Streaming Sparse Gaussian Process Approximations
Sparse pseudo-point approximations for Gaussian process (GP) models provide a suite of methods that support deployment of GPs in the large data regime and enable analytic intractabilities to be sidestepped. However, the field lacks a principled method to handle streaming data in which both the posterior distribution over function values and the hyperparameter estimates are updated in an online fashion. The small number of existing approaches either use suboptimal hand-crafted heuristics for hyperparameter learning, or suffer from catastrophic forgetting or slow updating when new data arrive. This paper develops a new principled framework for deploying Gaussian process probabilistic models in the streaming setting, providing methods for learning hyperparameters and optimising pseudo-input locations. The proposed framework is assessed using synthetic and real-world datasets.
This paper presents a variational and alpha-divergence approach specifically designed for the case of Gaussian processes with streaming inputs. 
The motivation given in the introduction is convincing. The background section is very well presented, sets up the methodology for section 3 and at the same time explains why a batch based approach would be inadequate. Section 3 which is the most important, is however more difficult to follow. It is not a problem to understand how one equation leads to the other, but many important details are left out of the description making it very difficult to understand in a conceptual level. In particular, it'd help to explain the concept behind starting from eq. (5). Further, what is p(a|b) in eq. (6)? It seems to be an important element for our understanding of the method, as it somehow involves the update assumption. I was also wondering whether it'd be possible to arrive at the same bound by starting with the joint distribution and simply applying Jensen's inequality together with (4). That would probably be a clearer presentation.
The extension to alpha-divergences and related conversation is interesting, although I would also expect to see a deeper comparison with [9] in the methodological level. This is important since the proposed method converges to [9] for alpha zero, which is what the authors use in practice. 
The experiments are structured nicely, starting from intuitive demonstration, moving to time-series and spatial data and finishing with classification data. The method does seem to perform well and run fast - it seems to be two time slower than SVI-GP, which still makes it a fast method. However, the experiments do not explore the hypothesis whether the SVI (batch-based) or the streaming based method performs better, simply because they use different approximations (we see that in time zero SVI is already much worse). Since the uncollapsed bound has been used for classification, I'd be keen to see it for the regression tasks too, to be able to compare with SVI-GP. Indeed, the SVI-GP seems to actually reduce the RMSE more than OSGP in Fig. 3.
Another useful property to explore / discuss, is what happens when the order of the data is changed. Further, the comparisons do not consider any other competitive method which is tailored to streaming data, such as [9] or [Gijsberts and Metta: Real-time model learning using incremental sparse spectrum GP Regression, 2013]. Finally, it'd be useful to either repeat the plots in Fig 2,3 with number of data in the x-axis (so all methods are aligned) or to simply put those and state the difference in running times.
Overall seems like a nice paper, which has good motivation and considers an important problem. However, the key points of the method are not demonstrated or explained adequately. I feel that re-working the presentation and experiments would significantly improve this paper. 
EDIT after rebuttal period: The authors have clarified most of my concerns and I have raised my score to reflect that.