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First provide a summary of the paper, and then address the following criteria: Quality, clarity, originality and significance. The authors propose an accelerated proximal block coordinate descent algorithm, describe its application to standard regularized loss minimization problems, and conclude with experiments on a smoothed SVM. On the question of clarity: I found the paper on the whole difficult to follow, with the authors showing a marked preference for writing equations in lieu of explanations. There are also numerous small grammatical errors. I'm not aware of other algorithms that are designed to work on block-coordinate problems (although single-coordinate algorithms are common enough), and have to question the advantage of this formulation, aside from being slightly more general. Given that the application considered in section 4 is single-coordinate (am I correct about this?), it might simplify the presentation to work from a single-coordinate formulation, and merely mention that block-coordinate updates are also possible.