Summary and Contributions: The paper proposes a Robust Risk Minimization based classifier approach, that is optimizing 0-1 loss (instead of surrogate losses) and considers uncertainty sets that does have the true distribution.
Strengths: The approach is theoretically motivated giving performance bounds and generalization bounds. Performance bounds have been shown to hold well in practice on two datasets. Empirically, the classifier motivated as such is shown to be competitive with the other classifiers used in practice. The contribution is novel in terms of the problem setting and its analysis.
Weaknesses: It is unclear from the paper as to why optimizing for the direct 0-1 loss is important as opposed to optimizing some surrogate loss. While one can say that directly optimizing 0-1 loss should in effect be better, but the empirical results does not seem to indicate this.
Correctness: The claims and the method look correct and empirical methodology seems correct as well.
Clarity: The paper is reasonably well written. However, the theorems are not motivated using any intuition. Particularly, in theorem 1, the conditions to be satisfied by mu_a* etc. seem to have been taken out of thin air, which makes it really difficult to grasp the intuition behind the result and get a feel for why the method would work.
Relation to Prior Work: The paper discusses the prior works in the space of RRMs and explains how it differentiates itself from prior works.
Additional Feedback: There were problems in installing cvxpy package to get the code running. It would be good to include some instructions in installing some of these packages or at least links where one can follow instructions to install these packages. Post Author Response: First of all, I thank the author for the detailed response, particularly to the questions raised by Reviewer 2, as that gave me lot more appreciation of the work than I had before. I am glad that you plan to include some intuition for the proofs, as it goes a long way for people like me to understand what is going on. I will retain the evaluation as before.
Summary and Contributions: The authors present an LP-based algorithm for classification that minimizes the worst-case expected 0-1 loss across a set of distributions, called "uncertainty set." The algorithm is constructed such that the uncertainty set includes the true data distribution with a high probability. By construction, their method provides an (optimized) upper bound and they also show that lower bounds can be constructed by solving a similar LP problem. Finally, they present experimental results that support the method. ============ Post Rebuttal: I have read the authors' feedback and I am generally satisfied. I had three primary concerns and two of them were addressed in the rebuttal. The only downside I still see is that the authors do not explore the subject of choosing the mapping Phi in their paper, which I believe to be important. However, I will keep my score.
Strengths: I find the paper to be novel. The approach the authors use to formulate classification problems as a minimization of an upper bound on the true risk using uncertainty sets is quite different from the conventional methods of minimizing the empirical risk using surrogate loss functions, such as in SVM or logistic regression. In addition, their method is non-parametric and elegant. The paper itself is very well-written and a pleasure to read. Also, the experimental results are promising, particularly Figure 1, which shows that the true expectation is indeed sandwiched between the upper and lower bounds claimed in the paper.
Weaknesses: Perhaps, the biggest weakness is computational efficiency. Current classification methods are formulated as a minimization of empirical loss so they are often solved in practice using variants of the stochastic gradient descent method, which is quite efficient and scales well to massive amounts of data and labels. The proposed minimax risk classifiers (MRC) requires solving a full LP, whose time complexity is O(m^3). Regarding the evaluation in Table 1, it is not clear if the authors evaluated competing methods fairly. There is no discussion of hyper-paramter tuning. For example, if you tune the value of k using a validation dataset, kNN on Haberman achieves 22% error rate (not 30%). At the same time, however, I acknowledge that the choice of the feature map in MRC can also be optimized as well, which the authors did not since they currently use a simple thresholding rule in their evaluation.
Correctness: Aside from the concern above regarding Table 1, the claims seem to be sound. I have not checked the proofs in the appendix but the claims appear to be reasonable (for example, you would expect the log(m) to appear in Theorem 3 by the union bound and expect the term d to appear as a scaling factor, and so on). The method is elegant and Figure 1 supports it quite well (which shows that the true expectation is indeed sandwiched between the two terms claimed in the paper).
Clarity: The paper is very well-written and a pleasure to read.
Relation to Prior Work: The relation to prior work is clearly discussed.
Additional Feedback: 1- It would an important addition if the authors show how the LP in (3) can be solved using variants of SGD. Otherwise, the current method cannot be applied to large amounts of data. In Table 1, all of the classification problems use less than 1,000 training examples. Please correct me if I am wrong about the number of training examples of each dataset. 2- There should be some discussion on what would be a good feature map. For example, I would expect a good set of m feature maps to be diverse (different from each other). Also, we would want the variance of the feature map to be as large as possible, depending on the available sample size, which puts a limit on variance as stated in Theorem 3. A few lines about this can be helpful such as at the Conclusion section. 3- Please mention the criteria used for choosing the six UCI datasets in Table 1.
Summary and Contributions: This paper presents minimax risk classifiers (MRCs) that do not rely on a choice of surrogate loss and family of rules. The goal of MRC is to find a classification rule that minimize the worst-case expected 0-1 loss with respect to a class of possible distributions. It first represents data, probability distributions and classification rules by matrices. The estimated classifier is cast as a linear optimization problem in which the uncertainty set is cast as the linear constraints. Some performance guarantees are proved, and numerical comparisons are conducted. ---- update: I have read the rebuttal. It is claimed that "the methods presented do not create or compute matrices describing probability distributions and classification rules", but the formula at the end of page 3 clearly indicates that p is used to describe the probability mass function for all possible data point x (there \ell is the expected value of the misclassification loss (1-h)). Perhaps p is not needed for the training of the classifier; however, the formula at the bottom of page 3 should clearly be correctly presented as the empirical risk instead of the population risk and the p therein should be the empirical probability distribution. This is at least sloppy writing. Moreover, the feature maps have not been satisfactorily explored. Some illustrative examples and a discussion on how to choose Phi will help a lot. Overall I believe this work has a lot of merits but I believe a thorough revision will make it much better. I have bumped my rating by 1.
Strengths: There is some novelty in the proposed method. Theoretical guarantees are provided.
Weaknesses: 1. The proposed method is inapplicable to data from absolutely continuous probability distribution. The number of possible values of a data point in this case will be infinite. However, the paper relies on the vectorization of the probability distribution. For truly real world continuous data, huge matrices will have to be created and computed. 2. The choice of the uncertainty set is somewhat arbitrary. How to guarantee that the true distribution is covered? 3. There has to be a trade off between the data fitting and the generalization error. This seems to be related to how the uncertainty set is defined. In the extreme case that the uncertainty set is chosen to be infinitely large, then the model will underfit the data. Insight about this trade off is lacking. 4. The linear program in Theorem 3 need to be explained intuitively. I understand that this is a main theorem but it would help the reader a lot if the authors can explain what are the objective and the constraints in (3). 5. How is the feature mapping chosen? How sensitive is the model to the feature mapping? 6. I am not very much impressed by the numerical study. Cross-validation or other careful tuning methods should be used for the other SOTA methods to compare with the current method.
Correctness: Looks alright.
Clarity: Some explanation to the linear program (3) in Theorem will be good.
Relation to Prior Work: Yes.