We introduce a bottleneck method for learning data representations based on channel deficiency, rather than the more traditional information sufficiency. A variational upper bound allows us to implement this method efficiently. The bound itself is bounded above by the variational information bottleneck objective, and the two methods coincide in the regime of single-shot Monte Carlo approximations. The notion of deficiency provides a principled way of approximating complicated channels by relatively simpler ones. The deficiency of one channel w.r.t. another has an operational interpretation in terms of the optimal risk gap of decision problems, capturing classification as a special case. Unsupervised generalizations are possible, such as the deficiency autoencoder, which can also be formulated in a variational form. Experiments demonstrate that the deficiency bottleneck can provide advantages in terms of minimal sufficiency as measured by information bottleneck curves, while retaining a good test performance in classification and reconstruction tasks.
This paper used the concept based on channel deficiency to derive a variational bound similar to variational information bottleneck. Theoretical analysis shows that this bound is an lower bound on the VIB objective. The empirical analysis shows it outperforms VIB in some sense. 
I think this paper's contribution is rather theoretical than practical. The experiments section can be improved in the following aspect:
-  Figure 2 are hard to read for different M's. It would be better if the authors can show the exact accuracy numbers rather than the overlapped lines
- I(Z;Y) vs I(Z;X) graph is typically used in a VIB setting. In the paper's variational deficiency setting, although plotting I(Z;Y) vs I(Z;X) is necessary, it would be also helpful for the authors' to plot Deficiency vs I(Z;X), because this is what new objective is trading-off. 
- Again, Figure 3, it is hard to see the benefits for increasing M from the visualizations for different clusterings. 
- How do the paper estimate I(Z;Y) and I(Z;X) for plotting these figures? Does the paper use lower bound or some estimators? It should be made clear in the paper since these are non-trivial estimations.
Last comment is that, although the concept of `deficiency` in a bottleneck setting is novel, the similar idea for tighter bound of log likelihood has already been pursed in the following paper:
- Yuri Burda, Roger Grosse, Ruslan Salakhutdinov. Importance Weighted Autoencoders. ICLR 2016
It was kind of surprising that the authors did not cite this paper given the results are pretty much the same. It would also be helpful for the authors to do a comparison or connection section with this paper. 
I like the paper in general, but given it still has some space for improvement, I would keep my decision as boarder line for now.