Paper:
Collaborative Deep Learning in Fixed Topology Networks
There is significant recent interest to parallelize deep learning algorithms in order to handle the enormous growth in data and model sizes. While most advances focus on model parallelization and engaging multiple computing agents via using a central parameter server, aspect of data parallelization along with decentralized computation has not been explored sufficiently. In this context, this paper presents a new consensus-based distributed SGD (CDSGD) (and its momentum variant, CDMSGD) algorithm for collaborative deep learning over fixed topology networks that enables data parallelization as well as decentralized computation. Such a framework can be extremely useful for learning agents with access to only local/private data in a communication constrained environment. We analyze the convergence properties of the proposed algorithm with strongly convex and nonconvex objective functions with fixed and diminishing step sizes using concepts of Lyapunov function construction. We demonstrate the efficacy of our algorithms in comparison with the baseline centralized SGD and the recently proposed federated averaging algorithm (that also enables data parallelism) based on benchmark datasets such as MNIST, CIFAR-10 and CIFAR-100.

Review:
This paper explores a fixed peer-to-peer communication topology without parameter server. To demonstrate convergence, it shows that the Lyaounov functions that is minimized includes a regularizer term that incorporates the topology of the network. This leads to convergence rate bounds in the convex setting and convergence guarantees in the non-convex setting.
This is original work of high technical quality, well positioned with a clear introduction. It is very rare to see proper convergence bounds in such a complex parallelization setting, the key to the proof is really neat (I did not check all the details). As we know, insight from convex-case proofs usually apply to the non-convex case, especially as the end convergence should be approximately convex. Thanks to the proof, one can predict the impact of the topology over convergence (through the eigenvalues). This proof should also open to further work. Experiments are also well designed, thought they do not cover the hypothetical situation where a parameter server could fail.
However, several critical limitations will impact the short term significance. Some of them could probably be addressed by the time of publication.
First, the loss in both training speed and convergence accuracy of CDSGD compared to plain SGD is very large, and is only mitigated with the use of a momentum. While the authors show how the algorithm can be modified to support both Polyak and Nesterov momentums, I could not find anywhere which momentum CDMSGD actually use! There is no explanation of why this momentum helps performance so much.
Second, compared to other data-distributed SGD approaches such as Federated Averaging, the main practical contribution of the paper is to get rid of the parameter server, at some efficiency cost on controlled experiments (figure 2.b). So this work significance would greatly improve if the authors clearly identified the reasons for not using a parameter server. In the introduction, they hint at better privacy preserving (if I understood right), but Federated Averaging also claims privacy preserving WITH a parameter server. In the experiments, final accuracy with momentum is better than Federated averaging: is this because of the decentralized approach or the use of a momentum? Note also that this paper does not seem to mention the main reason I have seen for using decentralized approach, which is robustness in case of failure of the central server.
Last, the stochastic Lyapunov gradient (eq 7) shows that the regularization can be very strong especially with small learning rates, and the effects are visible in the experiments. What this regularization amounts to would be worth studying, as this is a striking side-effect of the algorithm.
Potential issues:
Line 92: f_j(x)=1/N f(x) does not make sense to me. 
It should be  f_j(x)=1/N sum_i f^i_j(x) ??
Line 119: it implies that all eigenvalues are non-zero, which contradicts the fully connected topology where they are all zero but one.
Line 143: "Gradient of V(x) also has a Lipshitz": you either remove gradient or replace 'has' with 'is'.