Preventing feature co-adaptation by encouraging independent contributions from different features often improves classification and regression performance. Dropout training (Hinton et al., 2012) does this by randomly dropping out (zeroing) hidden units and input features during training of neural networks. However, repeatedly sampling a random subset of input features makes training much slower. Based on an examination of the implied objective function of dropout training, we show how to do fast dropout training by sampling from or integrating a Gaussian approximation, instead of doing Monte Carlo optimization of this objective. This approximation, justified by the central limit theorem and empirical evidence, gives an order of magnitude speedup and more stability. We show how to do fast dropout training for classification, regression, and multilayer neural networks. Beyond dropout, our technique is extended to integrate out other types of noise and small image transformations.