Paper:
Sequentially learning of tasks arriving in a continuous stream is a complex problem and becomes more challenging when the model has a fixed capacity. Lifelong learning aims at learning new tasks without forgetting previously learnt ones as well as freeing up capacity for learning future tasks. We argue that identifying the most influential parameters in a representation learned for one task plays a critical role to decide on what to remember for continual learning. Motivated by the statistically-grounded uncertainty defined in Bayesian neural networks, we propose to formulate a Bayesian lifelong learning framework, BLLL, that addresses two lifelong learning directions: 1) completely eliminating catastrophic forgetting using weight pruning, where a hard selection mask freezes the most certain parameters (BLLL-PRN) and 2) reducing catastrophic forgetting by adaptively regularizing the learning rates using the parameter uncertainty (BLLL-REG). While BLLL-PRN is by definition a zero-forgetting guaranteed method, BLLL-REG, despite exhibiting some small forgetting, is a task-agnostic lifelong learner, which does not require to know when a new task arrives. This feature makes BLLL-REG a more convenient candidate for applications such as robotics or on-line learning in which such information is not available. We evaluate our Bayesian learning approaches extensively on diverse object classification datasets in short and long sequences of tasks and perform superior or marginally better than the existing approaches.

Review:
The paper addresses the problem of lifelong learning of neural networks - a setting where learning is performed on a continuously arriving new tasks without having access to previously encountered data.
Authors propose a method that prevents catastrophic forgetting typical for naive application of stochastic gradient descent by preventing supposedly important weights to change (in either soft of hard manner), where the weight importance is assessed by its signal to noise ratio estimated from the corresponding (approximate) posterior distribution.
Authors evaluate their approach on a set of image classification datasets and find it superior to the PackNet baseline as well as few simpler ones.
The idea of using uncertainty estimates obtained from Bayesian training to adjust weight updates is natural and potentially very promising. 
However, to me this paper does not seem to investigate the idea sufficiently deep.
The weight pruning or hard masking variant of the method depends on a very important hyperparameter p (size of the mask) which is unclear how to set beforehand. 
I also struggle with understanding the weight regularisation or soft masking variant. 
Authors seem to get their inspiration in the idea of assumed density filtering, where the posterior for 1:T-1 is approximated and used a prior for task T (last sentence on page 5).
At the same time, in Algorithm 2, line 6 the prior is defined as the standard BBB mixture prior and not the approximate posterior from the previous task.
Quite oddly, parameters of the _approximate posterior_ are being quadratically regularalized to not deviate from parameters of the _approximate posterior_ from the previous task. 
This deviates from the original idea and requires additional justification.
Besides that, I find the way this regularisation is applied potentially problematic for the variance parameter (last term in eq. 6).
Here authors apply the regularisation to the parameter of the softplus transformation they use, but scale it with the inverse std deviation which is the “classical” parametrisation. The choice of parametrisation was not discussed, however, clearly different parametrizations may lead to very different results. 
On the experimental side, I have two major issues:
1. The datasets considered are very small, authors could consider using ImageNet, especially given that they already work with 224x224 images.
2. The only prior work used as a baseline is PackNet, while there is no reason why other established methods such as EWC are not applicable.
Minor comments:
The middle expression in eq. 5 seems to miss the -log p(D_T | D_{1:T-1}) term which does not change the latter expression (since it does not depend on parameters theta).
Page 3: “citestochastic methods”, a citation seems to be missing.