Paper:
For an explanation of a deep learning model to be effective, it must provide both insight into a model and suggest a corresponding action in order to achieve some objective. Too often, the litany of proposed explainable deep learning methods stop at the first step, providing practitioners with insight into a model, but no way to act on it. In this paper, we propose contextual decomposition explanation penalization (CDEP), a method which enables practitioners to leverage existing explanation methods in order to increase the predictive accuracy of deep learning models. In particular, when shown that a model has incorrectly assigned importance to some features, CDEP enables practitioners to correct these errors by directly regularizing the provided explanations. Using explanations provided by contextual decomposition (CD) (Murdoch et al., 2018), we demonstrate the ability of our method to increase performance on an array of toy and real datasets.

Review:
This paper presents a method intended to allow practitioners to *use* explanations provided by various methods. Concretely, the authors propose contextual decomposition explanation penalization (CDEP), which aims to use explanation methods to allow users to dissuade the model from learning unwanted correlations. 
The proposed method is somewhat similar to prior work by Ross et al., in that the idea is to include an explicit term in the objective that encourages the model to align with prior knowledge. In particular, the authors assume supervision --- effectively labeled features, from what I gather --- provided by users and define an objective that penalizes divergence from this. The object that is penalized is $\Beta(x_i, s)$, which is the importance score for feature s in instance $i$; for this they use a decontextualized representation of the feature (this is the contextual decomposition aspect). Although the authors highlight that any differentiable scoring function could be used, I think the use of this decontextualized variant as is done here is nice because it avoids issues with feature interactions in the hidden space that might result in misleading 'attribution' w.r.t. the original inputs.
The main advantage of this effort compared to work that directly penalizes the gradients (as in Ross et al.) is that the method does not rely on second gradients (gradients of gradients), which is computationally problematic. Overall, this is a nice contribution that offers a new mechanism for exploiting human provided annotations. I do have some specific comments below.
- I am not sure I agree with the premise as stated here. Namely, the authors write "For an explanation of a deep learning model to be effective, it must provide both insight into a model and suggest a corresponding action in order to achieve some objective" -- I would argue that an explanation may be useful in and of itself by highlighting how a model came to a prediction. I am not convinced that it need necessarily lead to, e.g., improving model performance. I think the authors are perhaps arguing that explanations might be used to interactively improve the underlying model, which is an interesting and sensible direction.
- This work, which aims to harness user supervision on explanations to improve model performance, seems closely related to work on "annotator rationales" (Zaidan 2007 being the first work on this), but no mention is made of this. "Do Human Rationales Improve Machine Explanations?" by Strout et al. (2019) also seems relevant as a more recent instance in this line of work. I do not think such approaches are necessarily directly comparable, but some discussion of how this effort is situatied with respect to this line of work would be appreciated.
- The experiment with MNIST colors was neat. 
- The authors compare their approach to Ross and colleagues in Table 1 but see quite poor results for the latter approach. Is this a result of the smaller batch size / learning rate adjustment? It seems that some tuning of this approach is warranted. 
- Figure 3 is nice but not terribly surprising: The image shows that the objective indeed works as expected; but if this were not the case, then it would suggest basically a failure of optimization (i.e., the objective dictates that the image should look like this *by construction*). Still, it's a good sanity check.