Paper:
Given an existing trained neural network, it is often desirable to learn new capabilities without hindering performance of those already learned. Existing approaches either learn sub-optimal solutions, require joint training, or incur a substantial increment in the number of parameters for each added task, typically as many as the original network. We propose a method called Deep Adaptation Networks (DAN) that constrains newly learned filters to be linear combinations of existing ones. DANs preserve performance on the original task, require a fraction (typically 13%) of the number of parameters compared to standard fine-tuning procedures and converge in less cycles of training to a comparable or better level of performance. When coupled with standard network quantization techniques, we further reduce the parameter cost to around 3% of the original with negligible or no loss in accuracy. The learned architecture can be controlled to switch between various learned representations, enabling a single network to solve a task from multiple different domains. We conduct extensive experiments showing the effectiveness of our method on a range of image classification tasks and explore different aspects of its behavior.

Review:
This paper proposes to adapt convnet representations to new tasks while avoiding catastrophic forgetting by learning a per-task “controller” specifying weightings of the convolution-al filters throughout the network while keeping the filters themselves fixed.
Pros
The proposed approach is novel and broadly applicable.  By definition it maintains the exact performance on the original task, and enables the network to transfer to new tasks using a controller with a small number of parameters (asymptotically smaller than that of the base network).
The method is tested on a number of datasets (each used as source and target) and shows good transfer learning performance on each one.  A number of different fine-tuning regimes are explored.
The paper is mostly clear and well-written (though with a few typos that should be fixed).
Cons/Questions/Suggestions
The distinction between the convolutional and fully-connected layers (called “classifiers”) in the approach description (sec 3) is somewhat arbitrary -- after all, convolutional layers are a generalization of fully-connected layers. (This is hinted at by the mention of fully convolutional networks.)  The method could just as easily be applied to learn a task-specific rotation of the fully-connected layer weights.  A more systematic set of experiments could compare learning the proposed weightings on the first K layers of the network (for K={0, 1, …, N}) and learning independent weights for the latter N-K layers, but I understand this would be a rather large experimental burden.
When discussing the controller initialization (sec 4.3), it’s stated that the diagonal init works the best, and that this means one only needs to learn the diagonals to get the best results.  Is this implying that the gradients wrt off-diagonal entries of the controller weight matrix are 0 under the diagonal initialization, hence the off-diagonal entries remain zero after learning?  It’s not immediately clear to me whether this is the case -- it could help to clarify this in the text.
If the off-diag gradients are indeed 0 under the diag init, it could also make sense to experiment with an “identity+noise” initialization of the controller matrix, which might give the best of both worlds in terms of flexibility and inductive bias to maintain the original representation. (Equivalently, one could treat the controller-weighted filters as a “residual” term on the original filters F with the controller weights W initialized to noise, with the final filters being F+(W\crossF) rather than just W\crossF.)
The dataset classifier (sec 4.3.4) could be learnt end-to-end by using a softmax output of the dataset classifier as the alpha weighting. It would be interesting to see how this compares with the hard thresholding method used here.  (As an intermediate step, the performance could also be measured with the dataset classifier trained in the same way but used as a soft weighting, rather than the hard version rounding alpha to 0 or 1.)
Overall, the paper is clear and the proposed method is sensible, novel, and evaluated reasonably thoroughly.