --
      Some notation updates are required, e.g., on p.2, U stands for a set of
      latent variables but on p.3 U is used to describe a set of observed
      variables. -- Figure 3 has wrong caption. Generally, Section 3 looks
      like it suffered last minute restructuring. -- Figure 2 should come
      later, around the place it is referred to. The algorithms should probably
      not be called figures. -- Thorough proofreading is necessary. There
      are multiple typos and missing references to figures and sections (in the
      main file and in the supplementary material).


Sparse nonnegative deconvolution for compressive calcium imaging: algorithms and phase transitions
We propose a compressed sensing (CS) calcium imaging framework for monitoring large neuronal populations, where we image randomized projections of the spatial calcium concentration at each timestep, instead of measuring the concentration at individual locations. We develop scalable nonnegative deconvolution methods for extracting the neuronal spike time series from such observations. We also address the problem of demixing the spatial locations of the neurons using rank-penalized matrix factorization methods. By exploiting the sparsity of neural spiking we demonstrate that the number of measurements needed per timestep is significantly smaller than the total number of neurons, a result that can potentially enable imaging of larger populations at considerably faster rates compared to traditional raster-scanning techniques. Unlike traditional CS setups, our problem involves a block-diagonal sensing matrix and a non-orthogonal sparse basis that spans multiple timesteps. We provide tight approximations to the number of measurements needed for perfect deconvolution for certain classes of spiking processes, and show that this number undergoes a "phase transition," which we characterize using modern tools relating conic geometry to compressed sensing.

[EDIT] I read the rebuttal and removed point 3 from my
      review since it has been sufficiently addressed. Although I'm still not
      100% happy with the responses to points 1 and 2 I upped the score to 8.
      The authors propose a framework for two-photon imaging based
      on compressed sensing. In the first half of the paper they use simulations
      to demonstrate the feasibility of estimating calcium traces, non-negative
      firing rates (closely corresponding to spike times in a high-SNR
      situation) and spatial intensity profiles of the imaged neurons based on
      random projections of the data. In the second part they show that in the
      noiseless case the quality of reconstruction undergoes a phase transition.
      They derive an approximation to the phase transition curve, which they
      show to be quite accurate under some conditions. I am generally
      enthusiastic about the approach since it could have a lot of potential to
      substantially improve the efficiency of two-photon imaging experiments,
      which generally suffer from the inherent trade-off between data quality
      and number of neurons that can be imaged. Although I am unable to verify
      the math, in particular in section 3, judging from the successful
      verification through simulations I believe it's correct. Besides
      the stylistic complaint that the paper is written in a rather technical
      way with a lot of jargon, my major concern is that the authors did not do
      a good job at convincing the reader that the approach will actually be
      feasible or even useful in practice. There are three main issues related
      to this concern: 1. Since the method is sensitive to noise (Fig.
      1) it is important to test it with realistic noise levels. In practice,
      noise levels are very high with calcium imaging due to shot noise, in
      particular so if one images fast as the authors propose. Although I don't
      know for sure what a typical SNR (as defined in line 189) would be for a
      real experiment, my guess would be that it's closer to 0 dB than to 20 dB
      as used in the Fig. 1 and I wouldn't be surprised if it's even
      substantially lower than 0 dB. Since this is a very fundamental problem,
      the authors should present some convincing evidence that the SNRs they use
      are realistic. 2. For estimating the locations, it looks like
      although the neurons were overlapping there was no background (neuropil)
      activity between the neurons and the neurons were actually covering the
      major fraction of the imaged space. However, in practice the situation is
      dramatically different. In a 3d volume cell bodies make up at most 10%
      (probably less) of the voxels while the remainder consists of neuropil
      (axons and dendrites), which display calcium activity as well (although
      usually highly correlated and with faster dynamics). Although I would
      guess that the result would be a few (large) singular values that could be
      excluded based on their spatial profile, I cannot judge whether the
      algorithm (P-NN) is really not affected in a major way by this. In
      addition, using SVD to identify neurons assumes they fire uncorrelated,
      which is unlikely to be true for most experiments. Since one of the main
      goals of imaging many neurons at the same time is to characterize their
      joint activity, this could be a major caveat and should at least be
      discussed. [3. removed since clarified by rebuttal] The
      above concerns would be somewhat less of a problem if the goal of the
      paper was to just lay out the basic mathematical framework, but then the
      authors should be more upfront about the various practical problems that
      may arise. Currently, the manuscripts reads somewhat like the method could
      be used to build an actual imaging setup, but I think we are still
      relatively far away from that goal.


Variational Policy Search via Trajectory Optimization
In order to learn effective control policies for dynamical systems, policy search methods must be able to discover successful executions of the desired task. While random exploration can work well in simple domains, complex and highdimensional tasks present a serious challenge, particularly when combined with high-dimensional policies that make parameter-space exploration infeasible. We present a method that uses trajectory optimization as a powerful exploration strategy that guides the policy search. A variational decomposition of a maximum likelihood policy objective allows us to use standard trajectory optimization algorithms such as differential dynamic programming, interleaved with standard supervised learning for the policy itself. We demonstrate that the resulting algorithm can outperform prior methods on two challenging locomotion tasks.

The paper presents a new model-based policy search
      algorithm that is based on the variational decomposition of policy search.
      I order to estimate the variational distribution, a linearized model is
      used to minimize the KL-term involved in determining the variational
      distribution. The resulting algorithm outperforms existing methods which
      is shown on challenging benchmark tasks. I personally think the
      approach might be promising and i also like the experiments, but I think
      there are crucial comparisons missing. First of all, the presented method
      is a model-based method, and, hence the authors should also compare
      against other model-based approaches. For example, the pilco framework [1]
      learns a GP model for long-term prediction with moment matching (which is
      actually a very similar approximation as the one performed by ddp). The
      policy is then updated by Bfgs. This algorithm is state of the art in
      model-based policy search and it is not clear to me why the presented
      algorithm should perform better than pilco. Furthermore, the authors claim
      that the ddp solution has several disadvantages to the learned policy.
      While this might be well the case, the authors should also show that in
      their evaluations. Maybe the generalization of the learned policy is
      better but the ddp solution has higher quality for a single task. That
      would be good to know. The comparison to the variant of power is also
      rather unfair as power does not use a model. However, as the model is
      assumed to be known, the model can be used to generate a vast amount of
      samples. In the limit of an infinite amount of samples, power should
      produce the same results as the ddp solution?. It would be interesting to
      know how many samples are needed to match the performance of the ddp
      solution. In theory, the presented method probably simplifies the
      computational demands in comparison to the samplebased version of power,
      but, in the limit of infinite samples, the result should be the same.
      Hence, the authors need to evaluate the computational benefits of the
      presented method. Furthermore, the relationship to existing state of the
      art stochastic optimal control (soc) algorithms, such as soc by
      approximate inference [2] or policy improvements by path integrals [3]
      should at least be discussed. Minor issues: - the choice of
      alpha_k seems to be very hacky to me. See [4] for a more sophisticated
      approach for choosing the temperature of the exponential transformation. -
      I do not really get why the exploration strategy of power is "random"
      while the exploration strategy of ddp is guided. Both use exactly the same
      exploration policy. While power uses samples to obtain the variational
      distribution, ddp uses approximation to determine a closed form solution
      for q. In the limit of infinite samples, power should even produce the
      better results as no approximations are used. - how does your method
      perform if the model is not known, but needs to be learned from data,
      e.g., by using a Gaussian Process? [1] Deisenroth et. al. : PILCO:
      A model-based and data-efficient approach to policy search [2] Rawik et.
      al.: Stochastic optimal control by approximate inference [3] Theoudorou
      et. al.: A Generalized Path Integral Control Approach to Reinforcement
      Learning [4] Kupscik et. al.: Data-Efficient Generalization of Robot
      Skills with Contextual Policy Search


One-shot learning and big data with n = 2
We model a "one-shot learning" situation, where very few observations y 1 , ..., y n ∈ R are available. Associated with each observation y i is a very highdimensional vector x i ∈ R d , which provides context for y i and enables us to predict subsequent observations, given their own context. One of the salient features of our analysis is that the problems studied here are easier when the dimension of x i is large; in other words, prediction becomes easier when more context is provided. The proposed methodology is a variant of principal component regression (PCR). Our rigorous analysis sheds new light on PCR. For instance, we show that classical PCR estimators may be inconsistent in the specified setting, unless they are multiplied by a scalar c > 1; that is, unless the classical estimator is expanded. This expansion phenomenon appears to be somewhat novel and contrasts with shrinkage methods (c < 1), which are far more common in big data analyses.

Update: The point of my simple method was to
      obtain a better understanding of the theoretical analysis. I ran the same
      experiments the authors ran from section 7, with d=500,5000,500000 and the
      simple method began to converge to the oracle method and at d=500 it
      certainly did not have an error greater than 500 (roughly 7). I ran all of
      the experiments with n=2. **edit** The focus of these experiments were on
      weak convergence as that was the theoretical analysis performed for n=2.
      For n=2 the simple method does perform very poorly for squared loss, but
      performs better for absolute loss. **end edit** I believe that
      this one-shot idea is very interesting, but the bias correction has not
      been thoroughly explored. The experiments are show a lot of promise, which
      is why I remain borderline with this paper. ==== ***
      This paper aims to provide an analysis for principle component
      regression in the setting where the feature vectors $x$. The authors
      let $x = v + e$ where $e$ is some corruption of the nominal feature
      vector $v$; and $v = a u$ where $a \sim N(0,\eta^2 \gamma^2 d)$ while
      the observations $y = \theta/(\gamma \sqrt{d}) \langle v,u \rangle +
      \xi$. This formulation is slightly different than the standard one
      because our design vectors are noisy, which can pose challenges in
      identifying the linear relationship between $x$ and $y$. Thus, using
      the top principle components of $x$ is a standard method used in order
      to help regularize the estimation. The paper is relevant to the ML
      community. The key message of using a bias-corrected estimate of $y$
      is interesting, but not necessarily new. Handling bias in regularized
      methods is a common problem (cf. Regularization and variable selection
      via the Elastic Net, Zou and Hastie, 2005). The authors present
      theoretical analysis to justify their results. I find the paper
      interesting; however I am not sure if the number of new results and
      level of insights warrants acceptance. *** The result is
      interesting, but lacks enough depth to provide a convincing argument
      for its use. The authors present a very specific setting for the
      design vectors: spiked covariance. The motivation of the bias
      correction is interesting, but how do the authors expect it should
      work under other models for $x$ and $y$? A simulation study of these
      methods could prove useful in order to provide insights into the
      behavior in the non-Gaussian setting. Furthermore, as the authors
      admit, the signal of $x$ is very strong in the direction of the
      regression vectors. It is conceivable that the signal $x$ is strong in
      multiple directions (especially in the high-dimensional setting). How
      would the bias-correction change when adapting to multiple PCA
      factors? *** Overall, the model being used to perform the
      analysis seems to simple to shed serious light on the problem. For
      example, one can show risk consistency in the simple setting of
      letting the largest principle component simply be equal $x_1$. That
      is, only perform PCA one the first example and regression on the
      second. The authors go through a lot of analysis from fundamentals
      without providing intuition for why the shrinkage occurs. In
      particular, the authors should explain why least squares (which
      produces consistent estimators) must be corrected in this context.
      Interestingly, the simple estimator of just taking $x_1$ to be the
      principle component and performing regression on the remaining
      examples performs better. Thus, rather than a shrinkage correction, a
      more reasonable solution seems to be to learn the PC vectors on one
      set of data and regress on the other. Can the authors provide more
      intuition for why they expect their proposed method to perform better
      in generic settings?


Annealing Between Distributions by Averaging Moments
Many powerful Monte Carlo techniques for estimating partition functions, such as annealed importance sampling (AIS), are based on sampling from a sequence of intermediate distributions which interpolate between a tractable initial distribution and the intractable target distribution. The near-universal practice is to use geometric averages of the initial and target distributions, but alternative paths can perform substantially better. We present a novel sequence of intermediate distributions for exponential families defined by averaging the moments of the initial and target distributions. We analyze the asymptotic performance of both the geometric and moment averages paths and derive an asymptotically optimal piecewise linear schedule. AIS with moment averaging performs well empirically at estimating partition functions of restricted Boltzmann machines (RBMs), which form the building blocks of many deep learning models.

This paper presents an alternative method for choosing
      intermediate distributions when performing AIS on distributions in the
      exponential family. Experimental results are promising. These results may
      be of significant practical benefit, as they could improve researchers'
      ability to make objective comparisons between complex probabilistic
      models. Overall I liked it. I did think the theoretical motivation for the
      proposed intermediate distributions was somewhat lacking. Equation
      3 -- cool! "high curvature" --> not high curvature. Rather,
      large information distance per theta distance. Section 5.1 -- why
      is the larger variance of 1437.89 *better*? I would expect larger variance
      to be worse. The experimental results were entirely for second
      order models, while the technique is proposed for arbitrary exponential
      family models. It would be nice to see experiments for other choices of
      sufficient statistic. page 7 - "substantially", say something
      quantitative. Some speculation follows: It's never really
      explained why interpolating between the sufficient statistics should be
      better than interpolating between the parameter values. Eqs. 4 and
      5 strongly suggest that the optimal path in parameter space is the
      geodesic (minimal distance) path between the two parameterizations, with
      the Fisher information as a metric. If I've done my algebra
      correctly, then $ds/d\theta = G$. (write down s = \int dx p(x) dE/dtheta,
      and take the gradient) If the path is linear in $s$, as you propose, then
      for small steps $\Delta s$, $\Delta \theta = d\theta / ds \Delta s =
      G^{-1} \Delta s$. An infinitesimal step along your proposed trajectory in
      terms of \theta looks like the step in s times the inverse Fisher
      information matrix. This is the same as moving towards the final \theta
      taking into account the information geometry (with metric = Fisher
      information matrix). That is, these updates have a functional form which
      is nearly identical to the natural gradient update. Interpreting
      your algorithm in terms of a natural gradient trajectory also opens up
      some possible approximations/extensions. eg, roughly in order of
      decreasing promise: - Apply the same technique to non-exponential
      family models, where the sufficient statistics are not defined, but the
      Fisher information still is. - Use well motivated metrics other than
      the Fisher information -- the matrix natural gradient is likely to best
      bet here (Amari, S.-I. (1998). Natural Gradient Works Efficiently in
      Learning. Neural Computation) - Use an approximation to G^-1, which
      can be computed much more cheaply (Sohl-Dickstein, J. (2012). The Natural
      Gradient by Analogy to Signal Whitening, and Recipes and Tricks for its
      Use. arXiv:1205.1828v1.) - Iterate over choices for the entire
      trajectory, in order to find the global shortest trajectory (with metric
      G), rather than just taking the local shortest steps --
      Finally, here's a plug for reproducible science. You can become
      more awesome by putting source code for experiments in the supplemental
      material! When source code is shared for computational experiments, it has
      a major positive impact on the usefulness of the authors' hard work for
      future researchers, on the credibility of the work, and on the number of
      citations earned. And as a reviewer it makes me more positively disposed
      towards a paper.


Distributed k-Means and k-Median Clustering on General Topologies
This paper provides new algorithms for distributed clustering for two popular center-based objectives, k-median and k-means. These algorithms have provable guarantees and improve communication complexity over existing approaches. Following a classic approach in clustering by [13], we reduce the problem of finding a clustering with low cost to the problem of finding a coreset of small size. We provide a distributed method for constructing a global coreset which improves over the previous methods by reducing the communication complexity, and which works over general communication topologies. Experimental results on large scale data sets show that this approach outperforms other coreset-based distributed clustering algorithms.

This paper proposes new distributed algorithms for
      k-means and k-median methods. Its main idea is to construct a global
      coreset in a distributed fashion using cost information from local
      approximations. This paper is well written with solid content. It proposes
      distributed protocol for k-means/median clustering based on coreset
      construction, provides theoretical guarantee on the approximation and
      upper bounds on the communication cost, performs a couple of experiments
      to show the proposed approach outperforms the existing ones.
      Having said, the paper also comes with several caveats. First of
      all, it is not very clear what is the main technical contribution of the
      paper, especially given the existing works in [5], [10] and [19]. Using
      coreset idea for approximating clustering algorithms is well studied
      (e.g., in [5],[10]), and a distributed clustering algorithm based on
      coreset is proposed in [19]. I clearly see that this paper contributes a
      distributed method for constructing coreset independent of the network
      topology, and develops a better algorithm than that in [19], by reducing
      the communication cost by a factor of sqrt(n). However, the distributed
      coreset construction in Algorithm 1 and the related bounds seems to be a
      straightforward application of results in [5]. It is nice to see
      the theoretical analysis on the algorithm and the upper bounds on coreset
      size and communication cost. However, the upper bounds seem to be too
      loose to use in practical applications. Given a dataset and its location
      over a distributed network, how could you determine the size of coreset?
      The bound for achieving this goal for k-means is along the line of
      O(kd/(e^4)), which could be huge even for moderate large k, d and relative
      small e. For example, for k=10, d = 100, e = 0.1, the size of coreset
      predicted by your bound is 10 million, which could be way more larger than
      the actual data size. In the experiments, the number of
      distributed sites is small (e.g., 10 for Pendigits, 25 for
      ColorHistogram). Have you tried experiments with larger number of
      distributed sites? In addition, the three datasets used are relative-low
      dimensional. Would this kind of approach work for high-dimensional sparse
      data?


Message Passing Inference with Chemical Reaction Networks
Recent work on molecular programming has explored new possibilities for computational abstractions with biomolecules, including logic gates, neural networks, and linear systems. In the future such abstractions might enable nanoscale devices that can sense and control the world at a molecular scale. Just as in macroscale robotics, it is critical that such devices can learn about their environment and reason under uncertainty. At this small scale, systems are typically modeled as chemical reaction networks. In this work, we develop a procedure that can take arbitrary probabilistic graphical models, represented as factor graphs over discrete random variables, and compile them into chemical reaction networks that implement inference. In particular, we show that marginalization based on sum-product message passing can be implemented in terms of reactions between chemical species whose concentrations represent probabilities. We show algebraically that the steady state concentration of these species correspond to the marginal distributions of the random variables in the graph and validate the results in simulations. As with standard sum-product inference, this procedure yields exact results for tree-structured graphs, and approximate solutions for loopy graphs.

Summary: The authors present a compilation
      procedure for implementing message passing for factor graphs over discrete
      random variables in a chemical reaction network. A chemical reaction
      network consists of a set of reactions, comprised of reactants, products
      and reaction rates. They use mass action kinetics to design and
      implement specific information flows through chemical species. The
      authors show how the sum-product algorithm can be mapped onto a set of
      reactions for computation of sum messages, product messages and belief
      recycling reactions. Furthermore, it is discussed how global equilibrium
      states can be achieved and how these correspond to the states BP would
      achieve in a graphical model. The paper is very well written,
      with very minor occasional typos or missing prepositions. However, the
      notation for the chemical reaction networks is unusual to the machine
      learning community and takes significant effort to read comfortably. As
      this is a byproduyct of the unusual topic, it is not a mistake of the
      authors, but needs to be kept in mind if targeting ML communities. Still,
      the authors try to find mathematical abstractions for chemical reactions
      and explain them cleanly. Content-wise, this is an unusual
      paper for the machine learning community. the mapping of a fundamental
      inference algorithm to chemical reaction networks is presented in a lot of
      detail, but can be confusing in terms of certain aspects: it is a bit
      unclear how exactly damped BP corresponds to what the authors are doing as
      they are stating and I would have appreciated a little more elaboration on
      it. Furthermore, the mentioned problem of unassigned probability mass in
      case of high reaction speeds is a recurrent topic in the paper and could
      use a dedicated paragraph where it is theoretically explained in one
      concentrated place to reduce confusion about it. The authors make the
      strong statement that arbitrary factor graphs can be implemented, but only
      show experiments for very low dimensional variables and comparably small
      graphs. A larger scale implementation of a complicated model and a more
      detailed quantitative comparison compared to the shown experiments would
      be welcome. The proposed experiments underline the suggested contribution,
      but are not very extensive or deep. Considering this is work at an early
      stage, it is reasonable that the authors suggest exploring noisy and
      uncertain settings. It would be exciting to see a non-toy experiment (i.e.
      labeling in an mrf) implemented through chemical reaction networks and
      compare it to real BP solutions, probably this should happen with the
      introduction of the Max-Product algorithm the authors suggest for the
      future. In total, the paper is meticulous in suggesting the framework
      of chemical reaction networks and mapping belief propagation to it, but
      the experiments appear a little lacking in real scope. Suggestions:
      a)more and more convincing experiments with more complicated and/or
      bigger graphs b)better theoretical explanation of damped BP in
      relation to this work c) discuss how reaction speeds can be
      implemented in reality with different kappas. I expect them to be
      regulated through chemical compounds, which would most likely lead to
      discrete subsampling of the speed-space. Would this lead to local minima
      or other problems during inference? Are the assumptions of the 'perfect
      chemical reaction network' based on arbitrary species realistic? Where's
      the catch if graphs get bigger and have largewr state spaces and
      hundreds/thousands of chemical species are needed to implement a problem.
      Does it scale? d) Discussions of more models. MaxProduct, continuous
      variables etc.... I found the paper to be highly original
      for the ML-community. While biological computation is a vibrant field with
      many recent contributions by Erik Winfree, Luca Cardelli, Andrew Phillips
      and more researchers stuydying multiple aspects of DNA computing and
      Biological Programming, the particular focus of this work in implementing
      a standard ML inference algorithm and mapping it cleanly to a chemical
      implementation is new to the best of my knowledge and highly significant.
      It will be of great value to the field of biological computation and
      computation in general if well-established and theoretically studied
      inference algorithms can be mapped to biological/hardware implementations,
      especially considering the possibilities of microscopic computation in the
      future.


Exact and Stable Recovery of Pairwise Interaction Tensors
Tensor completion from incomplete observations is a problem of significant practical interest. However, it is unlikely that there exists an efficient algorithm with provable guarantee to recover a general tensor from a limited number of observations. In this paper, we study the recovery algorithm for pairwise interaction tensors, which has recently gained considerable attention for modeling multiple attribute data due to its simplicity and effectiveness. Specifically, in the absence of noise, we show that one can exactly recover a pairwise interaction tensor by solving a constrained convex program which minimizes the weighted sum of nuclear norms of matrices from O(nr log 2 (n)) observations. For the noisy cases, we also prove error bounds for a constrained convex program for recovering the tensors. Our experiments on the synthetic dataset demonstrate that the recovery performance of our algorithm agrees well with the theory. In addition, we apply our algorithm on a temporal collaborative filtering task and obtain state-of-the-art results.

(*) The paper provides algorithms with recovery
      guarantees for the special problem of pairwise interaction tensor
      completion. The algorithms use a convex optimization formulation.
      Results are proven both for exact recovery, and for recovery in the
      presence of noise. In addition, there are also experimental results on
      simulated and real datasets, demonstrating the success of tensor
      decomposition in predicting unobserved entries. (*)
      Overall, the paper is original, provide a significant contribution to a
      hard problem of tensor decomposition, and looks technically impressive
      (I could not verify all the proofs, they are quite involved). It is
      also written quite clearly. The theoretical results showing robustness to
      noise, and the encouraging experimental results show that the author's
      approach has great potential. (*) It is not clear how restrictive
      is the requirement to factor a tensor as a pairwise interaction tensor
      (compared to more general tensors) A better explanations on this
      restriction is needed. (e.g. dimension, number of parameters needed, how
      much would one 'lose' if the true decomposition is more general, yet
      one uses the pairwise interaction decomposition?). (*) One
      weakness is that the authors present all of their results for
      3-dimensional tensors. This is fine for sake of presentation, but it is
      not clear how do the authors results generalize to higher-order
      tensors. Is this all straightforward, or would it be hard to extend the
      techniques used? a discussion of the applicability of the techniques
      to higher-order tensors should be added. (*) Another possible
      concern in the bounds is that the reconstruction guarantees provide bounds
      on the nuclear norm of the difference between the true tensor and the
      reconstructed tensor (e.g. Theorem 2). It would have been better to
      obtain a bound on a more natural L2 norm (Mean-squared error). Can this be
      deduced by the methods used by the authors? (*) The authors
      discuss the computational complexity of their algorithm (of each
      iteration) in page 6, lines 292-304. Would be also good to include the
      memory requirements (large high dimensional tensors may be to big to hold
      in memory. Is the entire tensor need to be stored, or each time only a
      small amount of data?), and the number of iterations required to
      converge (i. are there theoretical guarantees on the error in the convex
      program as a function of the number of iteration? ii. How many
      iterations were required for the experimental examples?). (*) It
      looks like there are many implementation details which are important for
      allowing scalability of the author's algorithm. It is nice that the
      authors give more details of the algorithms in the appendix, but it
      would be better to also provide the author's software implementation
      available for the community. (*) The authors should provide a
      reference for the movielens dataset used. Where was it downloaded from?
      (url? list a reference?), or alternatively, make the data available.
      (*) The description of the experimental results (Figure 1) looks
      wrong. In the text (lines 319-320) it is said that the y-axis is the ratio
      m/d, but this seems to be the x-axis (hard to tell, the axis labels
      are too small). What is the y-axis? the rank r? (also too small).
      Also, it is not clear to me why is recovery error stable as the number
      of observations is increased (table 1 (b)). I would expect the error
      to go to zero as more and more measurements are added (even if these
      are noisy measurements). In Figure 2 - what are the units? is RMSE
      somehow normalized to 1? (or maybe just coincidence) were the movie
      ratings normalized somehow? (*) I don't understand theorem 2
      (robustness to noise). It is stated that if the measurement error is
      \epsilon_1, and one solves an optimization problem allowing error
      \epsilon_2, then the overall reconstruction error is bounded (in
      probability) by \epsilon_1+\epsilon_2. If this is true, why wouldn't
      one want to decrease \epsilon_2 as much as possible (even take
      \epsilon_2=0?) This will reduce the reconstruction error! however,
      intuitively, it seems to me that under measurement noise you want to
      allow some noise \epsilon_2 and not take it to zero, otherwise you may
      increase the dimension, or fail to satisfy the constraints. So how is
      this consistent with the bound given by the authors? (*) Some of
      the proofs steps are unclear. For example, in page 7, lines 335-341: why
      and how did the authors go from 26 \mu_0 r / n_1 n_2, to an expression
      involving also division by n_1 n_3, and then back to 28 \mu_0 r / n_1 n_2?
      Minor: Page 2, line 88: 'show a quadratically ..' -> 'show
      that a quadratically ..' Page 3, line 146: 'sampled from' ->
      'sampled' Page 4, line 206: 'recover algorithm' -> 'recovery
      algorithm' Page 5, line 238: 'be the adjoint' -> 'the adjoint'


The Randomized Dependence Coefficient
We introduce the Randomized Dependence Coefficient (RDC), a measure of nonlinear dependence between random variables of arbitrary dimension based on the Hirschfeld-Gebelein-Rényi Maximum Correlation Coefficient. RDC is defined in terms of correlation of random non-linear copula projections; it is invariant with respect to marginal distribution transformations, has low computational cost and is easy to implement: just five lines of R code, included at the end of the paper.

The authors propose a non-linear measure of dependence
      between two random variables. This turns out to be the canonical
      correlation between random, nonlinear projections of the variables after a
      copula transformation which renders the marginals of the r.vs invariant to
      linear transformations. This work details a simple but seemingly
      powerful procedure for quantifying non-linear dependancies between random
      variables. Part of the key idea of the work is the use of non-linear
      random projections via the "random fourier features" of Rahimi and Recht
      (2007). The authors compare their new randomised dependence coefficient
      (RDC) against a whole host of other coefficients for measuring both linear
      and non-linear dependence. They also consider the suitability of the RDC
      for screening for variable selection in linear prediction problems. I
      think these are useful comparisons and experiments and they allow the
      reader to get a decent feel for the behaviour of RDC. The overall
      exposition of the paper is clear and each component part of the procedure
      is clearly described. In this respect the schematic diagram in figure 1 is
      particularly useful. Similarly, the comparison between other dependence
      coefficients combined with the empirical results is very illuminating and
      suggests that the performance of RDC is very promising. Although
      the authors choose to use RFFs, recently comparisons have been made with
      other schemes for generating random features have been made. Yang et al
      (2012) "Nystrom Method vs Random Fourier Features: A Theoretical and
      Empirical Comparison" flesh out some more of the theoretical properties of
      the difference between RFF and Nystrom features. One difference in
      particular is the difference in sampling scheme: Nystrom samples randomly
      in such a way that it takes the probability distribution of the data into
      account. In this sense it achieves a better estimate of the underlying
      kernel. If this is an important property of the proposed RDC, perhaps
      looking at Nystrom features would be interesting. It also would be
      interesting to see the effect of different types of non-linear projections
      where RFFs are limited to shift invariant kernels (with some other
      extensions), Nystrom features can be computed for any type of kernel
      function. Also, Yang et al show empirically that there is a large
      improvement in predictive performance when more random features are used -
      it would be interesting to see what happens when k is increased. I
      have some concerns about the theoretical treatment presented. The
      discussion of the bound in (10) seems to be skipping over the tradeoff
      involving k. It seems that in order to drive the LC/sqrt(k) term to be
      small (which could be quite slow?), ||m||_F could grow very large compared
      with n but this is hidden by referring to the dependancy as O(1/sqrt(n)).
      As it stands I'm not sure the analysis is sufficient to complement the
      good empirical performance. With regards to the properties of the
      RFF approximation for linear prediction, one possibility is that the
      approximation acts as a regulariser. For this reason, I am not sure that
      the type of generalisation bound used for prediction algorithms are
      completely appropriate to quantify the performance and analyse the
      behaviour of a dependence measure. One small point: the reference
      to Recht is incorrectly referred to as "Brecht" in the text.
      =============== I have read the author rebuttal and I am
      satisfied with the response. The authors should be sure to clarify the
      constraints in the CCA optimisation problem.


Causal Inference on Time Series using Restricted Structural Equation Models
Causal inference uses observational data to infer the causal structure of the data generating system. We study a class of restricted Structural Equation Models for time series that we call Time Series Models with Independent Noise (TiMINo). These models require independent residual time series, whereas traditional methods like Granger causality exploit the variance of residuals. This work contains two main contributions: (1) Theoretical: By restricting the model class (e.g. to additive noise) we provide general identifiability results. They cover lagged and instantaneous effects that can be nonlinear and unfaithful, and non-instantaneous feedbacks between the time series. (2) Practical: If there are no feedback loops between time series, we propose an algorithm based on non-linear independence tests of time series. We show empirically that when the data are causally insufficient or the model is misspecified, the method avoids incorrect answers. We extend the theoretical and the algorithmic part to situations in which the time series have been measured with different time delays. TiMINo is applied to artificial and real data and code is provided.

I read the author feedbacks: - Theorem 1 (i)
      (our main result) does not require acyclicity of the summary time graph.
      Note that acyclicity is only one of two sufficient conditions. I
      found that I did not reflect enough that the acyclic assumption is one of
      two sufficient conditions. Thanks. I changed the score up. I still
      don't understand why they tested their method only on data which are
      generated under the acyclic assumption on the summary time graph. That
      seems to be more limited cases. Summary of the
      paper: This paper considers a class of structural equation models for
      times series data. The models allow nonlinear instantaneous effects
      and lagged effects. On the other hand, Granger-causality based methods do
      not allow instantaneous effects and a linear non-Gaussian method TS-LiNGAM
      (Hyvarinen et al., ICML2008, JMLR2010) assumes linear effects.
      Then they gave Theorem 1, which shows the identifiability
      conditions of the model. The conditions are i) the functions that causally
      relates variables come from a class of functions called IFMOC considered
      in Peters et al. [2011b], ii) the faithfulness and iii) the summary time
      graph is acyclic (They define the summary time graph is a graph that
      contains all the nodes of the time series and an arrow X^i (X at time
      point i) and X^j, i ¥neq j, if there is an arrow from X^i_{t-k} to X_t^i
      in the full time graph for SOME k in lines 53-55). The theorem 1 is an
      extension of Peters et al. [2011b] to a time series case and a partial
      extension of Hyvarinen et al. (ICML2008, JMLR2010) to nonlinear cases.
      Hyvarinen et al. proves the identifiability of their model assuming
      linearity and non-Gaussian residuals (disturbances or external
      influences), in which the summary graph needs not to be acyclic.
      They further presented an estimation algorithm (Algorithm 1),
      which is a natural extension of Mooij et al. [2009] to time series cases.
      They use independence test to detect possible violations of model
      assumptions. Their algorithm can be applied (with small modifications) to
      shifted time series cases, which are often observed in fMRI data.
      They conducted artificial data experiments to test their
      algorithms under various settings including latent confounders with time
      lag but still assumed that the summary graph is acyclic. They compared
      their results with a Granger-causality-based linear method and a linear
      non-Gaussian method TS-LiNGAM (Hyvarinen et al., ICML2008). Their methods
      are better in some nonlinear cases. Finally, they tested some real
      datasets. Their methods worked better in some cases and were comparable to
      linear non-Gaussian TS-LiNGAM in some cases. Pros.: - It is
      interesting to extend TS-LiNGAM (Hyvarinen et al., ICML2008; JMLR2010) to
      nonlinear cases. - It is practically important to consider shifted
      time series cases. Cons.: - The assumption in Theorem 1 that
      the summary graph is acyclic seems too restrictive unless I miss something
      important. It means that it does not happen that X^1_t-1 causes X^2_t and
      X^2_t-1 causes X^1_t, doesn’t it?? They claim that their methods is a
      kind of safe methods since it outputs `I don’t know’ if independence
      between exploratory variables and residuals is statistically rejected,
      which implies some violations of the model assumptions. The idea is not
      very new. It can be found in many papers, e.g., Entner and Hoyer
      (AMBN2010), Tashiro et al. (ICANN2012), Shimizu and Kano (2008, JSPI) and
      Hoyer et al. (NIPS2009). Suggestions: Would it be possible to
      prove Theorem 1 without assuming that the summary graph is acyclic?
      - They claim that their method is a kind of safe methods since it
      outputs `I don’t know’ if independence between exploratory variables and
      residuals is rejected. The idea is not very new. For example, they should
      at least prove that explanatory variables and residuals are not
      independent if latent confounders exist to make their contribution
      clearer. Entner and Hoyer (AMBN2010) and Tashiro et al. (ICANN2012) proved
      that in linear non-Gaussian cases. Comments An extension
      of TS-LiNGAM to cases with latent confounders with time lag might be
      Kawahara, S. Shimizu, and T. Washio. Analyzing relationships among
      ARMA processes based on non-Gaussianity of external
      influences.Neurocomputing, 74(12-13): 2212-2221, 2011. It considers
      ARMA instead of AR model to model temporal dependency. Another extension
      of TS-LiNGAM to cases with instantaneous confounders would be Gao and
      H. Yang. Identifying structural VAR model with latent variables using
      overcomplete ICA. Far East Journal of Theoretical Statistics, 40(1):
      31-44, 2012. In the last year NIPS, TS-LiNGAM was extended to cases
      with variance-dependent external influences. Z. Chen, K. Zhang, and L.
      Chan. Causal discovery with scale-mixture model for spatiotemporal
      variance dependencies. In Advances in Neural Information Processing
      Systems 25 (NIPS2012), pp. xx-xx, 2012. Taking these developments in
      linear non-Gaussian models into account, the proposed nonlinear model
      assuming the sample time graph is acyclic might be a small contribution,
      although extending to nonlinear models certainly is an interesting topic.
      In Experiment 11, they say that “TiMINo does not decide since the
      model leads to dependent residuals. We nevertheless analyzed ...” I didn’t
      follow why they decided to continue to analyze the data even if the
      residuals were found to be dependent. - They say that “we now
      refer to entries larger than 0.5 as causation”. I didn’t follow why they
      choose the threshold 0.5.


What do row and column marginals reveal about your dataset?
Numerous datasets ranging from group memberships within social networks to purchase histories on e-commerce sites are represented by binary matrices. While this data is often either proprietary or sensitive, aggregated data, notably row and column marginals, is often viewed as much less sensitive, and may be furnished for analysis. Here, we investigate how these data can be exploited to make inferences about the underlying matrix H. Instead of assuming a generative model for H, we view the input marginals as constraints on the dataspace of possible realizations of H and compute the probability density function of particular entries H(i, j) of interest. We do this for all the cells of H simultaneously, without generating realizations, but rather via implicitly sampling the datasets that satisfy the input marginals. The end result is an efficient algorithm with asymptotic running time the same as that required by standard sampling techniques to generate a single dataset from the same dataspace. Our experimental evaluation demonstrates the efficiency and the efficacy of our framework in multiple settings.

The paper concerns extracting reliable information
      about binary matrices, given only their column and row sums. These sums
      are considered as constraints on the space of possible matrices. The goal
      is to estimate the entry-wise probability density functions and to do this
      simultaneously via implicit sampling, making the proposed algorithm much
      more efficient than existing techniques. The paper is well written and
      easy to follow. -- For sparse matrices, which is the case in the
      considered DBLP and NOLA datasets, the probability of having 1 is low, and
      the proposed approach computes low values as well. However, how useful is
      this information? It would be interesting to see how the algorithm
      performs on real or simulated matrices with density around 50%. Would we
      have many interpretable entries with probabilities close to 0 or close to
      1, or would we get probabilities around 0.5? -- Minor remarks:
      1. Is there a reference for the considered LS approach with \bar h
      being the average entry value (Section 4.1)? If \bar h is close to 0.5,
      wouldn't this approach select an approximation with entries as close to
      0.5 as possible, making the approximation less interpretable? 2. The
      legend of Figure 1(b) contains undefined labels. -- Further
      questions: 1. Is the approach generalizable to integer matrices?
      2. Is the proposed approach generalizable to the noisy case, i.e. how
      much would the entry-wise PDFs change if a given row or column sum is
      increased or decreased by one? 3. To speed up the proposed algorithm,
      columns having the same sum are considered to be the same and duplicates
      are removed from the computational step. Is it possible to extend the
      algorithm to be able to incorporate other types of prior knowledge? For
      example, a common assumption about data matrices is that they have low
      rank. (Redundant columns/rows lead to low rank matrices, but the low rank
      assumption is more general than that.)


Correlations strike back (again): the case of associative memory retrieval
It has long been recognised that statistical dependencies in neuronal activity need to be taken into account when decoding stimuli encoded in a neural population. Less studied, though equally pernicious, is the need to take account of dependencies between synaptic weights when decoding patterns previously encoded in an auto-associative memory. We show that activity-dependent learning generically produces such correlations, and failing to take them into account in the dynamics of memory retrieval leads to catastrophically poor recall. We derive optimal network dynamics for recall in the face of synaptic correlations caused by a range of synaptic plasticity rules. These dynamics involve well-studied circuit motifs, such as forms of feedback inhibition and experimentally observed dendritic nonlinearities. We therefore show how addressing the problem of synaptic correlations leads to a novel functional account of key biophysical features of the neural substrate.

The authors study the role of correlations between
      synaptic connections in a recurrent network induced by synaptic learning
      rules. The authors separate the contribution of learning rules by making
      the simplifying assumption that the activity patterns which drive learning
      do not themselves have correlations. They proceed to discuss two classes
      of learning rules, additive and cascade models, derive and examine the
      differences in the correlations induced by them and their effect on the
      accuracy, simplicity and plausibility of recall dynamics. The paper
      follows in the path of several recent papers discussing recall dynamics as
      probabilistic inference, is generally clearly written, interesting and has
      substantial results that should be of interest to the general NIPS
      community. Major comments The first part of the paper overlaps
      with previously published papers. Though this introduction is important
      and some relevant papers are referenced by the authors, it must be made
      much clearer where the novel parts are found and which results are already
      published. Minor comments In general, one can certainly
      appreciate trying to write in a more engaging, elevated form but some
      sentences are a bit too poetical. For instance, “assuaging an old enemy
      leads to…” is a bit much. Lines 39-41: this sentence is too vague
      and imprecise. As the authors themselves will state in a few pages, the
      perfect symmetry of Hopfield networks (an obvious synaptic correlation) is
      well known, and heavily studied issue. The authors should be more precise
      with what kinds of correlations have not been explored and in what
      context, or at least tone down this statement. Line 77-78: “recall
      is inherently a probabilistic inference…” that is not the only approach,
      rather the authors can claim that “recall can be seen as a probabilistic…”
      Footnote page 2 “the the”, one “the” is a typo Line
      390-392: “Statistical dependencies between synaptic efficacies are a
      natural consequence of activity dependent synaptic plasticity, and yet
      their implications for network function have been unexplored.” Again, this
      statement has to be made more specific considering the Hopfield network
      literature. Footnote #10: this statement should be put into the
      discussion at the point where the footnote was referenced. Line
      414-415: “the deeper is the cascade” should be “the deeper the cascade”
      Post-feedback response: My comments were mostly minor, the
      authors addressed them in their response and I am sure will be able to
      correct everything for the final version.


Estimating LASSO Risk and Noise Level
We study the fundamental problems of variance and risk estimation in high dimensional statistical modeling. In particular, we consider the problem of learning a coefficient vector θ 0 ∈ R p from noisy linear observations y = Xθ 0 + w ∈ R n (p > n) and the popular estimation procedure of solving the 1 -penalized least squares objective known as the LASSO or Basis Pursuit DeNoising (BPDN). In this context, we develop new estimators for the 2 estimation risk θ − θ 0 2 and the variance of the noise when distributions of θ 0 and w are unknown. These can be used to select the regularization parameter optimally. We establish high-dimensional consistency of our estimators for sequences of matrices X of increasing dimensions, with independent Gaussian entries. We establish validity for a broader class of Gaussian designs, conditional on a certain conjecture from statistical physics.
To the best of our knowledge, this result is the first that provides an asymptotically consistent risk estimator for the LASSO solely based on data. In addition, we demonstrate through simulations that our variance estimation outperforms several existing methods in the literature.

(*) The paper proposes new estimators for: (i.)
      the risk of a lasso estimator (ii.) the noise-level in the lasso
      model. Both estimators are purely from the data. Under some
      conditions, it is proven that these are consistent estimators of the
      quantities when the dimension p and the number of measurements n go to
      infinity. There are also a few experimental results showing the
      accuracy of the proposed estimators on simulated examples. (*)
      The paper provides important results for a fundamental problem of
      estimating the lasso risk from data, and thus offers potentially a
      significant contribution. An important practical implication is that the
      estimator could be used to optimize the regularization parameter \lambda
      of the LASSO. However, the writing of the paper is not clear enough
      and often sloppy. This makes it hard to follow the paper, and verify
      the correctness (and sometimes even the meaning) of the results, and
      prevented me from assigning a higher score for the paper. It looks
      like more effort should have been devoted to presentation. (*)
      The computer experiments part is encouraging but limited and not well
      explained. What is the distribution for the non-zero values of x_0?
      how was each point in the plot obtained? from a single run of the lasso?
      if so, why not average over multiple runs, to see also the standard
      deviation of the estimation error and of the estimators the authors
      provide? (*) Many things are not explained: - What are
      'irrepresentability conditions' in the introduction? - How does one
      initialize y and z in the AMP algorithm in eq. (3.1)? - I don't
      understand what is the joint empirical distribution of {\hat{y}_i, x_0,i}
      (first defined in line 270). x_0 is the signal, with coordinates for
      i=1,..,p. But \hat{y} is the the data, with i=1,..,n. Why is the same i
      used here? - In line 198 it is said : \eta_t' is the derivative ...'.
      Derivative with respect to what? y^t? - What is D in proposition 3.2
      (line 297)? - The authors provide a general estimator for a general
      design matrix, in definition 3. But when should one use each estimator? is
      it assumed that the inverse covariance matrix \Omega is known? or
      should one estimate it from the data? - It is often not clear when the
      authors write Expectations E[..], over what are these expectations taken.
      For example in equations (3.2), (3.4), (3.5). Is it over the choice of
      elements of the matrix A? of error variables Z? of the signal X? all of
      these together? (*) The authors provide estimators for the error
      for one specific estimator: the lasso. But could it be that other
      estimators give better MSE for the case of sparse vectors? what would a
      comparison of the MSE to minimax lower bounds yield? (e.g. E. J.
      Candès and M. A. Davenport. How well can we estimate a sparse vector?
      Applied and Computational Harmonic Analysis, 2011) (*) I have
      a problem in interpreting the main results. In theorem (4.1), the authors
      provide a consistency result for the estimator of the lasso risk, as
      both the dimension p and the number of measurements n go together to
      infinity (with their ratio being fixed). But, first, it looks like the
      result is given for a fixed iteration t - yet one would expect the
      number of iterations required for convergence to depend on n and p (and
      the data). How do the results apply to the actual LASSO solution?
      Second, from eq. (4.1) it looks like n and p approach infinity
      independently, which is confusing. Finally, it seems to me that in
      this limit, the lasso risk itself approaches some limit (i.e. some
      constant. For example in [Wai09] it is shown that under certain
      conditions the error goes to zero). If this is true, then what is the
      utility of the author's estimator in this limit if the error can be anyway
      calculated as a function of the distribution of A and the noise?
      (*) There are many typos and inaccuracies, some of which
      really hinder the reading and understanding: The last section in
      the abstract ('We compare our variance ..') is unclear and confusing.
      What are 'two naive LASSO estimator'? estimators of the error? also,
      why 'On the other hand'? Page 1, line 43: 'p < n' should be 'p
      > n'? Page 2, line 69: 'the the risk' -> 'the risk'.
      Page 2, line 82: 'data y,A' -> 'data y and A'? Page 2,
      line 100: 'underbraccdelta' ??? Page 3, line 132: 'SURE' is
      written before it was introduced/defined. Page 3, line 153: 'for
      each lambda' Page 3, line 155: \delta is first written here but
      defined only later. Page 4, lines 192-196: many indices in eq.
      (3.1) are wrong/missing (e.g. when is t used, when t+1,t-1? the iteration
      looks circular). Also, shouldn't the 2nd line be z+Ax? Page 5,
      line 260: 'provides' -> 'provide' Page 6, line 295: 'for'
      should be removed. Would be also good to define 'weakly differentiable'
      Page 9, line 473: Reference is messed up


A Determinantal Point Process Latent Variable Model for Inhibition in Neural Spiking Data
Point processes are popular models of neural spiking behavior as they provide a statistical distribution over temporal sequences of spikes and help to reveal the complexities underlying a series of recorded action potentials. However, the most common neural point process models, the Poisson process and the gamma renewal process, do not capture interactions and correlations that are critical to modeling populations of neurons. We develop a novel model based on a determinantal point process over latent embeddings of neurons that effectively captures and helps visualize complex inhibitory and competitive interaction. We show that this model is a natural extension of the popular generalized linear model to sets of interacting neurons. The model is extended to incorporate gain control or divisive normalization, and the modulation of neural spiking based on periodic phenomena. Applied to neural spike recordings from the rat hippocampus, we see that the model captures inhibitory relationships, a dichotomy of classes of neurons, and a periodic modulation by the theta rhythm known to be present in the data.

Summary: -------- The authors apply
      Determinantal Point Processes (DPPs) to the spiking activity of
      simultaneously recorded neurons. In addition to stimulus dependence, the
      resulting model captures pairwise competitive interactions of neurons.
      The authors apply the model to artificial data and hippocampal
      recordings. Comments: --------- 1) The application
      of DPPs to neural recordings is novel to best of my knowledge. The
      incorporation of instantaneous interactions into GLMs for neurons
      (even coupled ones) such that the resulting model remains tractable is
      an important open problem. Hence, I think the paper is timely and of
      interest to the NIPS neuroscience audience. 2) The paper is
      clearly written and DPPs are introduced quite gently, resulting in a
      very readable paper (exceptions below). 3) My main criticism is
      that the authors did not fully convince me that their model is
      actually an appropriate one for multi-cell recordings. As far as I
      know, most noise correlations that have been experimentally measured
      seem to be positive (at least in cortex) and could therefore not be
      captured by this DPP approach; please correct me if I'm wrong here. In
      any case, the authors should have argued more thoroughly and given
      appropriate citations that the scenario of exclusively competitive
      interactions is an important one in multi-cell recordings. 4)
      Section 4.3, application to Hippocampus data: Unfortunately, this
      paragraph does not fully convince me that the DPP based model is a
      good model for the data. Are the noise correlations (computed from the
      data) between pairs of excitatory / pairs of inhibitory neurons /
      pairs of exc-inhi really mostly negative (emphasize here is on noise
      correlations as GLMs can capture stimulus induced correlations)? The
      authors should describe the main result figures FIG4 and FIG3(a) in
      greater detail: why is this latent embedding sensible / what does this
      kernel matrix tell us about the data? (The additional space required
      could be obtained by scaling back the experiments on artificial data.)
      The fact that the method uncovers the theta oscillations is not very
      surprising (as this is just the GLM part of the model) and could be
      described more briefly. Table 1: The authors could have made a
      stronger point for the model if the table included more pairs of
      models of the type "GLM_component_1+...+GLM_component_n" and
      "GLM_component_1+...+GLM_component_n+Latent", as this allows for a
      direct comparison to figure out if adding the DPP part helps.
      5) Section 3.3: In the model, the stimulus dependence of each
      neuron is already captured by the weight vector $w_n$. Isn't the
      introduction of the of $w_\nu$ redundant? 6) The manuscript is
      missing a more in depth comparison between coupled GLMs and the DPP
      approach, eg: coupled GLMs are lacking an instantaneous coupling
      between the neurons, but this could be compensated for by binning the
      data on finer time scales etc. Minor Comments:
      --------------- 1) Either I don't fully understand the
      notation around eqn. 4 and 5 or it is is rather sloppy: I guess
      $\Pi^{(t)}$ should be defined as
      $diag(\sqrt{\lambda_1^{(t)}},\ldots)$, instead of the definition given
      in l157. Furthermore in eqn 5, $K^{(t)}_{S_t}$ probably only contains
      the term $k_\theta$ and not the GLM part $\lambda_n^{(t)}$, in
      contrast to eqn 4. If this is correct, then $K^{(t)}_{S_t}$ should be
      $K_{S_t}$ as the only time dependence would be via ${S_t}$.
      Furthermore, shouldn't the normalizer in eqn 5 be independent of
      $S_t$, i.e. contain eg $K_{\mathcal S}$ instead of $K_{S_t}$ (similar
      for $\Pi_{S_t}$)? 2) In the definition of $K_S$ in l131, it might
      be worthwhile to again emphasize that $n,n' \in
      S$.


Linear Decision Rule as Aspiration for Simple Decision Heuristics
Several attempts to understand the success of simple decision heuristics have examined heuristics as an approximation to a linear decision rule. This research has identified three environmental structures that aid heuristics: dominance, cumulative dominance, and noncompensatoriness. This paper develops these ideas further and examines their empirical relevance in 51 natural environments. The results show that all three structures are prevalent, making it possible for simple rules to reach, and occasionally exceed, the accuracy of the linear decision rule, using less information and less computation.

This work turns NIPS on its head: In NIPS, linear
      threshold classifiers are usually considered to be a baseline model,
      but in this work, linear threshold classifiers are considered too
      sophisticated to characterize human decision making, and various
      simpler alternatives are explored. The work appears to make two
      important contributions: (1) to show that for natural environments,
      various decision-making short cuts perform well, and (2) to introduce
      novel decision short cuts that are probabilistic generalizations of
      previously proposed short cuts. The paper is very well written
      and accessible. It should have more discussion of the implications of
      the results. The authors should also make their theoretical
      contribution clearer. Is approximate (cumulative) dominance is new to
      this work? The motivation for the approximation might be made clearer
      as well. I believe the reason is that the conditions in section 2
      often do not apply, but an approximate rule would be more often
      applicable, and as long as it is fairly accurate, it should boost
      performance. This appears to be the case in the results. One
      additional point about the exposition: The introduction fails to give all
      the background necessary to motivate the work to a typical NIPS attendee.
      Before discussing fast-and-frugal heuristics, the authors should discuss
      the empirical literature showing that simple linear-threshold classifiers
      are better than human decision makers (e.g.,
      http://www.sciencemag.org/content/243/4899/1668). If you start from this
      perspective, it motivates the treatment of the linear-threshold
      classifier as the target, and the question of what heuristics
      approximate this target. Without this background for the reader, I suspect
      the typical NIPS attendee will wonder the most sophisticated model of
      decision making is a linear-threshold classifier. The work claims
      to be a serious analysis of structure of natural environments, but I'm
      far from clear whether the 51 real-world data sets involve the type of
      decision making tasks that humans might typically make. It might be
      worth moving Table 1 to the supplementary materials (along with the
      page of references), and instead to present an argument for why these
      data sets are representative of human decision making tasks. I
      was hoping the authors would compare their 51 natural data sets to
      randomly generated data sets, where the weights and features are drawn
      from specified distributions. It would also be interesting to
      determine the distributional assumptions that lead to success of the
      various decision making short cuts. I am confused about the trade
      off in setting the cutoff value for the approximate dominance schemes:
      A high cutoff means that the approximate-dominance relationship
      predicts the correct decision, but also that fewer
      approximate-dominance relationships will hold between pairs. The trade
      off is not discussed. Was the chosen cutoff (.99) arbitrary, or are the
      results sensitive to this choice?


Generalized Random Utility Models with Multiple Types
We propose a model for demand estimation in multi-agent, differentiated product settings and present an estimation algorithm that uses reversible jump MCMC techniques to classify agents' types. Our model extends the popular setup in Berry, Levinsohn and Pakes (1995) to allow for the data-driven classification of agents' types using agent-level data. We focus on applications involving data on agents' ranking over alternatives, and present theoretical conditions that establish the identifiability of the model and uni-modality of the likelihood/posterior. Results on both real and simulated data provide support for the scalability of our approach.

This paper addresses the problem of identifying the
      type of each agents from his/her partial preference data, in order to use
      this information to better estimate the underlying preferences for each
      type. The authors propose a Generalized RUM to model the behavior of such
      clustered agents. A reversible jump MCMC technique is used to estimate the
      latent variables, including the types of the agents. A theoretical
      analysis of the identifiability of the model and uni-modality of the
      likelihood posterior are presented. Quality There are
      three contributions of this paper. The new GRUM model, theoretical
      analysis, and inference algorithm. The model is a generalization of RUM
      model to multiple agents with types, which is new. Theoretical guarantees
      are interesting, but have limitations discussed below in the significance
      section. The inference algorithm is quite standard and the numerical
      analysis is not impressive, either on the simulated data or the real world
      datasets. Only the performance of estimating the number of clusters is
      addressed, while the main problem is in clustering agents. A much more
      relevant numerical simulation would be simulating how the number of
      misclustering depends on problem parameters such as number of clusters,
      how much the matrix W differ between clusters, missing data, etc.
      Clarity Some claims could be better explained. On
      page 2, it is not clear what the authors mean by the first paragraph of
      section 1.1. Which aspects of the model eliminates unrealistic
      substitution patterns? and avoid the situation where removing the top
      choices result in the same alternative choice? On page 2, it is
      not clear from the numerical results that "the clustering of types
      provides a better fit to real world data". On page 5, in the
      definition of `nice' pdfs, $\phi^{(n)}(x)$ is used without proper
      definition, which makes the conditions difficult to understand. For
      instance, given \phi is a pdf, g_n should be non-negative. But
      g_n(x1)/g_n(x_2) converges to -1. The definition of `nice' cdf's
      is not intuitive and no explanation is given as to why the model might not
      be identifiable if noise cdf is not `nice'. On page 7, it is
      claimed that "It can be seen that GRUM with 3 types has significantly
      better performance than...". However, from the table, it seems like the
      gain is only marginal. How significant is the gain of 2~3 % in the log
      posterior? Originality This paper extends the definition
      of RUM model to the setting where there are multiple alternatives and
      multiple agents. The correlation between the agents are modelled via types
      that an agent belongs to. RJ-MCMC approach seems to be quite standard.
      Significance The main results on the theoretical
      guarantees are interesting, but the application is limited. For theorem 1,
      unimodality is only established when the types are known (as clearly
      explained in the paper). This limits the convergence of MCMC approach, and
      it is not clear how long one should run the MCMC in practice. This paper
      does not explain why the proposed problem is difficult. Why has this
      problem not been addresses so far, as the authors claim? Further, because
      there is no comparisons either in theoretical results or numerical
      results, it is difficult to judge how good the proposed algorithm is.


Structured Learning via Logistic Regression
A successful approach to structured learning is to write the learning objective as a joint function of linear parameters and inference messages, and iterate between updates to each. This paper observes that if the inference problem is "smoothed" through the addition of entropy terms, for fixed messages, the learning objective reduces to a traditional (non-structured) logistic regression problem with respect to parameters. In these logistic regression problems, each training example has a bias term determined by the current set of messages. Based on this insight, the structured energy function can be extended from linear factors to any function class where an "oracle" exists to minimize a logistic loss.

The authors address the issue of learning unary and
      pairwise parameters for discrete valued random fields in a structured
      output SVM setting. In order to enable learning with non-linear function
      classes, the authors remove the standard l2 regularization and replace it
      with an entropy regularization that decomposes over the potentials of the
      random field. This formulation results in a saddle-point problem (typical
      for structured prediction) in which parameters of an LP relaxation are
      optimized along with parameters of the (non-linear) discriminant function.
      The authors use a message passing formulation, and show that with their
      decomposable loss, and decomposable regularizer, the problem decomposes
      into problems that each resemble logistic regression. Any solver for a
      non-linear function class can now be applied. Experimental results are
      given on relatively simple denoising and horse segmentation datasets
      showing that potentials modeled with a multi-layer perceptron performed
      better than other linear and non-linear methods. Quality: The
      derivation is principled and complete, with sufficient references to the
      optimization literature. The idea is interesting, and (given a source code
      release) could result in various applications of non-linear function
      classes in learning the parameters of graphical models. The quality of the
      experimental validation is somewhat weak, however, given that the datasets
      are quite simple, and results only compare with variants of the proposed
      algorithm. Plenty of citations are given for competing methods, e.g. [18]
      which is very closely related to the proposed approach in its goal of
      enabling non-linear function classes for random field models.
      Clarity: The article is well written, and clear throughout.
      Originality: [18] have previously proposed a random forest
      framework for learning the parameters of a random field model. This work
      is more general and uses a different formulation to arrive at its
      objective function. Significance: This work is quite general
      in the functions that can be used, though a number of limiting assumptions
      are made (decomposable loss, margin rescaling, only consider discrete
      labelings). It is an interesting approach nevertheless. Additional
      comments: Sec 3 - Limiting assumptions about the problem setting are
      first introduced here: Eq (3) - only considers margin rescaling
      Around line 93 - \Delta decomposes over \alpha - e.g. Hamming distance
      Around line 94 - only considering discrete labelings - which is fine
      but not fully general in structured prediction Decomposable loss and
      discrete labelings should probably already be mentioned in the
      introduction Line 306 - typo in equation


Learning Stochastic Inverses
We describe a class of algorithms for amortized inference in Bayesian networks. In this setting, we invest computation upfront to support rapid online inference for a wide range of queries. Our approach is based on learning an inverse factorization of a model's joint distribution: a factorization that turns observations into root nodes. Our algorithms accumulate information to estimate the local conditional distributions that constitute such a factorization. These stochastic inverses can be used to invert each of the computation steps leading to an observation, sampling backwards in order to quickly find a likely explanation. We show that estimated inverses converge asymptotically in number of (prior or posterior) training samples. To make use of inverses before convergence, we describe the Inverse MCMC algorithm, which uses stochastic inverses to make block proposals for a Metropolis-Hastings sampler. We explore the efficiency of this sampler for a variety of parameter regimes and Bayes nets.

This paper is based on the observation that while
      doing posterior sampling in a Bayesian network, information can be
      collected about conditional relationships between variables that
      follow structure that is different from the given Bayes net. The idea
      is then to use this insight to learn conditional relationships for
      alternative Bayes net factorizations. This is useful either to (a) sample
      from these approximate alternative factorizations to get fast
      approximate samples by conditioning on the observations then sampling
      outwards following the directed structure of the alternative graph, or
      (b) using the learned conditional relationships to create proposals for
      Metropolis Hastings MCMC. The primary benefit of this setup is in
      the setting of amortized inference, where it is assumed that we are
      required to solve many posterior inference tasks over the same model
      but where the pattern of observations may change. There are three
      main steps: 1. Construct the inverse factorization networks. 2.
      Learn the parameters for the inverse factorization networks from
      posterior sampling runs. 3. Use the learned parameters to improve
      inference. The paper gives a heuristic algorithm for (1). Given
      the justification provided in the paper, (2) is relatively
      straightforward, and the paper suggests either using empirical
      frequency counts or learning an approximation via e.g. logistic
      regression. One inverse network is learned per variable. For (3), the
      paper uses the inverse networks to produce proposals for Metropolis
      Hastings MCMC. Block proposals can be generated by proposing jointly
      the assignments to the "last" k variables in the inverse network.
      Overall, I found the paper to have interesting ideas and to be an
      enjoyable read. The exposition is good and the ideas come through
      clearly. One work that I'd like to see cited and discussed is data
      driven MCMC: Tu, Zhuowen, and Song-Chun Zhu. "Image segmentation by
      data-driven Markov chain Monte Carlo." Pattern Analysis and Machine
      Intelligence, IEEE Transactions on 24.5 (2002): 657-673. Another
      concern is in how large the parent sets will be in the learned inverse
      networks. Specifically, on line 12 of Algorithm 1, we must choose a
      parent set that d-separates a variable from the rest of the graph. It
      would be nice to have some discussion about how large these sets are
      in practice (relatedly, some more motivation about what is going on in
      Algorithm 1, what tradeoffs were considered when developing the
      heuristic, etc. would be welcome). A related question is if
      keeping the parent sets as small as possible is important. It seems
      like the extreme opposite choice for the inverse factorizations would
      be something like the NADE model [1], which assumes no conditional
      independence. Have you experimented with different inverse
      factorization algorithms, and can you say what aspects of Algorithm 1
      are most important? [1] The Neural Autoregressive
      Distribution Estimator Hugo Larochelle and Iain Murray, Artificial
      Intelligence and Statistics, 2011


Documents as multiple overlapping windows into a grid of counts
In text analysis documents are often represented as disorganized bags of words; models of such count features are typically based on mixing a small number of topics [1,2]. Recently, it has been observed that for many text corpora documents evolve into one another in a smooth way, with some features dropping and new ones being introduced. The counting grid [3] models this spatial metaphor literally: it is a grid of word distributions learned in such a way that a document's own distribution of features can be modeled as the sum of the histograms found in a window into the grid. The major drawback of this method is that it is essentially a mixture and all the content must be generated by a single contiguous area on the grid. This may be problematic especially for lower dimensional grids. In this paper, we overcome this issue by introducing the Componential Counting Grid which brings the componential nature of topic models to the basic counting grid. We evaluated our approach on document classification and multimodal retrieval obtaining state of the art results on standard benchmarks.

The paper presents an LDA-style mixed membership model
      where the component distributions are counting grids. Each document is a
      Dirichlet mixture over windows into the grid. Each window is a uniform
      distribution over the cells contained in that window. Each cell in the
      grid is a distribution over the vocabulary. The method is evaluated using
      classification accuracy, embedding quality, AUC in a retrieval task, and a
      visualization example. I like this paper a lot. I think it's
      marginally above the accept threshold as is, and could be substantially
      improved with another round of revisions. I gave it an "incremental", but
      it's more like a 1.499. The introduction does a reasonably good
      job presenting the basic intuition of the paper, that words often appear
      in a smooth continuum rather than distinct topics, and that the
      single-window model is too restrictive for real documents. Figure 1 is a
      bit of a missed opportunity. 1a works for me, but I found 1b and 1c
      confusing. A simpler toy example might be better, especially if it could
      be merged with Figure 2b. I had a hard time moving from the
      introduction to section 2. In part, I felt there was a lot of technical
      terminology floating around, some of it redundant. What is a source, for
      example, or a bag? Are these cells and documents? I would pick a small set
      of concrete terms and use them consistently in both sections. Avoid
      "topic". In Sect 2: I would call pi_i(z) a distribution
      over the vocabulary, not a set of normalized counts. The latter may be its
      actual implementation in an algorithm, but here we're defining the model
      in the abstract. Give an example: pi_{1,1}("pizza") is the probability of
      the word "pizza" in the top left cell (or similar, if that's not correct)
      What's a sample? Is that a document? What is Z? The size
      of the vocabulary or the size of the grid? The variable z means "hidden
      variable" to me. I'm really looking for a toy example.
      Give an example for W, like 3x3. Actually, I might
      restrict the initial presentation to square windows in 2D grids, for the
      sake of simplicity of notation, and then comment that the model is easily
      extended to d-dimensional grids and arbitrary window sizes. I
      don't see why the number of non-overlapping windows is equivalent to the
      number of topics. The whole point is that the windows can overlap, no?
      Overlapping windows don't represent independent topics, but they do add
      modeling power, and that has to be taken into account for a fair
      comparison. Also, give a formula for kappa explicitly: it's not
      unambiguous right now (which ratio?). I would rewrite section 2
      around a step-by-step generative model. To generate a document, you first
      pick a distribution over windows. Then to generate each word, you pick a
      window, then pick a cell within the window, and finally pick a word from
      the cell. The connection to LDA (windows of size 1x1) would be
      good to point out here. It took a moment to figure out that U^W
      was a uniform distribution over a window. It looked like a matrix
      multiplication at first. "Locations in the grid generate similar
      features" -- does this mean that the words in neighboring cells are
      semantically similar (but the words themselves differ), or that the
      distributions themselves are similar? "The number of sources must
      not be specified..." First, I think this should be "need not be" -- "must
      not be" implies that you will get in trouble if you do! Second, I'm not
      sure what this means. The rest of the paper seems to assume that all
      dimensions are known in advance. Also, I'm still not sure what sources are
      (cells? windows?). I didn't follow the marginalization step. I'd
      like a bit more explanation. If space is needed, variational update
      equations are an excellent candidate for supplementary material.
      Distinguish between variables and distributions, and avoid
      referring to symbol alone: "the new updates for the cell distributions
      q(k) and the window distributions q(l) ..." The comparison to
      Dunson et al is reasonable, but it might be worth talking about what is
      lost by using a grid rather than a continuous space. Section 2.1
      seems like a tangent. Evaluation: I didn't see held-out
      likelihood tests. It's not trivial for LDA. How would they be computed for
      this model? I'd like to see a linear SVM used for document
      classification, in addition to the nearest neighbor method (I'm assuming
      that's BOW?). See my earlier concerns about using "capacity" as a
      measure of approximate # of topics. Why not just the number of grid cells?
      In some sense CCG is just a fancier correlated distribution over the
      cells, like CTM or PAM. The Euclidean embedding section is
      interesting. I'd like to see more of it. What are the visual
      words? Are they given with the data set? On the whole I buy the
      visualization aspect, because the word distributions are intrinsically
      embedded in a 2D space. I'd like to see more development of this idea. How
      would people use this embedding to learn something about the corpus? I
      didn't find Figure 5 particularly compelling for the fry/deep fry
      distinction, but the cluster of words that I associate with Indian recipes
      in the bottom right was interesting.