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Softmax is a standard final layer used in Neural Nets (NNs) to summarizeinformation encoded in the trained NN and return a prediction. However, Softmaxleverages only a subset of the class-specific structure encoded in the trainedmodel and ignores potentially valuable information: During training, modelsencode an array $D$ of class response distributions, where $D_{ij}$ is thedistribution of the $j^{th}$ pre-Softmax readout neuron's responses to the$i^{th}$ class. Given a test sample, Softmax implicitly uses only the row ofthis array $D$ that corresponds to the readout neurons' responses to thesample's true class. Leveraging more of this array $D$ can improve classifieraccuracy, because the likelihoods of two competing classes can be encoded inother rows of $D$.  To explore this potential resource, we develop a hybrid classifier(Softmax-Pooling Hybrid, $SPH$) that uses Softmax on high-scoring samples, buton low-scoring samples uses a log-likelihood method that pools the informationfrom the full array $D$. We apply $SPH$ to models trained on a vectorized MNISTdataset to varying levels of accuracy. $SPH$ replaces only the final Softmaxlayer in the trained NN, at test time only. All training is the same as forSoftmax. Because the pooling classifier performs better than Softmax onlow-scoring samples, $SPH$ reduces test set error by 6% to 23%, using the exactsame trained model, whatever the baseline Softmax accuracy. This reduction inerror reflects hidden capacity of the trained NN that is left unused bySoftmax.

Money on the Table: Statistical information ignored by Softmax can improve classifier accuracy