Paper:
ACCELERATING NONCONVEX LEARNING VIA REPLICA EXCHANGE LANGEVIN DIFFUSION
Langevin diffusion is a powerful method for nonconvex optimization, which enables the escape from local minima by injecting noise into the gradient. In particular, the temperature parameter controlling the noise level gives rise to a tradeoff between "global exploration" and "local exploitation", which correspond to high and low temperatures. To attain the advantages of both regimes, we propose to use replica exchange, which swaps between two Langevin diffusions with different temperatures. We theoretically analyze the acceleration effect of replica exchange from two perspectives: (i) the convergence in χ 2 -divergence, and (ii) the large deviation principle. Such an acceleration effect allows us to faster approach the global minima. Furthermore, by discretizing the replica exchange Langevin diffusion, we obtain a discrete-time algorithm. For such an algorithm, we quantify its discretization error in theory and demonstrate its acceleration effect in practice.

Review:
PROS:
- The text is very well written, with a good balance between mathematical details and intuitions.
- I really like the high-level description of the algorithms and proof techniques
CONS:
to be completely honest, I am not sure I have learnt anything new from the paper. 
1) the proof techniques are very standard
2) although there must be some small innovations, I thought that all the results had more or less been proven by Dupuis and co-authors:
a. large deviation principles
b. the larger the swapping rate, the better (which motivated Dupuis & al to consider the infinite swapping limit.)
and
c. Bakri & al methodology to prove convergence relying on the carre du champ is by now very standard and the proofs of the paper are only minor adaptations.
I must probably be missing something, and I encourage the authors to clarify what the main novelties are when compared to the several papers by Dupuis & al. 
REMARKS:
1) I do not really understand the emphasis on optimisation while all the proofs are related to the convergence to the stationary distributions.