In representational lifelong learning an agent aims to learn to solve novel tasks while updating its representation in light of previous tasks. Under the assumption that future tasks are 'related' to previous tasks, representations should be learned in such a way that they capture the common structure across learned tasks, while allowing the learner sufficient flexibility to adapt to novel aspects of a new task. We develop a framework for lifelong learning in deep neural networks that is based on generalization bounds, developed within the PAC-Bayes framework. Learning takes place through the construction of a distribution over networks based on the tasks seen so far, and its utilization for learning a new task. Thus, prior knowledge is incorporated through setting a history-dependent prior for novel tasks. We develop a gradient-based algorithm implementing these ideas, based on minimizing an objective function motivated by generalization bounds, and demonstrate its effectiveness through numerical examples.
The paper considers multi-task setting of machine learning. The first contribution of the paper is a novel PAC-Bayesian risk bound. This risk bound serves as an objective function for multi-task machine learning. A second contribution is an algorithm, called LAP, for minimizing a simplified version of this objective function. LAP algorithm uses several training tasks to learn a prior distribution P over hypothesis space. This prior distribution P is then used to find a posterior distribution Q that minimizes the same objective function over the test task. The third contribution is an empirical evaluation of LAP over toy dataset of two clusters and over MNIST.
While the paper has the title of "life-long learning", the authors admit that all experiments are in multi-task setting, where
the training is done over all tasks simultaneously. The novel risk bound and LAP algorithm can definitely be applied to life-long setting, where training tasks are available sequentially. But since there is no empirical evaluation in this setting, I suggest to adjust the title of the paper. 
 
The novel risk bound of the paper is an extension of the bound from [Pentina & Lampert, ICML 2014]. The extension seems to be quite significant. Unlike the bound of [Pentina & Lampert, ICML 2014], the new bound allows to re-use many different PAC-Bayesian complexity terms that were published previously. 
I liked risk bound and optimization sections of the paper. But I was less convinced by the empirical experiments. Since 
the paper improves the risk bound of [Pentina & Lampert, ICML 2014], I expected to see an empirical comparison of LAP and optimization  algorithm from the latter paper. To make such comparison fair, both optimization algorithms should use the same base algorithm, e.g. ridge regression, as in [Pentina & Lampert, ICML 2014]. Also I suggest to use the datasets from the latter paper.  
The experiment with multi-task learning over MNIST dataset looks interesting, but it is still a toy experiment. This experiment  will be more convincing with more sophisticated datasets (CIFAR-10, ImageNet) and architectures (e.g. Inception-V4, ResNet). 
Minor remarks:
Section 6, line 4: "Combing" -> "Combining"
Page 14, first equation: There should be "=" before the second expectation.