Paper:
While domain adaptation has been actively researched in recent years, most theoretical results and algorithms focus on the single-source-single-target adaptation setting. Naive application of such algorithms on multiple source domain adaptation problem may lead to suboptimal solutions. We propose a new generalization bound for domain adaptation when there are multiple source domains with labeled instances and one target domain with unlabeled instances. Compared with existing bounds, the new bound does not require expert knowledge about the target distribution, nor the optimal combination rule for multisource domains. Interestingly, our theory also leads to an efficient learning strategy using adversarial neural networks: we show how to interpret it as learning feature representations that are invariant to the multiple domain shifts while still being discriminative for the learning task. To this end, we propose two models, both of which we call multisource domain adversarial networks (MDANs): the first model optimizes directly our bound, while the second model is a smoothed approximation of the first one, leading to a more data-efficient and task-adaptive model. The optimization tasks of both models are minimax saddle point problems that can be optimized by adversarial training. To demonstrate the effectiveness of MDANs, we conduct extensive experiments showing superior adaptation performance on three real-world datasets: sentiment analysis, digit classification, and vehicle counting.

Review:
Quality:
The paper appears to be correct.
Clarity:
The paper is very clear
Originality:
The theoretical contribution extends the seminal work of Ben-David et al., the idea of using adversarial learning is not new, the novelty is mediaum
Significance:
The theoretical analysis is interested but for me limited, the idea of the algorithm is not new but as far as I know the first explicitly presented for multi-source. 
Pros:
-new theoretical analysis for multisource problem
-paper clear
-smoothed version is interesting
Cons
-Learning bounds with worst case standpoint is probably not the best analysis for multisource learning
-experimental evaluation limited in the sense that similar algorithms in the literature are not compared
-Extension a bit direct from the seminal work of Ben-David et al.
Summary:
This paper presents a multiple source domain adaptation approach based on adversarial learning.
The setting considered contains multiple source domains with labeled instances and one target domain with unlabeled instances. The authors propose learning bounds in this context that extend seminal work of Ben-David and co-authors(2007,2010) where they essentially consider the max source error and the max divergence between target and source with the presence of empirical estimate.
Then, they propose an adversarial algorithm to optimize this bound, with another version optimizing a smoothed version, following the approach of Ganin et al.(2016). 
An experimental evaluation on 3 known tasks is presented.
Comments:
Comments:
-I am not particularly convinced that the proposed theory explains best multi-source learning. In multi-source, you expect that one source may compensate the others when needed for classification of particular instances. The paper considers a kind of worst case by taking the max error over the sources and the max divergence between target and source, but not really representative of what happens for real problems in the sense that you do not take into account how the different sources interact.
The experimental results confirm this aspect actually.
Maybe, the authors could propose a learning bound that correspond to the smoothed version proposed in the paper and that works best.
The Hard version of the algorithm seems here to comply with the bound, while the algorithm that is really interesting is the smoothed version.
-Experimental evaluation is a bit limited, there is no comparison with other (deep learning methods) tackling multi-source scenarios (or equivalent), while I think it is easy to find related approaches :
-E. Tzeng, J. Hoffman, T. Darrell, K. Saenko. Simultaneous Deep Transfer Across Domains and Tasks. ICCV 2015.
-I-H Jhuo, D Liu, D.T. Lee, and S-Fu. Chang. Robust visual domain adaptation with low-rank reconstruction. In IEEE CVPR, 2012.
-Muhammad Ghifary, W. Bastiaan Kleijn, Mengjie Zhang, and David Balduzzi. Domain generalization for object recognition with multi-task autoencoders. In IEEE International Conference on Computer Vision (ICCV), 2015.
-Chuang Gan, Tianbao Yang, and Boqing Gong. Learning attributes equals multi-source domain generalization. In IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2016.
-R. Gopalan,R. Li,and R. Chellappa. Unsupervised Adaptation Across Domain shifts by generating intermediate data representations. PAMI, 36(11), 2014.
Note also this paper at CVPR'17: about Domain adversarial adaptation.
E. Tzeng, J. Hoffman, K. Saenko, T. Darrell.  Adversarial Discriminative Domain Adaptation, CVPR 2017.
-Nothing is said about the complexity of applying the algorithm on the different datasets (convergence, tuning, ...)
For the smoothed version, it could be interesting to see if the weights w_i associated to each source are related to each (original) source error and see how the sources are complementary. 
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After rebuttal
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The new results and experimental evaluation have improved the paper. I increased my score.