Venue: ICLR 2020 Conference

Reviews:

final_score: rating: 1: Reject
review_content: My understanding of this paper is that it proposes a combination of simple logical blocks that can efficiently learn logic rules implemented by a logic function. I think the authors define interpretability as the possibility to exactly express an unknown function in a composition of blocks, but despite several reading of this paper, I am not sure. A strong assumption in this paper seems to be that the target function of a supervised task can be exactly expressed via some compositional blocks. I would suggest a significant revision to clearly explain why WBNs are more interpretable and avoid any vague terminology.

I will list below some of my concerns:

- After reading several times this paper, I do not understand why this method is "more interpretable". The explanation of the papers are quite verbose. I tried to phrase my concern as this: could a standard CNN be more interpretable than this WBN because it simply uses linear operation? I'd like to see the reaction of the authors to this questions

For instance:
"This differs from the normal neural network because the WBN reveals the exact functions with the correct inputs and their ordering to construct the target function, instead of merely approximating them."
If the target function is precisely a cascade of linear operators and ReLU, then the objective of learning would be to recover exactly the linear operators and wouldn't consist in an approximation. Are the authors trying to tackle the nature of the objective functions? It is very unclear to me.

"It is different from other neural networks as it not only approximates a target function, but also constructs and reveals its structure."
I do not understand why the structure is less opaque than in standard CNNs or how it is revealed.

The authors must clearly define the notion of interpretability in the text, in an explicit and simple manner. As an active researcher in this field, I believe that this is quite difficult because everybody has its own interpretation of interpretability. Here, I would suggest to significantly rephrase this.

- I am quite confused in the notation.. For instance, sometimes \hat W is used, sometimes W...

- It is claimed that $\ell^1$ minimisation will allow to avoid... sparse and sharp operators: "This causes Wˆ (l) to be extremely sparse and sharp, and it can be an obstacle for shifting the function blocks from one ordering to another." This goes against my intuitions/knowledge, could the authors point me to a reference?

- Table 1: Do the mean square errors indicate an exact learning? If yes, this should be commented. Also, the Table is not discussed in the text...

- I do not understand why the CIFAR and MNIST experiments are relevant to this paper. Furthermore, the accuracy are very low.
review_title: Official Blind Review #1

final_score: rating: 1: Reject
review_content: This paper presents White Box Network (WBN), which allows for composing function blocks from a given set of functions to construct a target function. The main idea is to introduce a selection layer that only selects one element of the previous layer as an input to a function block. This allows for both introducing function priors as well as interpreting the learned function. The paper also presents a setting where each function block is a neural network that can be learned end-to-end using a PathNet style setting and shows positive transfer across MNIST and CIFAR classification tasks.

This presents an interesting technique to enforce learning composition of function blocks while learning the target function. This is important for both interpretability of the learned function as well as for introducing prior information about useful functions in a given domain. The extension to learnable functions (in the form of neural networks) for learning pathways for different tasks is also promising.

The biggest weakness of the paper is that it does not compare both theoretically as well as empirically with several closely related techniques such as RoutingNetworks[1], Modular Networks[2], Neural RAM[3], Compositional Recursive Learner[4], etc. (please find the references below). Routing Networks allow for selecting among a set of function blocks given some inputs. Module Networks similarly introduce modular layer that determines the appropriate modules given the inputs from the previous layers. Neural RAM learns to compose differentiable functions to learn a target function (similar to learning the LLD programs).

It would be good to describe the differences between the proposed approach in WBN and these approaches, as they all seem to propose a similar solution of learning target functions by learning to compose function blocks and reusing the learnt computations for transfer learning. It would also be important to empirically evaluate the related approaches to better understand the pros/cons of WBN compared to these approaches.

What is the biggest size LLD programs that can be learned by WBNs? It would be interesting to evaluate the scalability of the approach to understand the limits of learning complex target functions.

I was also curious if instead of providing the four pre-defined function block (Identity, not, and, or), what would the behavior be if they were all neural networks and also learnt in an end-to-end fashion somewhat similar to [4].

For the MNIST and CIFAR classification tasks, the function blocks are neural networks themselves. After training them using the PathNet like training, is the learned network more interpretable? It might be interesting to see if the selection layer and the function blocks learned some semantic information that might be easier to distill.

For a better comparison with Neural RAM, it might also be interesting to empirically evaluate the performance of WBNs on algorithm induction tasks such as the one used in [3].

[1] Clemens Rosenbaum, Tim Klinger, Matthew Riemer. Routing Networks: Adaptive Selection of Non-Linear Functions for Multi-Task Learning. ICLR 2018

[2] Louis Kirsch, Julius Kunze, David Barber. Modular Networks: Learning to Decompose Neural Computation. NeurIPS 2018

[3] Karol Kurach, Marcin Andrychowicz, Ilya Sutskever. Neural Random-Access Machines. ICLR 2016

[4] Michael Chang, Abhishek Gupta, Sergey Levine, Thomas Griffiths. Automatically Composing Representation Transformations as a Means for Generalization. ICLR 2019

Minor:

page1: ?. -> ?
page 2: questions and answering --> question answering
page 3: y^(l) represents (m(l)+n(l))-dimensional vectors --> should this be m(l) + 2*n(l)?
review_title: Official Blind Review #2

final_score: rating: 1: Reject
review_content: This paper investigates the question of identifying concise equations from data to understand the functional relations. In particular, a set of base functions are given in hand and the goal is to obtain the right composition of these functions which fits the target function. The main contribution of the paper is to introduce a selection layer, which enhances sparse connections in the network. Several experiments are conducted to show the effectiveness of the method. 

My main concern of the paper is about the novelty and the lack of comparison of existing methods. The framework of finding functional relations is set up in [1,2], the main contribution of the paper is a refine architecture with the introduction of the selection layer. However, this selection layer is nothing but incorporating a softmax function. The idea of combining softmax functions in the hidden layers is not novel neither, which could be found in [3,4]. As a result, I find the contribution of the paper very limited, which could be summarized as applying an existing technique on a specific problem. Moreover, in the experimental section, there is a lack of comparison with existing methods such as EQL[1,2] and I consider it a major omission. 

Overall, due to the novelty concern and the lack of comparison, I do not support publication of the paper.

[1] Sahoo et al.  Learning Equations for Extrapolation and Control
[2] Martius et al. Extrapolation and learning equations
[3] Graves et al.  Neural turing machines
[4] Graves et al.  Hybrid computing using a neural network with dynamic external memory
review_title: Official Blind Review #3

Meta-review: This paper presents White Box Network (WBN), which allows for composing function blocks from a given set of functions to construct a target function. The main idea is to introduce a selection layer that only selects one element of the previous layer as an input to a function block.  The reviewers were unanimous in their opinion that the paper is not suitable for publication at ICLR in its current form.  There were significant concerns about the clarity in writing, and reviewers have provided detailed discussion should the authors wish to improve the paper.
Decision: Reject