Venue: NeurIPS 2021 Conference

Reviews:

final_score: rating: 6: Marginally above the acceptance threshold
review_content: summary: This paper presents a new scalable classifier, named Rule-based Representation Learner (RRL), that can automatically learn interpretable rules for data representation and classification.  A new gradient-based discrete model training method, i.e., Gradient Grafting, is proposed as well that directly optimizes the discrete model. RRL's performance on structured data seems promising wrt experiments performed.



main_review: Novelty:
1. Although influenced by Wang et al [2020], the proposed gradient grafting technique based RRL is novel as it can handle vanishing gradient and large no of features.

Strengths:
1. Code available at https://github.com/anonymous-rrl/rrl
2. The layers are well explained.
3. The proposed gradient grafting technique seems to work quite well in the specified experiments.
4. Case study proves application in practical scenario.

Weakness:
1. Explanation on 'interpretability' part is missing while accuracy is established. Using rules increases expectation interpretation.
2. Experiments are sound, however the worst accuracy obtained should have been reported as well across datasets along with corner cases of failures.
3. Limitations of the paper (unstructured data) are only mentioned as one liner in Section 5. This needs expansion as a subsection including implementation level and formulation level limitations and assumptions.

Relevance to NeurIPS:
This paper is relevant to Neural Information Processing System conference, as we are moving slowly towards Neuro-symbolic AI papers.

Correctness:
Claimed formulations and results seem to be correct.

Clarity:
Paper is well written.

Related work Comparison:
Comparison is fine wrt state-of-the-art.

Effort: The amount of work done in writing paper, coding, experimenting, supplementary material is appreciable.

Suggestions:
1. It is better to do a Train-Eval-Test split to avoid biased results following guidelines in Andrew Ng's "Machine Learning Yearning" book.
2. Try to bring in the mathematical guarantees, upper-lower bound of the testing stage.
3. Apart from Figure 1 inner blocks, a pipeline of the whole process would have made the picture clear in a single figure.

Verdict: This paper falls in the above acceptance threshold area and there are still some concerns in terms of analysis of results in terms of interpretability. Differentiation with related work was expected to be more rigorous. However, this is a good work. Papers on rule based learning are rare wrt the general bulk of usual deep learning papers - so it is recommended to accept the paper to maintain flavour of NeurIPS, provided other reviewers agree.
limitations_and_societal_impact: There is no issue wrt negative Societal Impact, hence skipped.
review_title: null

final_score: rating: 7: Good paper, accept
review_content: summary: The paper introduces a gradient-descent and network-based training method to learn a rule-based interpretable classifier by training concurrently a continuous model and a discretized version.
main_review: A discrete model and a continuous one are trained concurrently, but the way the gradient information by the continuous model used to affect (support to train) the discrete model is not presented clearly.

Using k random upper bounds and k random lower bounds for each continuous attribute in binarization step maybe is not a good practice, since the bounds are selected randomly and continuous attributes can be different in ranges and number of distinct values. The hyper-parameter k is added to the set of inherent hyper-parameters of the network-based training methods that makes the algorithm more difficult to apply.

The RRL algorithm is stated to be scalable; however how its efficiency (running time) and consumed memory are that is not reported. The following features can make the algorithm less scalable in term of running time.
- To support processing directly continuous attributes, the way of binarization used by the RRL increases very much the dimension of input vectors (at Binarization layer).
- To support both CNF and DNF, RRL algorithm maintains concurrently Conjuction part and Disjunction part in each Logical layer that also makes Logical layer bigger.
- A discrete model and a continuous one are trained concurrently. 

How is the behavior of training loss of Gradient Grafting on the other 11 datasets, it could be nice if its training loss for the other 11 dataset is supplemented in Figure 3.

A model with high classification performance in F1-score does not imply it will be high in Accuracy metric. Choosing Accuracy or F1-score metric to evaluate classification performance of a model depends on that (TP, TN) or (FP, FN) is more important in a particular application rather than on whether the class distribution is balance or not. It could be more interesting to report the performance in accuracy metric to see how the ranking will be.

limitations_and_societal_impact: The authors have not addressed this question, nor would I see how they could have done that.
review_title: null

final_score: rating: 6: Marginally above the acceptance threshold
review_content: summary: This submission proposes a new rule-based learner for classification. The learner uses discrete logical learners to obtain the rules. In order to obtain differentiability, the authors also propose a continuous approximation to the discrete layers. After a discussion about avoiding vanishing gradients problem, the authors conduct numerical experiments on a set of datasets. 
main_review: The authors propose a new network learner using discrete and continuous layers. Although their main objective is to come up with an interpretable machine learning approach, they also obtain a method that performs remarkably well according to the numerical experiments. I believe the strength of the proposed method is not its interpretability but its accuracy as it beats (or performs quite close to) several sophisticated algorithms including SVM, XGBoost and Random Forest.  

When it comes to the interpretability-accuracy trade-off, I have several questions and comments:

1) The number of obtained rules with the proposed approach is still quite large. Do you obtain less number of rules than the ones obtained with the interpretable models like CRS and CORELS?

2) Do the authors use the resulting set of rules (with the weights for classification) after training, or do they use the trained model with the continuous weights for classification? I guess this is related to whether they use the discrete or the continuous weights in the logical layers during testing.

3) How does the random bound selection in the binarization layer affect the interpretation? Given different bounds for two trained models on the same dataset, could the interpretations be completely different?

4) In Figure 2, the performance of CART is not good for bank-marketing dataset. But its performance looks quite good in Table 1. What is the difference?

5) The authors report the weights with absolute values. But clearly some of the weights could be negative. How can one interpret negative weights? 

6) In Section 4.5, the authors state that "However, in practice, we can ignore these weight rules and only focus on rules with large weights first." However, in this case the accuracy of the resulting set of rules does not necessarily match the accuracy obtained with all the rules.  Therefore, this statement is not exactly true.

7) The submission claims to  propose a scalable rule-based learner but the related work considered in rule-based methods do not involve some recent work after 2017. For instance 
Wei et al., (2019), Generalized linear rule models.
Lin et al., (2020) Generalized and scalable optimal sparse decision trees

Overall, the numerical experiments show that the authors have come up with a quite accurate algorithm. However, as I listed above, the arguments supporting the interpretability of the algorithm are not convincing.
limitations_and_societal_impact: Yes they have.
review_title: null

final_score: rating: 6: Marginally above the acceptance threshold
review_content: summary: The authors propose a new classifier - called Rule-based Representation Learner - that learns interpretable non-fuzzy rules. The model consists of 3 main modules: 
- the binarization layer
- the logical layer(s)
- the linear layer. 
Further, the authors propose a new training algorithm called Gradient Grafting.
The model was tested tested on 13 datasets and compared with six interpretable models and five complex models.
main_review: **Novelty**: novel. The authors propose a new method that automatically learns non-fuzzy rules for data representation and classification. 

**Clarity**: clear. The paper is easy to follow. 

**Quality**: the paper seems sound, however I have the following concerns: 

- the authors propose Gradient Grafting, and their experiments show that it converges. However, the authors do not provide any intuition on why this new training method should work better than the others.

- The authors, while implicitly using them, never mention the t-norms. According to [1], a fuzzy logic can be defined upon a certain t-norm (triangular norm) representing an extension of the Boolean conjunction. A t-norm is a function $T : [0,1]^2 \to [0,1]$ that, for every $a,b,c \in [0,1]$, satisfies the following properties: 
$$
T(a,b) = T(b,a),  \qquad
T(a,T(b,c)) = T(T(a,b),c), 
$$
$$
a \le b \to T (a, c) \le T (b, c)  \qquad
T(a,1) = a, \qquad
T(a,0) = 0.
$$
So, if we consider the product t-norm we can represent:
$$
x \wedge y, \qquad x \cdot y, 
$$
$$
\neg x, \qquad 1-x. 
$$
$$
x \vee y = \neg (\neg x \wedge \neg y), \qquad x+y -xy = 1 - (1-x)(1-y)
$$
Which are exactly the functions used in Eq. (2) in the paper. The authors should add a discussion on the relationship between Eq. (2) and t-norms in order to justify why they model conjunction and disjunction in the proposed way. Further, they should discuss whether the improved logical activation functions retain the properties of the t-norms. 

- Why do the authors need to add the small constant $\epsilon$ in Eq. 5?

- the authors mention the following "trick": '' One trick to deal with it is replacing $\frac{\partial \mathbb{P}(v)}{\partial v}$ with $\frac{P(P^2(v))}{v}$ for $P$ can lower the speed of approaching 0 while keeping the value range and tendency.'' Has this been tested? 

- the authors compare the models by average rank, however, as proposed by Demsar himself in [2], when comparing multiple models models over multiple datasets the Friedman test + post-hoc test should be used to verify the statistical significance of the results. 

- Parameter settings not specified. It would be highly beneficial to have a table stating exactly the number of layers and the number of neurons per layer for each dataset (and hence the final number of rules obtained). The interpretability of the model itself highly depends on this. Further, it would also help the reader to include in Table 4 in the appendix not only the average length of the rules, but also the minimum and maximum length of the rules.  

- I am not sure I understand the graphs in Fig. 2. Regarding RRL: the authors plot how the F1-score varies according to the number of edges. Regarding the other models: what does the $x$ axis represent for the other models? For example, consider the bank-marketing dataset why is CART's F1-score plotted on left of the graph, while XGBoost F1-score gets plotted on the right?

[1] Petr Hajek. Metamathematics of fuzzy logic, volume 4. Springer Science & Business Media, 2013.

[2] Demsar, J. (2006). Statistical comparisons of classifiers over multiple data sets. Journal of Machine Learning Research, 7, 1–30.

limitations_and_societal_impact: Yes
review_title: null

Meta-review: This paper presents a new scalable classifier, named Rule-based Representation Learner (RRL), that can automatically learn interpretable rules for data representation and classification. A new gradient-based discrete model training method, i.e., Gradient Grafting, is proposed as well that directly optimizes the discrete model. RRL's performance on structured data seems promising wrt experiments performed.

Overall, the reviewers agree that there is merit in the proposed approach and they all support the acceptance of the manuscript. I also looked at aspects of the manuscript and found the experimental evaluation very well organized and thorough. Moreover, I particularly like the results properly showing the trade-offs of complexity and performance and the comparison with other methods according to these two measures. This can convincingly, and in a quantitative manner show, that the proposed method can produce short explanations (i.e., small number of rules/edges) while still performing relatively well. Moreover, when allowing complex explanations, the model performs provides state-of-the-art results when compared to other approaches that produce less interpretable predictors. I believe that approaches like the proposed ones will be important to get AI accepted with application fields, such as medicine or finance.

In summary, there are strong arguments supporting the acceptance of the manuscript and I recommend accepting the manuscript.
Decision: Accept (Poster)