Paper:
Machine learning models are usually tuned by nesting optimization of model weights inside the optimization of hyperparameters. We give a method to collapse this nested optimization into joint stochastic optimization of both weights and hyperparameters. Our method trains a neural network to output approximately optimal weights as a function of hyperparameters. We show that our method converges to locally optimal weights and hyperparameters for sufficiently large hypernets. We compare this method to standard hyperparameter optimization strategies and demonstrate its effectiveness for tuning thousands of hyperparameters.

Review:
*Summary*
The paper proposes to use hyper-networks [Ha et al. 2016] for the tuning of hyper-parameters, along the lines of [Brock et al. 2017]. The core idea is to have a side neural network sufficiently expressive to learn the (large-scale, matrix-valued) mapping from a given configuration of hyper-parameters to the weights of the model we wish to tune.
The paper gives a theoretical justification of its approach, and then describes several variants of its core algorithm which mix the training of the hyper-networks together with the optimization of the hyper-parameters themselves. Finally, experiments based on MNIST illustrate the properties of the proposed approach.
While the core idea may appear as appealing, the paper suffers from several flaws (as further detailed afterwards):
-Insufficient related work
-Correctness/rigor of Theorem 2.1
-Clarity of the paper (e.g., Sec. 2.4)
-Experiments look somewhat artificial
-How scalable is the proposed approach in the perspective of tuning models way larger/more complex than those treated in the experiments?
*Detailed comments*
-"...and training the model to completion." and "This is wasteful, since it trains the model from scratch each time..." (and similar statement in Sec. 2.1): Those statements are quite debatable. There are lines of work, e.g., in Bayesian optimization, to model early stopping/learning curves (e.g., Domhan2014, Klein2017 and references therein) and where training procedures are explicitly resumed (e.g., Swersky2014, Li2016). The paper should reformulate its statements in the light of this literature.
-"Uncertainty could conceivably be incorporated into the hypernet...". This seems indeed an important point, but it does not appear as clear how to proceed (e.g., uncertainty on w_phi(lambda) which later needs to propagated to L_val); could the authors perhaps further elaborate?
-I am concerned about the rigor/correctness of Theorem 2.1; for instance, how is the continuity of the best-response exploited? Also, throughout the paper, the argmin is defined as if it was a singleton while in practice it is rather a set-valued mapping (except if there is a unique minimizer for L_train(., lambda), which is unlikely to be the case given the nature of the considered neural-net model). In the same vein, Jensen's inequality states that Expectation[g(X)] >= g(Expectation[X]) for some convex function g and random variable X; how does it precisely translate into the paper's setting (convexity, which function g, etc.)? 
-Specify in Alg. 1 that "hyperopt" refers to a generic hyper-parameter procedure.
-More details should be provided to better understand Sec. 2.4. At the moment, it is difficult to figure out (and potentially reproduce) the model which is proposed.
-The training procedure in Sec. 4.2 seems quite ad hoc; how sensitive was the overall performance with respect to the optimization strategy? For instance, in 4.2 and 4.3, different optimization parameters are chosen.
-typo: "weight decay is applied the..." --> "weight decay is applied to the..."
-"a standard Bayesian optimization implementation from sklearn": Could more details be provided? (there does not seem to be implementation there http://scikit-learn.org/stable/model_selection.html to the best of my knowledge)
-The experimental set up looks a bit far-fetched and unrealistic: first scalar, than diagonal and finally matrix-weighted regularization schemes. While the first two may be used in practice, the third scheme is not used in practice to the best of my knowledge.
-typo: "fit a hypernet same dataset." --> "fit a hypernet on the same dataset."
-(Franceschi2017) could be added to the related work section.
*References*
(Domhan2014) Domhan, T.; Springenberg, T. & Hutter, F. Extrapolating learning curves of deep neural networks ICML 2014 AutoML Workshop, 2014
(Franceschi2017) Franceschi, L.; Donini, M.; Frasconi, P. & Pontil, M. Forward and Reverse Gradient-Based Hyperparameter Optimization preprint arXiv:1703.01785, 2017
(Klein2017) Klein, A.; Falkner, S.; Springenberg, J. T. & Hutter, F. Learning curve prediction with Bayesian neural networks International Conference on Learning Representations (ICLR), 2017, 17
(Li2016) Li, L.; Jamieson, K.; DeSalvo, G.; Rostamizadeh, A. & Talwalkar, A. Hyperband: A Novel Bandit-Based Approach to Hyperparameter Optimization preprint arXiv:1603.06560, 2016
(Swersky2014) Swersky, K.; Snoek, J. & Adams, R. P. Freeze-Thaw Bayesian Optimization preprint arXiv:1406.3896, 2014
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Update post rebuttal
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I acknowledge the fact that I read the rebuttal of the authors, whom I thank for their detailed answers.
My minor concerns have been clarified. Regarding the correctness of the proof, I am still unsure about the applicability of Jensen inequality; provided it is true, then it is important to see that the results seem to hold only for particular hyperparameters, namely regularization parameters (as explained in the new updated proof). This limitation should be exposed transparently upfront in the paper/abstract. 
Together with the new experiments and comparisons, I have therefore updated my rating from 5 to 6.