Paper:
Weight-sharing-the simultaneous optimization of multiple neural networks using the same parameters-has emerged as a key component of state-of-the-art neural architecture search. However, its success is poorly understood and often found to be surprising. We argue that, rather than just being an optimization trick, the weight-sharing approach is induced by the relaxation of a structured hypothesis space, and introduces new algorithmic and theoretical challenges as well as applications beyond neural architecture search. Algorithmically, we show how the geometry of ERM for weight-sharing requires greater care when designing gradientbased minimization methods and apply tools from non-convex non-Euclidean optimization to give general-purpose algorithms that adapt to the underlying structure. We further analyze the learning-theoretic behavior of the bilevel optimization solved by practical weight-sharing methods. Next, using kernel configuration and NLP feature selection as case studies, we demonstrate how weight-sharing applies to the architecture search generalization of NAS and effectively optimizes the resulting bilevel objective. Finally, we use our optimization analysis to develop a simple exponentiated gradient method for NAS that aligns with the underlying optimization geometry and matches state-of-the-art approaches on CIFAR-10.

Review:
I have not worked in the optimization filed and I am only gently followed the NAS field. I might under-valued the theoretical contribution.
This work provides  theoretical analysis for the NAS using weight sharing in two aspects: 
1) The authors give non-asymptotic stationary-point convergence guarantees (based on stochastic block mirror descent (SBMD) from Dang and Lan (2015)) for the empirical risk minimization (ERM) objective associated with weight-sharing. Based on this analysis, the authors proposed to use  exponentiated gradient to update architecture parameter, which enjoys faster convergence rate than the original results in Dang and Lan (2015). The author also provided an alternative to SBMD that uses alternating successive convex approximation (ASCA) which has similar convergence rate. 
2) The author provide generalization guarantees for this objective over structured hypothesis spaces associated with a finite set of architectures.
My biggest concern is the validity of the proposed exponentiated gradient update, at least empirically. We indeed observed slightly improvement in test error over DARTS on the CIFAR10 benchmark but how reproducible the results are? Can you compare at least on the other benchmark (PENN TREEBANK) used in Liu et al 2019? Also, comparing to first order DARTS, search cost is the same and this is hard to justify the better convergence rate for EDARTS. In addition, the results on feature map selection is not very encouraging as the gap to the successive halving is significant.
The author proposed ASCA, as an alternative method to SBMD. Why we need such alternative? What is the advantage of ASCA comparing to SBMD? When should I use ASCA and when SBMD? How do they empirically different? 
Then I feel some wording can be improved. For example, "while requiring computation training …”,  “…which may be of independent interest”.