All three reviewers have favourable opinion towards this paper. There are some minor questions or comments, but they can be addressed without requiring another round of reviewing. Therefore, I recommend acceptance of this work. I encourage the authors to incorporate the reviewers' comments and concerns as much as possible.

All four reviewers recommended acceptance of this paper. Please incorporate the reviewer comments, particularly the missing references.

The paper concerns a stochastic variational approach towards mutual information maximisation with applications to reinforcement learning. The authors present a treatment of MI following a bound taken form earlier work by Barber&Agakov and using it stochastically by mini-batch descent. In order to estimate the MI they introduce a novelty and use neural networks to predict parameters for factors in state transition models as shown in Equation 6. This replaces the need for an explicit generative model of the data. The authors use this algorithm in the context of empowerment, which is a measure that can be used to connect MI with reinforcement learning.

Correctness: The discussion on baseline's for POMO to me are a bit misleading. This is somewhat of a nit though. First, the use of "traditionally" is incorrect. Earliest work (including the REINFORCE paper if I recall correctly) make use of a rolling average baseline. Newer works do use more complicated baselines, but for a reason!

Three reviewers support accepting the paper, one argues for rejection. From the reviews, rebuttal and discussion, the consensus seemed to be that the paper has an interesting new idea and good empirical results. The debate was around how much novelty there is, and how likely it is for the idea to be useful in the future, which are slightly more subjective concerns. I recommend acceptance, and I hope future work will show that this was a valuable stepping stone. I still recommend that the authors revise the paper according to the reviewer's suggestions, in particular in terms of not making overstated claims and giving the reader broader context.

Correctness: I like the ideas and concepts of'diversity' and'realness' on the sphere (which is projected by simple L2-normalization), but it is non-trivial to say that proposed objective function actually minimizes some'distance' between real and fake probability distribution. SphereGAN implements IPMs as their objective function and shows the equivalence relation between minimizing Wasserstein distance in hyper-sphere and minimizing objective functions, but this kind of analysis is not dealt in proposed method even if SphereGAN is main baseline method. Thus authors needs to clarify what to minimize. The proposed method uses L2-normalization as a projection onto hyper-sphere which induces information loss as it is not one-to-one (All the conventional features lying in same lay started at origin is projected to same point in hyper-sphere). The stereo-graphic projection not only admits single fixed point where north pole ('center' in the paper) can be rotated transitively on the hyper-sphere.

This paper proposes a new GAN training technique based on intuition about the hypersphere. It attains state-of-the-art IS and FID scores on a few datasets. Reviewers were initially confused and concerned about its similarity to SphereGAN but were convinced it should be accepted after the rebuttal.

Additional Feedback: There were many thinks liked about the paper, including the idea of having an ML-advisor algorithm be e-close and alpha-accurate. I also liked the interpretation of a Hedge algorithm with an advisor give on page 5. In some ways the Hedge formalization, though, seems to minimize the use of predictors to just giving a prior. As someone interested in this area I find that a not especially hopeful or compelling message, but perhaps for some problems that's the right methodology. The experiments don't seem that useful.

It is shown how one can integrate predictions, typically coming from a machine learning algorithm, into this framework using a multiplicative weight algorithm. The paper has been positively evaluated by all the reviewers, with a uniform score of 6. The reviewers liked the overall idea of using ML predictions to improve the performance of online algorithms while still keeping the worst case guarantees, as well as incorporation of the multiplicative weights algorithm. On the other hand, the novelty of the paper seems a bit weak from a methodological perspective: the authors apply a well-known Hedge algorithm with ML predictions just incorporated within the prior. Also, contribution comparing to [13] seems somewhat incremental.

Clarity: *** Derivations in Section 3 *** While the theorems across Section 3.1 seem reasonable I would have liked some a more self-contained presentation of theorems together with proofs. Assumption 2 (Bounded adversary power) is a bit strange, and while the experimental implementation (with the norm ball around s) seems reasonable for many environments, this should probably be defined in a better way. The authors refer to the Appendix a lot and in my opinion such derivations are necessary for the reader to follow along. I cannot really follow how the authors get there. Add Plots (similar to Appendix I, Figure 12).