Paper:
Incremental Variational Sparse Gaussian Process Regression
Recent work on scaling up Gaussian process regression (GPR) to large datasets has primarily focused on sparse GPR, which leverages a small set of basis functions to approximate the full Gaussian process during inference. However, the majority of these approaches are batch methods that operate on the entire training dataset at once, precluding the use of datasets that are streaming or too large to fit into memory. Although previous work has considered incrementally solving variational sparse GPR, most algorithms fail to update the basis functions and therefore perform suboptimally. We propose a novel incremental learning algorithm for variational sparse GPR based on stochastic mirror ascent of probability densities in reproducing kernel Hilbert space. This new formulation allows our algorithm to update basis functions online in accordance with the manifold structure of probability densities for fast convergence. We conduct several experiments and show that our proposed approach achieves better empirical performance in terms of prediction error than the recent state-of-the-art incremental solutions to variational sparse GPR.

Review:
The author propose a novel variational approximation for Gaussian Process regression, by parametrizing the approximate posterior with a set of inducing functions, rather than inducing points. This approach makes on-line learning more feasible, because the the inducing points in previously used approximations are usually a subset of the data, which would not be available in an on-line setting. The approach is developed by using the dual representation of a GP in a RKHS. The exposition is mostly understandable, and the results are justifying the effort. 
This is a very interesting, well written paper that tackles a relevant topic for the NIPS audience. Minor suggestions for improvement below:  - line 86, eqn.4: I assume f_{\tilde{X}} and f_X are the function values at the inducing points and the training data? please define. - line 110: "...whereas p(\tilde{X})..." do you mean p(f_{\tilde{X}) ? - line 126: "...of minimizing (8)..." do you mean maximizing? - lines 149-150: "The mean and the covariance functions...can be equivalently represented as a Gaussian distribution." this sentence made no sense to me. How can a distribution represent two functions? - line 156: "...such that \Sigma \geq 0..." should this be \tilde{\Sigma}? - line 172, eqn. (12): can the derivative of this expression, paricularly of the square root of the difference of determinants, lead to numerical instability when the difference btw. the matrices gets small? Please comment. - line 179: "...without referring to the samples.." this point is so important that it would be helpful for the reader if you wrote down the sample-independent form of the posterior here, so one does not have to piece it together from eqns 10-11 and the explanations that around these equations. line 188: "...with error \epsilon_t..." please define what this error is. line 191, eqn 16: the step from the second to the third line is puzzling, and the reader has to work through a whole paragraph before s/he is referred to the appendix where you explain the step.  Please put that reference closer to where you need it. - line 244: "...inducing functions to 512": ho did you come up with this number? what happens if you use less or more? Please also write down the inducing functions.  Proofs: - line 348: This derivation requires a number of auxiliary results which are proven afterwards. This is confusing for the reader. Please prove the auxiliary results first. Also, I fould the step from line 3 to 4, and from line 5 to 6 could do with more explanation. - line 349: please explain how you arrive at the first identity. - line 350: please explain how you arrive at the first identity. - line 354: second line of equations. please explain why the numerator can be rewritten like the denominator.  - line 354: last line of equations. I think there should be no p(f_X|f_{\tilde{X}) in the numerator. - line 367: what is \bar{\mu}?