Vanishing gradients pose a challenge when training deep neural networks, resulting in the top layers (closer to the output) in the network learning faster when compared with lower layers closer to the input. Interpreting the top layers as a classifier and the lower layers a feature extractor, one can hypothesize that unwanted network convergence may occur when the classifier has overfit with respect to the feature extractor. This can lead to the feature extractor being under-trained, possibly failing to learn much about the patterns in the input data. To address this we propose a good classifier hypothesis: given a fixed classifier that partitions the space well, the feature extractor can be further trained to fit that classifier and learn the data patterns well. This alleviates the problem of under-training the feature extractor and enables the network to learn patterns in the data with small partial derivatives. We verify this hypothesis empirically and propose a novel top-down training method. We train all layers jointly, obtaining a good classifier from the top layers, which are then frozen. Following re-initialization, we retrain the bottom layers with respect to the frozen classifier. Applying this approach to a set of speech recognition experiments using the Wall Street Journal and noisy CHiME-4 datasets we observe substantial accuracy gains. When combined with dropout, our method enables connectionist temporal classification (CTC) models to outperform joint CTC-attention models, which have more capacity and flexibility.
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Update review
After reading the authors response I would like to keep my score as is.
I still see many unclear statements, and most importantly I feel that more analysis of the proposed method should have been done here. 
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This paper proposed a Top-Down method for neural networks training based on the good classifier hypothesis. In other words, after obtaining a classifier that performs well on the test set, keep fine-tuning / re-learning the data representation.
The authors provide character error rate results for the task of Automatic Speech Recognition using WSJ and CHiME-4 datasets.
Although being an interesting research idea, several issues in this paper make it not yet ready for publication at ICLR. 
First, the paper is poorly written; there are many claims the authors are making without providing experiments/proofs/citations. 
For example: "...since the feature extractor learns more slowly, then potentially the classifier may overfit the feature extractor before the feature extractor is able to learn much about the underlying pattern of the data...". 
Or: "...We suggest that the reason for this is that when all layers are trained on the noisy dataset jointly, the middle layers overfit the bottom-most layers much faster than the bottom-most layers are able to learn input features..."
Next, since there is no theoretical/mathematical explanation of the proposed approach, I expect the authors to run an analysis on the results to better understand the effect of using such an approach. For instance, under which settings this method is most efficient? In what layer should  I start the fine-tuning? Is it better to reinitialize the bottom layers or fine-tune them? Does the proposed approach applicable to different domains? i.e. vision/nlp/other speech/signal processing tasks?  Does the proposed approach applicable to different models or only for the proposed one?
Lastly, although it is not the main point in this paper since all results are reported on ASR, did the authors tried to compute WERs too? That way, people can compare results with other ASR models. The baseline seems relatively weak, at least in Table 1.
Minor comments: 
The complexity of the algorithm is written to be O(n). However, this assumes training the model takes O(1) or did I miss something?
Can the authors provide more details/insights regarding the delta differences in Table 1? Did the authors use the same initializations? Did the authors try different ones?