Title: Top-down training for neural networks. Abstract: 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.  