Minimax Optimal Algorithms for Unconstrained Linear Optimization
We design and analyze minimax-optimal algorithms for online linear optimization games where the player's choice is unconstrained. The player strives to minimize regret, the difference between his loss and the loss of a post-hoc benchmark strategy. While the standard benchmark is the loss of the best strategy chosen from a bounded comparator set, we consider a very broad range of benchmark functions. The problem is cast as a sequential multi-stage zero-sum game, and we give a thorough analysis of the minimax behavior of the game, providing characterizations for the value of the game, as well as both the player's and the adversary's optimal strategy. We show how these objects can be computed efficiently under certain circumstances, and by selecting an appropriate benchmark, we construct a novel hedging strategy for an unconstrained betting game.

SUMMARY AND RELATED WORKS This paper is about
      online linear optimization with an unconstrained comparator set (e.g., the
      whole R^n space). Instead of trying to be competitive with the best linear
      predictor x \in R^n, the aim is typically to be competitive with the best
      penalized comparison vector. In this setting, and assuming that the convex
      hull of the set of moves of the adversary is a full-rank polytope in R^n,
      the authors provide stochastic techniques to derive the value of the game,
      as well as the optimal strategies for the player and the adversary. These
      abstract results are then applied to the one-dimensional case (a
      straightforward extension to arbitrary dimension with
      coordinate-decomposable games is also provided, but no thorough
      application was worked out). Three 1-dimensional examples are studied in
      details: linear optimization with a quadratic regularizer, non-stochastic
      betting, and the standard online linear optimization problem on [-1,1]
      (where the authors recover known results from their new general
      technique). This paper is a continuation of a series of papers
      that addressed adversarial online learning problems from a stochastic
      viewpoint (Abernethy et al. COLT 2008, Abernethy et al. COLT 2009, Rakhlin
      et al. NIPS 2012). It extends those works to the case of unconstrained
      comparator sets. Note however that unconstrained online convex
      optimization is definitely not new: papers like Duchi et al. (Composite
      Objective Mirror Descent, COLT 2010) already provided regret upper bounds
      for unconstrained comparison vectors. The novelty here is that this
      setting is studied from an exact minimax viewpoint. OVERALL
      ASSESSMENT My overall opinion about the paper is quite positive
      for the following reasons: PROS 1) The paper is well
      written and pleasant to read (all necessary intuitions are provided when
      needed). Mathematics seem correct. 2) Though not ground-breaking, this
      paper makes a further step towards the understanding of unconstrained
      linear optimization. 3) The application to non-stochastic betting is
      really nice, with a tight analysis and a simple algorithm. CONS
      4) There are not sufficiently many applications of the abstract
      results: Theorems 7 and 9 are nice for a warm-up, Theorem 8 is
      interesting, but I would have also expected a study of other forms of
      penalized regret like the ones already encountered with online mirror
      descent. In particular, can the techniques of the present paper be adapted
      to multi-dimensional games that are not coordinate-decomposable? Most
      Bregman-divergence-based regularizations are indeed of this form. 5)
      Contrary to what is announced on line 326, it seems that the constraint
      that the player cannot bet more than what he owns is not imposed to the
      game. I understand that it is possible to make sure that he loses no more
      than 1$ at the end of the game, but what can be said on the whole path
      from t=1 to t=T? OTHER COMMENTS 6) Related Work (Page 2):
      references about unconstrained regret upper bounds with Online Mirror
      Descent are missing. 7) Lines 105-107: the proposed 'alternative' is
      not new: actually, the comparison of the loss to the best penalized loss
      is known in statistics as an 'oracle inequality' from quite a long time
      (see, e.g., the references in "Concentration Inequalities and Model
      Selection", Massart, 2007). Parallely, in the machine learning community,
      some works like the one of Vovk ("Competitive on-line statistics", 2001)
      already began to formalize such ideas in the previous decade. 8) Line
      141: hull(G)=hull(G') 9) Line 206: is the Lipschitz assumption really
      needed? Is boundedness sufficient? 10) Statement of Theorem 3: I
      suggest to move the fact that [we can find a full-rank set of corners
      {g_1,...,g_{n+1}} of the polytope hull(G)] before the statement of Theorem
      3. This is indeed straightforward in all subsequent applications.
      (Furthermore, the term 'distinct' points is too weak, full-rank would be
      more precise). 11) Line 262: it is implicitely assumed that V_{t+1} is
      Lipschitz (or bounded? -> see above). 12) Line 288: the random
      variable B was not defined earlier. 13) Theorem 7: recall that g_t \in
      [-1,1]. 14) Title of Section 3.2: what does "all feasible sets" stand
      for? 15) Line 341: please explain "bet everything every round ... 2^T"
      16) Theorem 7 and the equation on line 683 are actually
      straightforward by writing that B is the sum of independent Rademacher
      variables. 17) Theorem 8: I think that the first exponential contains
      (G_t-\tau)/sqrt{T} (no "-1" term), and that the exponent of the last term
      is \tau-1 instead of \tau (the reason is that the authors used V_t instead
      of V_{t+1}). 18) Several typos : argmin -> min (line 117), that
      that -> that (line 121), Theorems 1 and 2 -> Corollary 2 (line 244),
      V_t -> V_{t+1} (line 286), g_{1:T} -> g_{1:t} (lines 288 and 289),
      -\partial L -> \partial L (line 292), *2* B sqrt{T} (line 338 + cf
      consequences in the following lines), |G| \leq sqrt{T} (line 355), loss
      functions -> losses (line 411) ADDITIONAL COMMENT AFTER THE
      AUTHORS' REBUTTAL I thank the authors for their detailed answer.
      It seems that point 5) can be taken care of. However, I regret that the
      pratical implications of the abstract results are too narrow in the
      submitted version, and that the amount of work needed to address the
      (interesting) applications mentioned in the rebuttal is too large. The
      proposed modified paper should have clear chances of acceptation in a
      later conference.


Generalizing Analytic Shrinkage for Arbitrary Covariance Structures
Analytic shrinkage is a statistical technique that offers a fast alternative to crossvalidation for the regularization of covariance matrices and has appealing consistency properties. We show that the proof of consistency requires bounds on the growth rates of eigenvalues and their dispersion, which are often violated in data. We prove consistency under assumptions which do not restrict the covariance structure and therefore better match real world data. In addition, we propose an extension of analytic shrinkage -orthogonal complement shrinkage-which adapts to the covariance structure. Finally we demonstrate the superior performance of our novel approach on data from the domains of finance, spoken letter and optical character recognition, and neuroscience.

The authors focus on the problem of the estimation of
      covariance matrices using empirical estimators. To avoid overfitting in
      high-dimensional cases they shrink the covariance matrix towards a
      multiple of the identity matrix. The optimal amount of shrinkage \lambda
      can be estimated empirically by minimizing the expected error of the
      shrinkage estimate. Then they show that existing results about the
      consistency of the estimate of \lambda depend on an assumption which does
      not hold in practice for most real-world datasets. The authors then
      introduce a new wider set of assumptions and proof that they guarantee
      consistency. Also, they show that in certain cases there will be no
      asymptotic shrinkage of the empirical estimate of the covariance matrix.
      As a solution that they propose is then to perform the shrinkage in the
      orthogonal complement to the eigen vector with largest eigen value. Since
      a dataset may have multiple large eigenvalues they iterate the procedure
      and automatically select the number of retained eigen directions. A series
      of experiments with synthetic and real-world data validate the proposed
      approach. Quality: The paper seems to be sound, with the
      experiments validating the theoretical results. I did not verified the
      proofs of the main theorems since these are in the supplementary material,
      which I am not required to examine. The experimental evaluation seems to
      be thorough. Clarity: The paper is clearly written and
      well organized. However, the main document does not describe how the
      orthogonal complement shrinkage is actually performed and the motivation
      for this approach is also not strong enough in the paper. The authors
      should clarify how this technique is implemented in practice.
      Originality: I find interesting and worth on its own the
      contribution of the authors to show that the existing shrinkage techniques
      will not produce enough shrinkage in some cases. The proposed orthogonal
      complement shrinkage is novel up to my knowledge. However, it is a bit
      incremental since it represents only a modification of already existing
      approaches. Significance: In my opinion, the most
      significant contribution is to point out the failure of the shrinkage
      method in several relevant cases, as well as the theoretical contributions
      contained in the paper. The orthogonal complement shrinkage seems to be
      less significant than these other elements.


Dirty Statistical Models
We provide a unified framework for the high-dimensional analysis of "superposition-structured" or "dirty" statistical models: where the model parameters are a superposition of structurally constrained parameters. We allow for any number and types of structures, and any statistical model. We consider the general class of M -estimators that minimize the sum of any loss function, and an instance of what we call a "hybrid" regularization, that is the infimal convolution of weighted regularization functions, one for each structural component. We provide corollaries showcasing our unified framework for varied statistical models such as linear regression, multiple regression and principal component analysis, over varied superposition structures.

This paper provides bounds for the learning of
      M-estimators minimizing a loss function with superposition-structured
      regularizers. This type of statistical model features a parameter vector
      which is a superposition of several vectors, each of them having a
      specific structure enforced by its own regularizer. Examples of such
      models have been the subject of very recent results (see in particular the
      referenced papers from Chandrasekaran et al.). The present work provides
      bounds for the learning of the parameter vector by extending the framework
      of decomposable regularizers proposed in (Negahban et al. 2012). While the
      general result is provided in Theorem 1, the resulting bound involves
      several unknown constants which are specific to the statistical model. The
      remaining of the paper studies several examples of such models and derives
      explicit bounds for each of them. Similarly to (Negahban et al. 2012), the
      resulting bounds emphasize the good behavior of structured regularizers
      for high-dimensional problems when the target vector lies in an
      appropriate structured subspace. Pros: The type of statistical
      models ("dirty" models) for which bounds are derived has triggered
      recently considerable interest from the statistical community, elegantly
      providing convex relaxations to challenging problems. These problems
      moreover touch areas directly relevant to machine learning (graphical
      models, regression,...) and have clear potential applications. The
      paper provides important insights in the factors determining the
      performance of the superposition structured M-estimator. In particular,
      the concept of Structural Incoherence is introduced (assumption C4) to
      account for the interactions between the regularizers associated to each
      structure. As stated in the paper, the provided bounds seem to
      represent considerable improvement with respect to the state of the art
      and are more general. Finally, the authors convincingly illustrate the
      application of their general methodology in 3 interesting examples of
      sparse statistical models. Cons: Although the authors claim their
      result is very general, the assumptions on the model seem in practice
      quite constraining: in particular, the family of appropriate penalties,
      that are both norms and decomposable (as a sum of penalties over
      orthogonal subspaces), seems restricted to immediate generalizations of
      the l1-norm. It would be useful to discuss the possibility/consequences of
      including additional regularizers such as the squared Euclidian norm
      (typically: can this framework deal with elastic net penalties?). One
      other issue is the applicability of the bounds to practical cases. As far
      as I understand, the assumptions required to prove these bounds are
      numerous and involve several unknown parameters. In particular, in the
      first two presented examples (Proposition 2, Corollary 2, Proposition 3),
      conditions for which assumptions hold are not established, especially all
      results assume the C-linearity condition (7) to hold without further
      justification. Corollary 4 assumes a (simpler) C-linearity condition (12)
      as well. Moreover, in all examples, bounds depend on \bar{\kappa} or
      directly on the curvature parameter \kappa_{\mathcal{L}}, for which no
      range of acceptable values is provided. Is it possible to show for that
      these assumptions hold with some degree of generality and for some values
      of the parameters (curvature…), that might depend on the specific model.
      Although very well written, the paper is also very dense and some
      definitions/notations are difficult to understand without referring to
      (Negahban et al. 2012), see the following comments for details.
      Comments: -line 142: proposition 1 is important, please provide at
      least references for the concept of infimal convolution. Although the
      result seems intuitive, please make sure all necessary assumptions are
      correctly listed. -line 157: please mention that the target parameter
      is the minimizer of the population risk -line 168: the concept of
      restricted set is introduced, however without referring to other readings,
      it is difficult to understand how these sets are defined for each
      structure and used. Please provide some explanation. -Corollaries 2, 3
      and 4 do not mention whether the bound is obtained with high probability,
      please clarify. Typos: -line 230: "subspaces are chosen so
      that $\prod$...$", I guess the projector on the subspace *orthogonal* to
      $\mathcal{M}$ should vanish. -line 303: "we are now ready *to* derive"


Speedup Matrix Completion with Side Information: Application to Multi-Label Learning
In standard matrix completion theory, it is required to have at least O(n ln 2 n) observed entries to perfectly recover a low-rank matrix M of size n × n, leading to a large number of observations when n is large. In many real tasks, side information in addition to the observed entries is often available. In this work, we develop a novel theory of matrix completion that explicitly explore the side information to reduce the requirement on the number of observed entries. We show that, under appropriate conditions, with the assistance of side information matrices, the number of observed entries needed for a perfect recovery of matrix M can be dramatically reduced to O(ln n). We demonstrate the effectiveness of the proposed approach for matrix completion in transductive incomplete multi-label learning.

The paper studies the matrix completion problem in the
      context of collaborative filtering or multi-label learning. The
      contribution over existing work is that the authors account for side
      information on the rows and columns of the matrix in their model and show
      how exploiting side information can bring about improvements in sample
      complexity over the standard matrix completion formulation. In particular
      the authors study a model for matrix completion where the matrix of
      interest M can be decomposed as M = A Z_0 B^T with A and B is the known
      side information . When A and B are low dimensional this formulation has
      fewer degrees of freedom since the unknown matrix Z_0 is much smaller than
      M itself which intuitively explains why one should expect better sample
      complexity guarantees. The authors analyze a very natural
      extension of the usual nuclear norm minimization program. They show that
      the sample complexity required to guarantee unique minimizer is O(\mu^2 r
      (r_a + r_b) \log n) for a n \times n matrix of rank r with Z_0 of
      dimension r_a \times r_b and where \mu is related to the usual incoherence
      parameter used in the literature. This implies that if the side
      information is low dimensional (r_a, r_b small) the sample complexity is
      sublinear in the matrix dimensions, which is an improvement over standard
      results. The authors complement the theoretical results with
      simulations and experiments in the context of multi-label learning. The
      simulations show that their algorithm is both computationally and
      statistically more efficient that singular value thresholding, one of the
      standard matrix completion algorithms. Their algorithm also outperforms a
      number of baselines on several data sets in the multi-label learning
      application. The paper studies an interesting problem and
      contributes to the body of work on matrix completion by capturing side
      information. The results are interesting and the paper is well written.
      The proof techniques are novel but not particularly innovative given the
      work on matrix completion. In light of this, I think the contribution is
      somewhat small. In particular, it would be nice to know the necessary
      conditions in the noiseless case and the guarantees for the noisy version
      of the matrix completion problem. The experiments are fairly convincing
      but could be presented better. Graphs in lieu of tables would lead to
      improved readability while also saving space. Questions: 1. In
      the main paper there is no discussion about the role of \Omega_0 and
      \Omega_1 or any discussion about the condition \Omega_1 \ge q \Omega_0. It
      would be nice to provide some intuition as to why such a condition
      necessary and regimes under which it is satisfied. 2. Lemma 4 in the
      appendix requires that \Omega_0 \le |\Omega| \le \Omega_1, but in the main
      theorem there is no upper bound on the sample complexity. This seems like
      an error, but I'm more concerned with why there is an upper bound on
      |\Omega| in the first place. Is this condition necessary? What is the
      intuition for it? 3. Why are there test instances held out of the
      experiments? Why not just randomly sample the label matrix and test on the
      unobserved entries of that matrix? 4. The paper title is misleading --
      Speedup implies computationally, which was not the focus of the paper. I
      read the paper thinking it would be much more of an algorithmic paper than
      it was.


Least Informative Dimensions
We present a novel non-parametric method for finding a subspace of stimulus features that contains all information about the response of a system. Our method generalizes similar approaches to this problem such as spike triggered average, spike triggered covariance, or maximally informative dimensions. Instead of maximizing the mutual information between features and responses directly, we use integral probability metrics in kernel Hilbert spaces to minimize the information between uninformative features and the combination of informative features and responses. Since estimators of these metrics access the data via kernels, are easy to compute, and exhibit good theoretical convergence properties, our method can easily be generalized to populations of neurons or spike patterns. By using a particular expansion of the mutual information, we can show that the informative features must contain all information if we can make the uninformative features independent of the rest.

This paper is devoted to the problem of finding
      relevant stimulus features from the neural responses. The main idea is to
      minimize the information between a certain set of features and the neural
      responses combined with the remaining stimulus features in order to
      separate the relevant from irrelevant features. The method builds on the
      prior work by Fukumizu and colleagues, but unfortunately makes an
      additional restriction that greatly diminishes the potential use of this
      method. Specifically, the irrelevant and relevant dimensions need to be
      statistically independent from each other within the stimulus ensemble.
      While the authors argue that this is not a crucial assumption “because
      many distributions are Gaussian for which the dependencies between U
      [informative features] and V[the uninformative features] can be removed by
      pre-whitening the stimulus data before training”, this assumption
      essentially eliminates the novel contribution from this method. This is
      because with correlated Gaussian stimuli one can use spike-triggered
      covariance (STC) to find all features that are informative about the
      neural response in one simple computational step that avoids the
      non-convexity and complexity of the proposed optimization. Furthermore,
      the STC method has recent been shown to apply to elliptically symmetric
      stimuli (Samengo & Gollisch J Computational Neuroscience 2013).
      Finally, the proposed method has to iteratively increase the number of
      relevant features, whereas this is not needed in the STC method.
      In terms of practical implementations, the method was only
      demonstrated in this paper on white noise signals. Even in this simple
      case, the performance exhibited substantial biases. For example, in Figure
      1, right panel, a substantial component along the irrelevant subspace
      remained after optimization. Note that it is not correct to claim that
      (lines 286-888, and also lines 315-316) that “LID correctly identifies the
      subspace (blue dashed) in which the two true filters (solid black) reside
      since projections of the filters on the subspace closely resemble the
      original filters. “ The point of this method is to *eliminate* irrelevant
      dimensions, and this was not achieved. The superior convergence
      properties of the proposed techniques relatively to spike triggered
      average/covariance were not demonstrated. In fact, the method did not even
      remove biases in the STA, which would be possible using the linear
      approximation (Figure 3). Note that the Figure 3 description was
      contradictory: stimuli were described as white noise, yet the footnote
      stated that the data were not whitened. The use of information
      minimization for recovering multiple features and correlated stimuli was
      demonstrated in Fitzgerald et al. PLoS Computational Biology 2011,
      “Second-order dimensionality reduction using minimum and maximum mutual
      information models.” Line 234-235: it is not correct that MID
      (also STC) “requires data that contains several spike responses to the
      same stimulus” – one can optimize the relative entropy based on a sequence
      of stimuli presented only once in either MID/STC methods. Lines
      236-237: MID/STC can be used with spike patterns by defining events across
      time bins or different neurons. The abstract was written in a
      somewhat misleading manner, because it is not the information that is
      being minimized, but a related a quantity in combination with crucial
      assumptions that are not spelled out in the abstract. For example, that
      kernels need to factorize. The last statement in the abstract is also not
      clear “… if we can make the uninformative features independent of the
      rest.” Here, it is not clear that this implies that inputs are essentially
      required to be uncorrelated if any of the input features can be
      informative for one of the neurons in a dataset.


Statistical Active Learning Algorithms
We describe a framework for designing efficient active learning algorithms that are tolerant to random classification noise and differentially-private. The framework is based on active learning algorithms that are statistical in the sense that they rely on estimates of expectations of functions of filtered random examples. It builds on the powerful statistical query framework of Kearns [30]. We show that any efficient active statistical learning algorithm can be automatically converted to an efficient active learning algorithm which is tolerant to random classification noise as well as other forms of "uncorrelated" noise. We show that commonly studied concept classes including thresholds, rectangles, and linear separators can be efficiently actively learned in our framework. These results combined with our generic conversion lead to the first computationally-efficient algorithms for actively learning some of these concept classes in the presence of random classification noise that provide exponential improvement in the dependence on the error over their passive counterparts. In addition, we show that our algorithms can be automatically converted to efficient active differentially-private algorithms. This leads to the first differentially-private active learning algorithms with exponential label savings over the passive case.

The authors propose a class of active learning
      algorithms, termed 'active statistical' algorithms that is supposed to
      combine the noise robustness, and differential privacy of the standard
      statistical query (SQ) framework, with the active learning setting in
      which one wishes to minimizes the number of labels one requires by
      actively (adaptively) selecting which data points to label. As
      defined, the active SQ algorithms asks queries that consist of two parts:
      a 'filter' f:X->[0,1] representing the probability that a point X
      should be included in the current query, and the function g:X x
      {-1,1}->[-1,1], which is a function of a point-label pair.
      Additionally, as part of the query, there are two tolerance parameters,
      t1,t2, and the guarantee is that provided the measure of examples defined
      by f is at least t1, one receives a noisy estimate of E[g(x,y)} to within
      t2, where the expectation is defined over the measure induced by f.
      In general, I doubt this new model will be as helpful, or have as
      much mathematical structure as the general SQ model. That said, it does
      seem to be `the right' model for capturing noise tolerance in an active
      learning setting (which yields differential privacy), and the algorithmic
      results, while not earth-shattering, provide a nice illustration of the
      power of this model. It would be nice to include a little more
      discussion of the relation between the general SQ framework and the
      proposed model--specifically, the fact that the proposed model of queries
      can be implemented in the SQ framework, just with a tolerance that scales
      according to the measure of the portion of the distribution in
      consideration for each query.


Optimal integration of visual speed across different spatiotemporal frequency channels
How do humans perceive the speed of a coherent motion stimulus that contains motion energy in multiple spatiotemporal frequency bands? Here we tested the idea that perceived speed is the result of an integration process that optimally combines speed information across independent spatiotemporal frequency channels. We formalized this hypothesis with a Bayesian observer model that combines the likelihood functions provided by the individual channel responses (cues). We experimentally validated the model with a 2AFC speed discrimination experiment that measured subjects' perceived speed of drifting sinusoidal gratings with different contrasts and spatial frequencies, and of various combinations of these single gratings. We found that the perceived speeds of the combined stimuli are independent of the relative phase of the underlying grating components. The results also show that the discrimination thresholds are smaller for the combined stimuli than for the individual grating components, supporting the cue combination hypothesis. The proposed Bayesian model fits the data well, accounting for the full psychometric functions of both simple and combined stimuli. Fits are improved if we assume that the channel responses are subject to divisive normalization. Our results provide an important step toward a more complete model of visual motion perception that can predict perceived speeds for coherent motion stimuli of arbitrary spatial structure.

COMMENTS BASED ON REVIEWER DISCUSSIONS AND AUTHOR
      REBUTTAL: I agree with the other reviewers that more could be done
      to constrain the specifics of the cue integration mechanism. However, I
      believe that if the data set is expanded, allowing the models to be better
      constrained, then the paper is appropriate and interesting for the NIPS
      community. I have left my quality score as it was, but I agree
      with the other reviewers that the paper merits a ``1'' rather than a ``2''
      for impact score. ORIGINAL REVIEW: Summary: This paper
      extends an existing model for the perception of visual speed that uses a
      Bayesian observer model acting on the activity of independent
      spatiotemporal frequency channels. Previously, the model accounted for
      illusions of perceived speed by postulating the Bayes-optimal combination
      of noisy sensory representations with a prior for slow speeds. Here this
      model is extended to operate on spatiotemporal filter outputs, allowing
      the model to account for changes in perceived speed as a function of
      spatial frequency. The revised model is shown to provide a good fit to
      human speed discrimination data. Strengths: The model is
      elegantly simple and provides a good fit to data. The model is a useful
      extension of previous work, providing a more general and
      neurophysiologically plausible version. The behavioral experiment is
      straightforward and clearly makes the case for the addition of the
      normalization mechanism. The paper is concise and clearly written.
      Weaknesses: -- ``The perception of combined stimuli
      can therefore be fully predicted by optimal combining responses from
      individual spatiotemporal frequency channels.'' This statement
      (``fully predicted'') seems too strong to me for two reasons: 1) while
      bias is predicted really well, threshold fits are less impressive and 2)
      no new data has been predicted (all data were used in the model fits). It
      would be interesting to see the cross-validated area under the ROC between
      the two models and the Gaussian fits to give the reader a better sense of
      the scale of the prediction improvement. On thresholds, while it's
      hard to say from three data points, is there a plausible reason for my
      impression that the discrimination thresholds for combined stimuli seem to
      decline more steeply before plateauing (e.g. power law or exponential)
      rather than the more linear predictions of the model? That is, the
      threshold predictions at high contrasts are good, but thresholds at low
      contrasts might be under-predicted whereas thresholds at medium contrasts
      are over-predicted. Would the ``Foley function'' with the exponent of the
      numerator set to 2.4 (instead of 2 in your CRF) provide a better fit to
      those data due to the accelerating contrast response early in the
      function? If I understand correctly, the channel variances
      \sigma^2 correspond to (or can be transformed to give) the bandwidths of
      the hypothetical channels. Can the authors comment on whether the
      bandwidths estimated in their model fits correspond to bandwidths of
      spatiotemporally-oriented filters measured elsewhere (e.g. physiology,
      psychophysical masking paradigms)? Line 367: ``The model also
      fully supports previous accounts on the shape of the prior for speed: the
      slope of the linear approximation of the prior probability density is
      negative and conforms to previously reported values.'' Where in the
      manuscript is the slope of the prior reported (or any fitted model
      parameters for that matter)? A small table of best-fitting model
      parameters and their standard errors would be of use here. -- How
      easily can the two-channel model derivation be extended? Will
      multi-channel (> 3) responses require numerical approximations?
      Minor points: Line 85: grammar... "proposing that the
      visual system treats the responses in individual spatiotemporal frequency
      channel as independent cues". Line 103: grammar... "only those unit"
      should be "units" Line 107: grammar... "An moving stimulus" should be
      "A moving stimulus" Line 131: typo "r2," Line 242: typo "agains
      data" I found the paragraph beginning Line 256 unclear, and I
      didn't really understand the stimulus combinations until reading the
      results. I think it would be clearer to write something like: ``There
      were four simple stimuli: a 0.5 c/deg grating presented at 22.5% contrast,
      and three 1.5 c/deg gratings presented at 7.5, 22.5 and 67.5% contrast
      respectively. Combined stimuli were superpositions of the lower and higher
      frequency simple stimuli, in either a ``peaks-add'' or ``peaks-subtract''
      configuration.'' Line 261:``We used a broadband stimulus regulated
      by an adaptive staircase as reference.'': it would be useful to explicitly
      state here that it is the speed of the reference that is under adaptive
      control. Line 315: ``we plot the peaks-add conditions as black
      squares, and the peaks-subtract conditions as black triangles.'' I see no
      black triangles in the figure; I assume these were separated into separate
      panels. Line 320: ``independent on phase'' should be ``independent
      of phase'' Line 375: ``This decrease is nicely reflected in the
      measured discrimination thresholds for the combined stimuli where the
      threshold for both individual gratings were of the same size (Fig. 4b)'':
      I think this figure reference must be wrong because 4b refers to bias, not
      thresholds. Line 377: ``predicts that the perceived speed are''
      should be ``speeds are'' or ``speed is''. Line 378: ``sensory
      system in general processes'' should be ``systems in general processes''
      or ``system in general process''. Same for ``question'' on line 380.


DESPOT: Online POMDP Planning with Regularization
POMDPs provide a principled framework for planning under uncertainty, but are computationally intractable, due to the "curse of dimensionality" and the "curse of history". This paper presents an online POMDP algorithm that alleviates these difficulties by focusing the search on a set of randomly sampled scenarios. A Determinized Sparse Partially Observable Tree (DESPOT) compactly captures the execution of all policies on these scenarios. Our Regularized DESPOT (R-DESPOT) algorithm searches the DESPOT for a policy, while optimally balancing the size of the policy and its estimated value obtained under the sampled scenarios. We give an output-sensitive performance bound for all policies derived from a DESPOT, and show that R-DESPOT works well if a small optimal policy exists. We also give an anytime algorithm that approximates R-DESPOT. Experiments show strong results, compared with two of the fastest online POMDP algorithms. Source code along with experimental settings are available at http://bigbird.comp. nus.edu.sg/pmwiki/farm/appl/.

---Response to Author Response--- Thank you
      for the clarifications. Just a couple of points: - While I do
      agree that characterizing the class of problems that have small optimal
      policy trees would be interesting, I think the paper could do a better job
      of describing the approach's strengths and weaknesses, even without fully
      answering this theoretical question. The authors must surely have some
      intuitive examples of problems where this algorithm should do very well,
      and problems on which it should do poorly. Sharing this intuition would
      significantly strengthen the paper in my opinion. - Relatedly,
      whether DESPOT is better at exploiting certain kinds of prior information
      was only part of my question. It is also important to address whether
      DESPOT *requires* prior knowledge to do well. From that perspective I
      don't think empirical comparison is difficult; you can simple try out
      DESPOT with an uninformed default policy (e.g. uniform random). If it does
      well, it would alleviate concern about the need for a good default policy.
      It if does poorly, it will be illustrative of problems for which DESPOT
      would not be a good choice. Either way it would increase the knowledge
      transmitted by the paper without requiring much additional effort or
      space. ---Summary of paper--- The paper presents an online
      POMDP planning algorithm that uses both classical search methods and Monte
      Carlo sampling to select an action. The main idea of the algorithm is to
      pre-sample random events (a la PEGASUS) in the roll-out tree, and then
      compute a good policy for the determinized future (values at the leaves
      are estimated using Monte Carlo samples of a default policy). The policy
      search is regularized to prefer "smaller policies" (policies that fall
      back on the default policy after a small number of steps) and the
      determinized tree can be pruned using a heuristic branch-and-bound
      algorithm to reduce the computational cost of finding a policy.
      Performance bounds are given in terms of the number of samples, depth of
      the tree, *and* the size of the optimal policy on the tree. Experiments
      show the algorithm performs well in several large POMDPs in comparison to
      contemporary competing methods. ---Quality--- As far as I
      can tell the analysis is technically correct. The bounds derived give
      insight into the algorithm's properties (though it's not clear how much
      purchase the bounds have once the DESPOT is approximated) and for the most
      part match the intuitions given. The main thing I would have liked
      to see in this paper is more of a discussion of the strengths and
      weaknesses of this approach, particularly in comparison to the UCT-like
      approach (POMCP). The introduction claims that R-DESPOT has better
      worst-case behavior, but this is only the case if the optimal policy tree
      is "small." I think the paper would be much stronger if it contained more
      discussion of what the implications of a "small" policy are. In what types
      of problems is this likely to be the case and when is it catastrophically
      not the case? I am most concerned that this may depend heavily on the
      choice of default policy (because a "small" policy is one that mostly
      defers to the default policy), meaning that for this method to perform
      well, one must already have a high quality policy available. I think that
      trying to exploit the existence of a simple optimal policy is sensible,
      but it's not clear to me that this is a good way to think about simple
      policies -- it would help to have a better idea of what the assumption
      means in terms of the problems it does/does not encompass.
      Relatedly, and perhaps more importantly, it's not clear to me that
      the experiments are really a fair comparison. R-DESPOT is given a
      domain-specific default policy in each experiment. While I can tell that
      the authors have tried to keep these policies intuitive and simple to
      specify, they nevertheless seem to constitute prior knowledge that is not
      supplied to the other methods. It is hard, therefore, to tell how much of
      the performance gap is due to the method, and how much is due to the
      choice of default policy. I don't necessarily think this fatally
      undermines the results. I think it would be reasonable, for instance, to
      claim that R-DESPOT can exploit prior knowledge that other methods can't,
      or that would be a challenge to encode for other methods. If that is the
      claim, however, it needs to be stated explicitly and justified at least in
      intuition. Even if it is difficult to supply comparable prior knowledge to
      the other methods, I would have at least liked to have seen the
      performance of R-DESPOT with a uniform random default policy, even if it
      performed poorly, since this would be informative about whether R-DESPOT
      is appropriate for a problem in which I do not have an informed default
      policy. ---Clarity--- I found this paper to be clear and
      very readable. I also found the mathematics in the supplementary materials
      to be refreshingly well-presented and easy to follow. A small
      suggestion: at first I found the paragraph at the end of Section 3 to be a
      bit out of place. Once in Section 4 I quickly realized that it was meant
      to foreshadow the R-DESPOT algorithm. This could be clarified with a
      simple "We will make use of this idea in the next section in order to
      develop a more practical algorithm" or similar. Type-o near bottom
      of p. 1: behavoior ---Originality--- Though the method
      presented here makes use of several existing ideas, it combines them in a
      novel and interesting way. I felt the method was well-contextualized in
      the relevant literature. ---Significance--- The problem of
      planning efficiently in high-dimensional partially observable worlds is
      both important and challenging. The algorithm presented here is rooted in
      theoretical performance bounds and performs well empirically. Though I do
      still have questions regarding the need for prior knowledge, in some
      domains prior knowledge is available. The main potential for impact that I
      see is that the algorithm represents a seemingly viable alternative
      approach to UCT-like algorithms (as well as a different style of analysis)
      that may serve as inspiration for further algorithmic developments in this
      area.


Reshaping Visual Datasets for Domain Adaptation
In visual recognition problems, the common data distribution mismatches between training and testing make domain adaptation essential. However, image data is difficult to manually divide into the discrete domains required by adaptation algorithms, and the standard practice of equating datasets with domains is a weak proxy for all the real conditions that alter the statistics in complex ways (lighting,  pose, background, resolution, etc.)  We propose an approach to automatically discover latent domains in image or video datasets. Our formulation imposes two key properties on domains: maximum distinctiveness and maximum learnability. By maximum distinctiveness, we require the underlying distributions of the identified domains to be different from each other to the maximum extent; by maximum learnability, we ensure that a strong discriminative model can be learned from the domain. We devise a nonparametric formulation and efficient optimization procedure that can successfully discover domains among both training and test data. We extensively evaluate our approach on object recognition and human activity recognition tasks.

This paper presents a new technique for domain
      adaptation for computer vision datasets, which typically present multiple
      aspects, e.g. viewpoint, illumination, level of clutter, compression, etc.
      Instead of simply equating domains with datasets, which ends up mixing
      those aspects together, the proposed technique automatically partitions
      the set of all images over all datasets into domains. The partitioner is
      driven by two principles: making separate domains in feature space, and
      making them so that a good discriminative classifier can be trained on
      each of them (to classify the original classes, not to separate the
      domains). This technique is more likely to partition according to the
      underlying aspects rather than datasets. Originality and
      significance: There is only little work on automatically defining
      domains, and this paper proposes a good idea towards automatically
      discovering useful domains, that will support training better classifiers
      for the original problem. On the negative side, I am not fully
      convinced of the proposed optimization method (section 2), as it's not
      clear how closely it solves the original problem (2)+(3). Moreover, the
      two proposed driving criteria are not well integrated yet: the maximal
      learnability criterion is only used as a 'wrapper around' the maximal
      separability criterion, in order to determine the number of domains.
      Essentially, it acts as a post-hoc validation score deciding how good is
      the domain partitioning learned for a given number of domains, but the
      partitioning itself is made based on the separability criterion alone. An
      integrated process would instead produce the partitioning that maximises
      some goal function including both criteria directly. Despite these
      shortcomings, which might be due to the fact that this type of work is
      still at quite exploratory stages, I feel that the paper is a step in the
      right direction for the community and should be accepted.
      Quality and clarity: The paper is well written, but lacks
      figures to illustrate the concepts presented. Experiments:
      On the positive side, the experiments show a significant advantage in
      using the domains produced by the proposed method, over just using
      datasets as domains, and over the very recent domain discovery technique
      [19] (sec. 4.2). The idea of including the test set in the
      'reshaping' process is interesting, but not clearly presented (sec. 4.3).
      Also, this corresponds to an imputation setting (i.e. all test data is
      available at the same time), but this is not stated clearly. On
      the negative side, the image descriptor used is very simple and outdated:
      just one bag-of-words of sparse SURF features for the entire image, and
      with just 800 codebook entries. This is really weak nowadays. I recommend
      the authors to use a spatial pyramid of bag-of-words, computed on _dense_
      SURF features. Also, it is not clear what similarity measure is used to
      compare image descriptors, hopefully X^2 or an intersection kernel? The
      Euclidean distance is not suitable for comparing histograms. As a next
      step, a Fisher Vector representation could help further and finally place
      the image representation in this paper at the level of modern systems. The
      following papers might help the authors: Zhang IJCV 2006; Jurie CVPR 2005;
      Lazebnik CVPR 2006; Perronin ECCV 2010.


Accelerating Stochastic Gradient Descent using Predictive Variance Reduction
Stochastic gradient descent is popular for large scale optimization but has slow convergence asymptotically due to the inherent variance. To remedy this problem, we introduce an explicit variance reduction method for stochastic gradient descent which we call stochastic variance reduced gradient (SVRG). For smooth and strongly convex functions, we prove that this method enjoys the same fast convergence rate as those of stochastic dual coordinate ascent (SDCA) and Stochastic Average Gradient (SAG). However, our analysis is significantly simpler and more intuitive. Moreover, unlike SDCA or SAG, our method does not require the storage of gradients, and thus is more easily applicable to complex problems such as some structured prediction problems and neural network learning.

This work proposes a new algorithm that perturbs a
      constant-step size stochastic gradient iteration. The perturbation is
      based on a previous full-gradient evaluation, and allows the method to
      achieve a linear convergence rate that is faster than the rate obtained by
      deterministic gradient methods. The paper proposes a nice
      approach, that does indeed address the memory problems with using
      alternate approaches like SDCA and SAG. There is a nice discussion of how
      all three algorithms essentially force the variance of the estimator to go
      to zero as the solution is approached. After equation (6):
      Normally, strong-convexity is defined with gamma strictly greater than 0.
      Also, the gradient method has a linear convergence rate with eta < 2/L.
      The citation style is wrong, and the bibliography needs to be
      cleaned up. Note that the SAG and SDCA papers were published in NIPS'12
      and ICML'13 conferences, respectively. Is is probably worth
      explicitly noting somewhere the requirement that eta < 1/2L in Theorem
      1. The proof is indeed much simpler than the SAG proof. However,
      in the strongly-convex case I don't think that it is obviously simpler
      than the SDCA proof. But it is probably worth pointing out that the SDCA
      method is for a special case of (1). Because the method requires
      full passes through the data, it doesn't really seem appropriate to call
      it a stochastic gradient method. Indeed, on the first pass through the
      data the method doesn't change the parameters, so it will start out a full
      pass behind the basic SGD method. A better name might be a hybrid method
      or a mixed method. Related to the above, it is worthwhile to add a
      sentence to the paper contrasting the new approach with the
      closely-related approaches that use growing sample-sizes, like Friedlander
      and Schmidt ["Hybrid Deterministic-Stochastic Methods for Data Fitting"]
      and Byrd et al. ["Sample size selection in optimization methods for
      machine learning"]. To complement Section 4, it might be
      worthwhile to try to write the new method as well as SAG as special cases
      of a more general algorithm. (In SAG, \nabla \phi_{i_t}(\tilde{w}) is
      replaced by \nabla \phi_{i_t} (w_{i_t}) where w_{i_it} is some previous
      iteration, and mu is updated at each iteration rather than after every m
      iterations). It seems like there are numerous variations on this family
      that would give linear convergence. The paper should say how eta
      and m were set in the experiments. Although the algorithm does
      indeed achieve a rate similar to SDCA and SAG in terms of number of
      examples, the rate achieved appears to be slower in general. For example,
      using the example on page 5 where L/gamma=n, it requires m=8n to have
      alpha=1/2 with the step size of 1/4L. So the algorithm requires 10 passes
      through the data on each iteration to achieve this rate (or 9 passes if a
      memory is used). If you are allowed to do 10 passes through the data of
      SDCA in this setting, you would get a faster rate of alpha=(1-1/2n)^10n
      for the same number of points visited as the new algorithm. (the
      rate of SDCA is 1 - n\lambda/n(n\lambda+L) = 1-1/2n if lambda=L/n)
      The last sentence of Section 3 should be accompanied by a citation
      or more details. Otherwise, it should be left out. Regarding
      Section 4, note that there are other works discussing the relationship
      between primal gradient methods and dual coordinate-wise methods. For
      example, see Bach's "Duality between subgradient and conditional gradient
      methods". I found the experimental evaluation of the proposed
      method to be quite weak, only comparing to the SGD method in a regime
      where 200 or 500 passes through the data set are performed. This does not
      seem like the regime where SGD would be the method of choice, particularly
      for the convex objective. Given this setting, and given that the method is
      a hybrid-type strategy, it would make more sense to also compare to
      competitive methods (like an L-BFGS or non-linear conjugate gradient code)
      and other hybrid methods (such as those cited above), since these would be
      the logical competitors. For the first experiment, it would also
      be informative to compare directly to SDCA or SAG, and simply say that
      their memory requirements do not allow them to be used in the second
      experiment. For Figure 3, it would be informative to plot the
      sub-optimality on a log-scale, to empirically verify that the method is
      converging linearly.


Generalized Denoising Auto-Encoders as Generative Models
Recent work has shown how denoising and contractive autoencoders implicitly capture the structure of the data-generating density, in the case where the corruption noise is Gaussian, the reconstruction error is the squared error, and the data is continuous-valued. This has led to various proposals for sampling from this implicitly learned density function, using Langevin and Metropolis-Hastings MCMC. However, it remained unclear how to connect the training procedure of regularized auto-encoders to the implicit estimation of the underlying datagenerating distribution when the data are discrete, or using other forms of corruption process and reconstruction errors. Another issue is the mathematical justification which is only valid in the limit of small corruption noise. We propose here a different attack on the problem, which deals with all these issues: arbitrary (but noisy enough) corruption, arbitrary reconstruction loss (seen as a log-likelihood), handling both discrete and continuous-valued variables, and removing the bias due to non-infinitesimal corruption noise (or non-infinitesimal contractive penalty).

This paper develops a step towards the first useful
      formulation of a denoising autoencoder, a particular flavor of neural
      network which implicitly captures the structure of the data by reproducing
      the input at its output, as a generative model - i.e. as a probabilistic
      model defining a distribution over the data. In particular, this paper
      develops a methodology for drawing samples from the data distribution
      implicitly captured by the denoising autoencoder. The paper does not
      develop a formulation of denoising autoencoders as generative models
      directly nor a method to extract the probability under the model of a
      novel data case. Nevertheless, this is a step towards the solution to a
      longstanding problem in the autoencoder and deep learning literature.
      Restricted Boltzmann machines, have existed as generative models of the
      data for many years but have remained difficult to use as such in
      practice, particularly for non-binary valued data. Thus, this work may
      offer a useful alternative for continuous valued data. From the
      revelations of the paper, the authors develop a new training algorithm for
      denoising autoencoders which they claim is better than existing methods -
      this is however not empirically validated. The paper is well written,
      structured and clear. The methodology and formulation appears to be
      correct and the paper offers some interesting discussion and intuition.
      The empirical validation is somewhat weak given the claims made in the
      paper. For example, the empirical justification of the walkback algorithm
      introduced in the paper is purely qualitative and only on one example.
      Given the wide practical applicability of denoising autoencoders, and a
      vast literature of corresponding quantitative empirical benchmarks, it
      seems strange to not see a quantitative comparison between denoising
      autoencoders trained with the walkback algorithm compared to those
      without. Also, the only quantitative empirical result (line 404) would
      suggest that the walkback algorithm in fact does not improve
      (quantitatively) at all over the standard procedure despite some
      additional cost. This is unfortunate as it seems that the walkback
      algorithm should really be empirically validated for the claim that it
      reduces spurious modes to be justified. Thus it seems unclear whether the
      sampling algorithm introduced in the paper can be used to actually improve
      autoencoder training. On line 350, the authors state that they
      observe in the empirical evaluation that the walkback algorithm is more
      efficient than the standard training algorithm - however, this is nowhere
      to be found in the empirical evaluation section. Quality: The
      overall quality of ideas, writing and concepts is high. The empirical
      evaluation could be significantly more thorough. Clarity: The
      paper is extremely clear, well written and structured.
      Originality: The work is certainly original and attempts to
      address a heretofore unsolved problem with denoising autoencoders. The
      algorithm for sampling from the data density captured by the autoencoder
      is neat and novel. Significance: This is a possibly
      significant step towards solving a longstanding problem in the autoencoder
      literature. However, some of the theoretical aspects of the paper are not
      sufficiently empirically validated. It is especially not clear from the
      empirical validation whether the 'walkback algorithm' is actually
      practically useful. The ability to draw samples from the data distribution
      is very neat but it would be useful to have some significant empirical
      justification of the practical utility of this. Much remains to be solved
      to use denoising autoencoders as generative models - thus the title may be
      somewhat misleading. It seems the paper claims much more than is actually
      justified by empirical evaluation.


Pass-Efficient Unsupervised Feature Selection
The goal of unsupervised feature selection is to identify a small number of important features that can represent the data. We propose a new algorithm, a modification of the classical pivoted QR algorithm of Businger and Golub, that requires a small number of passes over the data. The improvements are based on two ideas: keeping track of multiple features in each pass, and skipping calculations that can be shown not to affect the final selection. Our algorithm selects the exact same features as the classical pivoted QR algorithm, and has the same favorable numerical stability. We describe experiments on real-world datasets which sometimes show improvements of several orders of magnitude over the classical algorithm. These results appear to be competitive with recently proposed randomized algorithms in terms of pass efficiency and run time. On the other hand, the randomized algorithms may produce more accurate features, at the cost of small probability of failure.

This paper proposes improvements to the classical
      pivoted QR algorithm ("QRP") which lead to time and space advantages
      useful for feature selection in the large p setting (many variables). The
      paper is well-written for the most part, and the presentation is clear
      enough. The algorithm proposed in the paper follows from what might be
      called low-hanging (or even "obvious") improvements to QRP, in the sense
      that anyone working with large datasets in practice would begin to think
      about these and related ideas. But, as in many practical settings, the
      devil is in the details, and *details matter* particularly in large-scale
      implementations. This is where the submission shines. The authors have
      been careful about their CPU and memory bookkeeping at every step, and
      additionally give a few nice low-level suggestions (e.g., efficient data
      structures to consider; stability of orthogonalization subroutines is
      discussed, etc.). On the whole this paper provides a detailed, thorough
      picture for practitioners working with large datasets, and this is perhaps
      its most salient selling point: one can go from the paper directly to
      implementation. I believe this is enough of an advantage to overcome
      concerns about novelty and the other complaints outlined below.
      Other comments: - The interpolative decomposition is closely
      related, but it not cited let alone reviewed or compared. This is perhaps
      a major omission. Yes, the ID is also related to CUR decompositions, which
      are mentioned, but let's not assume the reader knows this, or by
      extension, how it ultimately relates to the IQRP algorithm proposed in the
      paper. See: - Work from Mark Tygert (NYU) - N. Halko, P. G.
      Martinsson, and J. A. Tropp, "Finding Structure with Randomness:
      Probabilistic Algorithms for Constructing Approximate Matrix
      Decompositions," SIAM Review, 2011. - The ID and other randomized
      techniques are reasonably efficient. So, if we're going to compare
      randomized algorithms to IQRP, the efficiency advantage of the latter
      seems to critically hinge on c being less than k. Why should c be
      substantially less than k? There is no discussion really exploring this
      critical, even key, distinction. The experiments suggest it may be true,
      but the empirical evaluation is limited, and might make it hard to form a
      concrete opinion as to whether one can expect c << k in a given
      practical setting going forward. Note that this is apparently what
      underpins the question as to whether to use IQRP over randomized ID, for
      instance. - Experiments: major improvements are needed. For what
      is mainly an applications/practical paper, the empirical evaluation is a
      bit underwhelming. Could you show QRP perhaps on the same plots at a
      minimum? Or comparisons to other, more competitive baselines one would
      actually consider if faced with making a design decision? - What
      is meant by "randomization algorithms can produce better features"? This
      statement is never really made precise. Also you say "IQRP may be less
      accurate". In terms of what? It would be interesting to see the accuracy
      on the regression/classification problems themselves, using the features
      selected by the various algorithms. Ultimately, this is what we're after,
      at least for the datasets considered in the paper. - There are
      other, related, applications for the pivoted QR decomposition that may
      help your case. For example, finding nearly dependent columns in forward
      stepwise regression techniques (and computing related statistics: F-stats,
      t-stats etc).


Approximate Bayesian Image Interpretation using Generative Probabilistic Graphics Programs
The idea of computer vision as the Bayesian inverse problem to computer graphics has a long history and an appealing elegance, but it has proved difficult to directly implement. Instead, most vision tasks are approached via complex bottom-up processing pipelines. Here we show that it is possible to write short, simple probabilistic graphics programs that define flexible generative models and to automatically invert them to interpret real-world images. Generative probabilistic graphics programs (GPGP) consist of a stochastic scene generator, a renderer based on graphics software, a stochastic likelihood model linking the renderer's output and the data, and latent variables that adjust the fidelity of the renderer and the tolerance of the likelihood. Representations and algorithms from computer graphics are used as the deterministic backbone for highly approximate and stochastic generative models. This formulation combines probabilistic programming, computer graphics, and approximate Bayesian computation, and depends only on generalpurpose, automatic inference techniques. We describe two applications: reading sequences of degraded and adversarially obscured characters, and inferring 3D road models from vehicle-mounted camera images. Each of the probabilistic graphics programs we present relies on under 20 lines of probabilistic code, and yields accurate, approximately Bayesian inferences about real-world images.

This paper proposes a general method for solving
      image-based computer vision tasks using a generative probabilistic model
      that uses a graphics program to generate images. The method takes the
      standard Bayesian approach to frame the inference of the target variables,
      and uses Metropolis-Hastings to perform the inference. This framework is
      implemented for a CAPTCHA and a road-finding application, with favorable
      results reported for each one. The primary contribution of this
      paper is a proposal for using graphics programs as a key element of a
      generative model for image-based tasks. While their claim that there are
      no previous real-world image interpretation frameworks that combine
      computer graphics among the other elements they list (last paragraph of
      Section 1) seems accurate, their proposed system does not seem to qualify
      as such a framework unless it's under a restricted interpretation.
      Specifically, the CAPTCHA problem is "real-world" in one sense, but
      CAPTCHAs are generated by a synthetic process and are therefore
      qualitatively a very different type of problem than such real-world
      problems as object recognition or detection in natural images. This
      distinction is important, because one of the main reasons that
      discriminative approaches have gained favor is due to their ability to
      deal with complex, noisy features found in natural images. The authors
      would presumably agree that their case is severely weakened if their
      framework is limited to synthetic images such as CAPTCHAs. As for
      the road-finding problem, it certainly qualifies as a real-world problem
      and deals with natural images, but it is a strain to consider what they
      use to be a graphics program. The elements that the model generates are
      simply the properties (height, width, and lane size) of the lines that
      form the lanes of the road, and the "renderer" is simply an assignment of
      pixels to the regions delineated by these properties rather than an actual
      realistic rendering of the scene. This is perfectly reasonable, but it is
      no more of a computer graphics-based approach than the many previous
      approaches that align geometric elements to a scene (e.g., [3], [8], Hedau
      et al. ICCV '09, Lee et al. NIPS '10, Gupta et al. ECCV '10, and many
      similar projects by some of the authors of those papers). As such it
      should be seen as another work in this line of research, not a unique and
      novel direction that it would be if it were applied using a generative
      model that produced something closer to an actual image than geometric
      lines. The authors point out that many other works require a good
      amount of application-specific engineering, which they do not require.
      While this is true, they also admit that such engineering would be a
      useful addition. In fact, in the Introduction section they point to [17,
      20] as examples of methods that rely on custom inference strategies --
      specifically, MCMC proposals that incorporate bottom-up cues -- in
      contrast to their more general approach, and then in the Discussion
      section they say that discriminatively trained proposals would likely be
      important to extend their approach to more complex scenes. Novelty
      aside, the proposed framework does have attractive properties and the
      paper is well written, and the recent line of generative models for
      computer vision problems seems to be worth pursuing and has yielded
      promising results. While the probabilistic aspect is relatively
      straightforward, it is simple and easily adjusted according to the needs
      of the application (such as incorporating rotation), and the use of
      Metropolis-Hastings is elegant and general. The authors report favorable
      results on two interesting applications. It would have been better to
      compare the system's performance on the CAPTCHAs to further
      state-of-the-art approaches. Furthermore, while they compare directly to
      [1] on the road-finding application, it would be helpful to provide a
      means of comparison to the other results reported in the dataset's
      original paper ([4]). As a technical note, it seems that the first
      equation should contain an integral over X, which is latent and does not
      appear on the left hand side.


Sparse Additive Text Models with Low Rank Background
The sparse additive model for text modeling involves the sum-of-exp computing, whose cost is consuming for large scales. Moreover, the assumption of equal background across all classes/topics may be too strong. This paper extends to propose sparse additive model with low rank background (SAM-LRB) and obtains simple yet efficient estimation. Particularly, employing a double majorization bound, we approximate log-likelihood into a quadratic lower-bound without the log-sumexp terms. The constraints of low rank and sparsity are then simply embodied by nuclear norm and ℓ 1 -norm regularizers. Interestingly, we find that the optimization task of SAM-LRB can be transformed into the same form as in Robust PCA. Consequently, parameters of supervised SAM-LRB can be efficiently learned using an existing algorithm for Robust PCA based on accelerated proximal gradient. Besides the supervised case, we extend SAM-LRB to favor unsupervised and multifaceted scenarios. Experiments on three real data demonstrate the effectiveness and efficiency of SAM-LRB, compared with a few state-of-the-art models.

This paper presents a model inspired by the SAGE
      (Sparse Additive GEnerative) model of Eisenstein et al. The authors use a
      different approach for modeling the "background" component of the model.
      SAGE uses the same background model for all; the authors allow different
      backgrounds for different topics/classification labels/etc., but try to
      keep the background matrix low rank. To make inference faster when using
      this low rank constraint, they use a bound on the likelihood function that
      avoids the log-sum-exp calculations from SAGE. Experimental results are
      positive for a few different tasks. Quality: The paper is
      solid. The idea of using a low-rank background matrix makes a lot of sense
      for SAGE, and the use of the double majorization bound and robust PCA
      learning algorithm seem like good ideas. It works in practice in terms of
      classification accuracy and perplexity, doing at least as well as SAGE but
      is simultaneously faster. All in all, I think it's a nice contribution.
      The paper is clear and the approach is sound. I had some questions, which
      are listed below, but in general they are minor things. I would
      have liked to have seen a little bit more analysis/perspective on why it's
      working better than SAGE, especially if the reason happens to relate to
      the use of multiple background vectors with the low-rank constraint. One
      could also try SAGE with multiple background vectors, right? Presumably
      this would not work as well, because something like the low-rank
      constraint would be needed to help bias the learning, but I would prefer
      more discussion on what exactly the multiple background vectors buys you
      in practice. Clarity: For the most part, the paper is
      clearly-written. It does draw upon lots of other work which is not fully
      described for space reasons, but the core ideas are mostly self-contained.
      I did have some questions here and there, which perhaps the authors can
      clarify. I deduce from Algorithm 1 that \nu is set by hand to be
      0.05. But is \mu also set by hand? Were the same values used for all
      experiments or were they tuned for each experiment separately? How
      sensitive is the performance to the particular values used? Giving some
      more perspective on this point with more experimental results would be
      very helpful. In Figure 2, what is lb-quad-loc? I couldn't find it
      in the text. I suspect it's lb-quad-j, which is in the text. I guess
      lb-quad-j should be changed to lb-quad-loc. The figures are very difficult
      to understand in black-and-white; please make them easier to read, both by
      increasing the size of the fonts used and using a representation that can
      be understood in black and white. Around line 140, "bound proposed
      by Jaakkola and Jordan [6]" -- I think you meant [14] there instead of
      [6]. Originality: This paper is not extremely original,
      but it's certainly not trivial either. I think that several members of the
      NIPS community could have come up with something like this if they took
      the time to think about it, but some points in here are rather neat, like
      the improvement through optimization of the variational parameters in the
      bound and the connection to robust PCA. Significance: SAGE
      has been used by a few research groups and seems to be generally known by
      researchers, so having an improvement to SAGE has the potential for
      significant impact. I like also that there are some simple ways to extend
      the approach due to its clean formulation (e.g., different norms could be
      considered). At the end of the day, it's incremental work, but it's done
      well and should get published at some point.


Sparse Overlapping Sets Lasso for Multitask Learning and its Application to fMRI Analysis
Multitask learning can be effective when features useful in one task are also useful for other tasks, and the group lasso is a standard method for selecting a common subset of features. In this paper, we are interested in a less restrictive form of multitask learning, wherein (1) the available features can be organized into subsets according to a notion of similarity and (2) features useful in one task are similar, but not necessarily identical, to the features best suited for other tasks. The main contribution of this paper is a new procedure called Sparse Overlapping Sets (SOS) lasso, a convex optimization that automatically selects similar features for related learning tasks. Error bounds are derived for SOSlasso and its consistency is established for squared error loss. In particular, SOSlasso is motivated by multisubject fMRI studies in which functional activity is classified using brain voxels as features. Experiments with real and synthetic data demonstrate the advantages of SOSlasso compared to the lasso and group lasso.

Comments: This paper proposes a new method for
      multi-task learning where the features in different tasks are similar but
      not identical. The main contribution is the proposed regularizer for
      Sparse Overlapping Sets lasso, where the group configuration can be
      automatically learned. Experiments with real and synthetic data
      demonstrate the advantages of SOSlasso compared to the lasso and group
      lasso. Overall, this paper is interesting. However, compared to
      earlier works, the novelty and improvements are not very clear enough.
      Several key problems need further clarified in this submission. In
      addition, connection between proposed framework and more practical
      applications is unclear. The paper presentation should be further
      improved. Some additional comments: 1. The equation (3) is
      interesting point in this paper. However, it is not an easy optimization
      problem. It seems that w_G is already non-overlapping. Why do the variable
      need to be duplicated as in [5]? Another issue is how to fix the group
      size of each group G, for the cases with overlapping and non-overlapping
      respectively? The readers need know more details on how to exactly solve
      this problem efficiently. Without it, it is hard to evaluate its
      advantage. 2. The reviewer has the feeling that Section 3 contains
      a significant amount of existing results, while the new things offered in
      this section seem to be simply extended from those existing works. It is
      suggested to present this section in a more efficient way to be
      differentiated clearly from prior works. 3. The experiments are
      not convincing. a) The synthetic data in Section 5.1 seems to be
      typical overlapping group sparse data, why does the Glasso method perform
      worse? b) In Fig.2(b), when alpha = 0, all coefficients are zeros. Why
      the MSE of Glasso is not zero as the other two algorithms? c) In
      Section 5.2, the proposed SOSlasso is compared with standard solvers lasso
      and Glasso. However, there is no comparison against the state-of-the-art
      methods of fMRI analysis and overlapped group lasso methods. Without such
      experiments, it is hard to validate the effectiveness of the proposed
      method on this application. 4. The connection between the method
      and fMRI application is missing. It is very difficult for readers know how
      to apply the proposed method on fMRI analysis, especially for those
      without any fMRI background. It is suggested to describe the problem in
      fMRI analysis briefly. The author should clarify which data is sparse or
      group sparse, and what is the data matrix $Phi$. 5. In table 1,
      the norm values of some examples are provided. Why not present the
      mechanism of the proposed regularizer, which can automatically select the
      best groups? 6. In p2 line 84, there should be a period at the
      end. 7. In p3 line 110-111, “Uppercase and lowercase” changes to
      “Lowercase and uppercase”. 8. In p3 line 119, “non zero” changes
      to “non-zeros”. 9. In p6 line 317, “$G \in G$” is a typo.
      10. Please define MSE, AWGN etc.


Reservoir Boosting : Between Online and Offline Ensemble Learning
We propose to train an ensemble with the help of a reservoir in which the learning algorithm can store a limited number of samples. This novel approach lies in the area between offline and online ensemble approaches and can be seen either as a restriction of the former or an enhancement of the latter. We identify some basic strategies that can be used to populate this reservoir and present our main contribution, dubbed Greedy Edge Expectation Maximization (GEEM), that maintains the reservoir content in the case of Boosting by viewing the samples through their projections into the weak classifier response space. We propose an efficient algorithmic implementation which makes it tractable in practice, and demonstrate its efficiency experimentally on several compute-vision data-sets, on which it outperforms both online and offline methods in a memory constrained setting.

This paper proposes a new boosting method that
      represents a tradeoff between online and offline learning. The main idea
      of the method is to maintain a reservoir of training examples (of fixed
      size) from which to train the weak learners. At each boosting iteration,
      new examples are added to the reservoir and then a selection strategy is
      used to reduce the reservoir to its original fixed size before the weak
      learner is trained. Several naive selection strategies are proposed but
      the main contribution of the paper is a more sophisticated selection
      strategy whose goal is to remove examples from the reservoir so that a
      weak learner trained on the reduced set will minimize the error computed
      on the whole set before reduction. The resulting algorithm is applied on
      four computer vision datasets, where it is shown to outperform several
      other online boosting methods. The idea of using a reservoir is
      original and very interesting. The derivation of the GEEM algorithm from
      this initial idea is technically sound and the authors have been very
      careful in designing an efficient solution. Experiments are done on only
      four datasets but these datasets represent real computer vision problems
      of practical interest. GEEM is compared with many different methods,
      including some well-chosen baselines for the reservoir framework. The
      authors seem to have taken great care to put other methods in the best
      conditions for the comparison and despite this, the performance of GEEM is
      clearly superior to all other methods. While I think that the
      reservoir idea is very interesting per se, it's not clear which particular
      learning setting or constraints the authors want to address with this idea
      in the paper. Do they want to reduce computing times, to reduce memory
      usage, to obtain the best possible predictive accuracy, or a combination
      of those? This lack of a clear definition of the problem makes the
      assessment of the significance of the method difficult. It clearly works
      better than other online methods in terms of accuracy, but computing times
      and memory usage seem worse. In terms of predictive performance, GEEM is
      also probably worse than offline boosting (but this baseline is actually
      not provided). I think that the paper should more clearly define the
      specific constraints the method is targeting and then make the experiments
      better reflect such constraints. At the moment, only test accuracies are
      compared in the experiments and the authors have only put a constraint on
      the number T of weak classifiers. In such conditions, offline boosting is
      probably the best method. Some design choices also are not totally
      convincing. In Section 3.1, the idea of selecting the subset B so that it
      is as representative as possible of the entire set A does not seem so
      natural to me. Ideally, the subset B should be selected so that the weak
      learner that will be obtained from it will improve as much as possible the
      current model. Making it as close as possible to the weak learner trained
      from A does not make sense when A is not a good subset to train a weak
      learner. So, a more natural selection criterion from B should not
      necessarily depend on A. The choice of a perfect model h* in 3.3 is
      justified but why not instead simply look for the optimal weak learner
      from A and then project it on all Bs when doing the greedy search. The
      computational cost of this operation would probably be negligible with
      respect to the cost of the computation of the covariance matrix.
      The paper is well written and mostly clear but it would be very
      difficult however to re-implement the algorithm given its complexity and
      also to reproduce the experiments as the settings of all tested algorithms
      are mostly unknown (see some questions below). The authors will however
      share their implementation upon publication. Other comments -
      I find it difficult to see how logitboost could be combined with the
      algorithm in Table 2, since logitboost as defined in (Friedman, Hastie,
      and Tibshirani, 1998) uses weak learners in the form of regressors and
      updates both the weights and the responses of these examples. Giving a
      description of the whole algorithm could help. - It's not clear how
      comparable are the different algorithms in the experiments. In particular,
      are all algorithms actually seeing the exact same number of examples, ie.
      is the number of calls to the filter function the same in each case? -
      In the algorithm of Table 2, why is h^t built from R_t and not from
      R_{t-1} U Q_{t-1}? Since R_t is selected so that the weak classifier built
      from it will look as much as possible like the weak classifier built from
      R_{t-1} U Q_{t-1} and since the computational cost for deriving R_t is
      higher than the computational cost for selecting h^t from R_{t-1} U
      Q_{t-1}, I don't see any reason not to train h^t from R_{t-1} U Q_{t-1}.
      - The authors show that the FIX method needs 10 times more examples at
      each iteration than GEEM to reach similar accuracy. However, FIX is forced
      to always use the same examples at each iteration while GEEM samples 100
      or 250 new examples at each iteration, meaning that at the end, GEEM has
      seen 25,000 or 62,500 fresh examples in total. So, I'm not sure the
      comparison is really fair. I would be interested to see the results of the
      application of offline boosting on the same number of examples as seen by
      GEEM, to see what we loose by working with a fixed reservoir. - Also,
      why is FIX using adaboost while GEEM uses logitboost? - The size r
      seems to be a sensible parameter that should not be chosen too small (as
      the weak learner is trained on r examples). The algorithm can thus not be
      turned into a real online method. I would like to see a study of the
      impact of the value of this parameter, in particular to see how large r
      has to be to reach a reasonable accuracy.


On model selection consistency of M-estimators with geometrically decomposable penalties
Penalized M-estimators are used in diverse areas of science and engineering to fit high-dimensional models with some low-dimensional structure. Often, the penalties are geometrically decomposable, i.e. can be expressed as a sum of support functions over convex sets. We generalize the notion of irrepresentable to geometrically decomposable penalties and develop a general framework for establishing consistency and model selection consistency of M-estimators with such penalties. We then use this framework to derive results for some special cases of interest in bioinformatics and statistical learning.

This paper proves sparsistency for a class of
      M-estimators with what is called ``geometrically decomposable penalty''.
      Geometric decomposability is I believe a novel concept introduced
      by the authors. It is similar to decomposability in the Negabhan et al.
      sense but specially suited for sparsistency analysis, i.e. primal dual
      witness. It seems that decomposable norms are also geometrically
      decomposable, though the authors did not elaborate on the connections
      between these two concepts. The paper is well written. The
      concepts are clearly explained except for a few minor points. I am not
      clear what is the $\Phi$-norm that suddenly appeared in line 223 and
      Theorem 3.4. Also, it would be nice to remind the reader that $m$ is the
      restricted strong convexity constant in the theorems. The results
      look reasonable. I have not checked the proof in detail. I think, except
      for the introduction of the geometric decomposability concept, the proof
      follows the Wainwright 2009 Sharp Thresholds paper and is formulaic; the
      authors should correct me if I am wrong here. The authors also use the
      sparsistency results to prove consistency, the process of which I think
      also follows standard proof techniques. The authors could
      elaborate more on the results in section 4. For instance, is there any
      aspect of the theorems that is especially significant or interesting?
      Also, when one uses equation (3.3) to derive the irrepresentability
      condition of the generalized lasso, how does that condition compare with
      the condition of vanilla lasso? Does the condition imply any constraints
      on the matrix D? Is there a way to intuitively interpret the
      irrepresentability condition for the exponential family results? The
      condition there seems to involve only the log-partition function
      $A(\theta)$. The paper also studies sparse M-estimation with
      homogenous linear equality constraints. The picture there seems a bit
      murky however; linear equality constraints should improve the rate of
      convergence since it provides additional information but the theorems
      don't explicitly show such improvements. The experimental section
      is OK. The lines in Figure 1.2 do not seem to line up that well to me.
      Also, it'd be nice to have some experimental results on generalized lasso.
      This is a fairly good paper. The sparsistency analysis does not
      seem very difficult or novel; the results, though not too surprising, are
      solid; and the notion of geometric decomposability introduced in the paper
      should be useful. Some miscellaneous notes: * type on equation
      (2.2), one of the ``A'' should be ``I'' * I am not sure why the
      authors mentioned isotonic regression. Isotonic regression has linear
      inequality constraints yes but it is a nonparametric method. There are
      $O(n)$ parameters in isotonic regression so restricted strong convexity
      will not hold I think; do the authors assert that their framework applies
      to sparse isotonic regression as well?


Probabilistic Principal Geodesic Analysis
Principal geodesic analysis (PGA) is a generalization of principal component analysis (PCA) for dimensionality reduction of data on a Riemannian manifold. Currently PGA is defined as a geometric fit to the data, rather than as a probabilistic model. Inspired by probabilistic PCA, we present a latent variable model for PGA that provides a probabilistic framework for factor analysis on manifolds. To compute maximum likelihood estimates of the parameters in our model, we develop a Monte Carlo Expectation Maximization algorithm, where the expectation is approximated by Hamiltonian Monte Carlo sampling of the latent variables. We demonstrate the ability of our method to recover the ground truth parameters in simulated sphere data, as well as its effectiveness in analyzing shape variability of a corpus callosum data set from human brain images.

The paper "Probabilistic Principal Geodesic Analysis"
      introduces and discusses a probabilistic formulation of Principal
      Geodesic Analysis. Principal Geodesic Analysis has been introduced
      before as a generalization of standard PCA to non-linear manifolds,
      i.e., to spaces with (potentially) other intrinsic geometries than
      Euclidean spaces. The authors explicitly point out and discuss earlier
      PGA results. It is argued that the contribution of the paper is a first
      fully probabilistic formulation of PGA and accompanying solutions: In
      analogy to probabilistic PCA, probabilistic PGA (PPGA) formulates the
      problem as a probabilistic generative model. In this way, the solution
      can be interpreted as maximum likelihood optimization. The authors
      provide an approximate gradient optimization to find the optimal
      parameters of the PPGA generative model. As a proof of concept, the
      numerical results for (A) artificial sphere data and (B) for shape
      estimation with corpus callosum data are provided. Evaluation:
      The paper contributes to a very important research direction:
      statistics for data within non-Euclidean spaces. A generative
      formulation of PPCA within a non-Euclidean space is certainly
      interesting and potentially of high relevance. I also agree with the
      authors that formulations of latent variable models for non-Euclidean
      spaces is an interesting future research subject. The
      generative formulation of PPGA also makes sense and builds up on previous
      work on PGA. The problems I have with the paper are (A) with the
      formulation of the inference and learning scheme, and (B) with the
      experiments. Formulation of inference and learning: The paper
      first makes the impression of providing solutions for general PPGA.
      While this may be the case for the probabilistic formulation of PPGA
      generative model, the main contribution is rather the inference and
      learning procedure. The latter is, however, only really provided for
      symmetric spaces. This should be very clearly spelled out, in the
      abstract and maybe even in the title. The inference and learning
      procedure, and the experiments, do rely on symmetric spaces. In this
      respect, please check formula (4). The text says the rhs is
      proportional to the posterior probability but then the normalizer of
      equation (2) seems to be missing. The normalizer would be relevant for
      derivatives w.r.t. \mu and \tau it seems. The independence of \mu
      already seems to be used for the gradient w.r.t. \mu but is only true
      for symmetric spaces. The paper at least gives the impression that at
      this point symmetric spaces are not assumed. For the \tau derivation,
      symmetric spaces are explicitly assumed. A lot of more general
      considerations like the locality of log map etc could presumably be
      simplified if the paper focussed on symmetric spaces right from the
      start. Experiments: The experiments should be clearly labelled
      as proof of concept. They basically say that the algorithm seems to
      work in principle. There is no comparison with other methods. It would
      have been interesting to compare with non-probabilistic PGA. As the
      methods are essentially equivalent for finding the principle geodesic
      directions, results should be similar. PPGA can potentially provide
      more information (other parameters, maybe a likelihood estimate) but
      non of this is shown or discussed. Also potential local optima are not
      discussed and other issues about the stability and complexity of the
      algorithm. More generally: Work out and discuss in what
      situations the approach really has a (potential) advantage over least
      square formulations of PGA, or over projection methods such as azimuthal
      equidistant projections. When does the generative formulation really
      pay of? I think it can. Other points: I would cite
      Roweis for PPGA alongside the other citations Optimization of the
      PPCA likelihood is convex, can anything be said about PPGA
      optimization (maybe for specific manifold properties). "requires
      we compute", "evaulation", "coule", "the as points" After
      author response: The author response has clarified major points. I
      am also confident that the promised changes and discussions will
      improve the manuscript. Especially an earlier mentioning of the
      constraint (of the algorithm) to symmetric spaces will make this a clearer
      paper. I suggested an explicit mentioning in the abstract because this
      is an important point for the paper. The authors promise to point out
      the constraint in the introduction, I still think it should be pointed
      out in the abstract; why not? Symmetric spaces are still large and
      interesting and potential later confusions can be removed. Eqn. 4 was
      not complete and will be corrected - which will avoid related
      confusions. In summary, yes, interesting paper, accept.