Systems, methods, and apparatus, including computer programs encoded on a computer storage medium for processing a network input through a neural network having one or more initial neural network layers followed by a softmax output layer. In one aspect, the methods include obtaining a layer output generated by the one or more initial neural network layers and processing the layer output through the softmax output layer to generate a neural network output. Processing the layer output through the softmax output layer includes determining, for each possible output value, a number of occurrences in the layer output values; for each possible output value occurring in the layer output values, determining a respective exponentiation measure; determining a normalization factor for the layer output by combining the exponentiation measures in accordance with the number of occurrences of the possible output values; and determining, for each of layer output values, a softmax probability value.
Please help me write a proper abstract based on the patent claims. CLAIM: 1. A method of processing a network input through a neural network having one or more initial neural network layers followed by a softmax output layer to generate a neural network output for the network input, the method comprising: obtaining a layer output generated by processing the network input through the one or more initial neural network layers, wherein the initial neural network layers are implemented by a processing system that performs computations specified by the one or more initial neural network layers using quantized arithmetic such that output values generated by the processing system can take only values from a predetermined finite set of values, and wherein: the layer output has a plurality of layer output values, and each layer output value is a respective one of a predetermined finite set of possible output values; and processing the layer output through the softmax output layer to generate the neural network output for the network input, comprising: determining, for each possible output value in the predetermined finite set of possible output values, a count of occurrences of the possible output value among the plurality of layer output values that are in the layer output; for each possible output value that occurs at least once in the plurality of layer output values, determining a respective exponentiation measure of the possible output value; determining a normalization factor for the layer output by combining the exponentiation measures in accordance with the counts of occurrences of the possible output values; and determining, for each of the plurality of layer output values, a softmax probability value from the respective exponentiation measure for the layer output value and the normalization factor for the layer output. 2. The method of claim 1, wherein obtaining the layer output comprises: receiving a plurality of initial layer output values from the processing system, the plurality of initial layer output values being unmapped output values of the one or more initial neural network layers that are each a respective one of the predetermined finite set of values; obtaining mapping data defining a mapping from the predetermined finite set of values to the predetermined finite set of possible layer output values; and determining, for each initial layer output value, a layer output value based on the mapping data. 3. The method of claim 2, wherein the mapping data specifies a scaling factor for scaling each of the plurality of initial layer output values to generate the layer output values. 4. (canceled) 5. (canceled) 6. The method of claim 1, wherein each of the finite set of possible output values map to a respective value of an integer data type. 7. The method of claim 1, wherein the network input comprises one or more floating point values, and the method further comprising: generating a modified network input from the network input by converting the one or more floating point values to fixed point values; and providing the modified network input as input to the processing system for processing by the one or more initial floating point values. 8. The method of claim 1, wherein determining the respective exponentiation measure of the possible output value comprises: exponentiating Euler's number by a multiplication of each respective possible output value. 9. The method of claim 1, wherein each softmax probability value is determined by dividing each respective exponentiation measure by the normalization factor. 10. The method of claim 1, wherein each of the finite set of possible output values is an output of a mapping function. 11. The method of claim 1, wherein each of the finite set of possible output values is an output of a compression function. 12. A system comprising one or more computers and one or more storage devices storing instructions that are operable, when executed by the one or more computers, to cause the one or more computers to perform operations for processing a network input through a neural network having one or more initial neural network layers followed by a softmax output layer to generate a neural network output for the network input, the operations comprising: obtaining a layer output generated by processing the network input through the one or more initial neural network layers, wherein the initial neural network layers are implemented by a processing system that performs computations specified by the one or more initial neural network layers using quantized arithmetic such that output values generated by the processing system can take only values from a predetermined finite set of values, and wherein: the layer output has a plurality of layer output values, and each layer output value is a respective one of a predetermined finite set of possible output values; and processing the layer output through the softmax output layer to generate the neural network output for the network input, comprising: determining, for each possible output value in the predetermined finite set of possible output values, a count of occurrences of the possible output value among the plurality of layer output values that are in the layer output; for each possible output value that occurs at least once in the plurality of layer output values, determining a respective exponentiation measure of the possible output value; determining a normalization factor for the layer output by combining the exponentiation measures in accordance with the counts of occurrences of the possible output values; and determining, for each of the plurality of layer output values, a softmax probability value from the respective exponentiation measure for the layer output value and the normalization factor for the layer output. 13. The system of claim 12, wherein obtaining the layer output comprises: receiving a plurality of initial layer output values from the processing system, the plurality of initial layer output values being unmapped output values of the one or more initial neural network layers that are each a respective one of the predetermined finite set of values; obtaining mapping data defining a mapping from the predetermined finite set of values to the predetermined finite set of possible layer output values; and determining, for each initial layer output value, a layer output value based on the mapping data. 14. (canceled) 15. (canceled) 16. The system of claim 12, wherein each of the finite set of possible output values map to a respective value of an integer data type. 17. The system of claim 12, wherein the network input comprises one or more floating point values, and the operations further comprising: generating a modified network input from the network input by converting the one or more floating point values to fixed point values; and providing the modified network input as input to the processing system for processing by the one or more initial floating point values. 18. A non-transitory computer storage medium encoded with instructions that, when executed by one or more computers, cause the one or more computers to perform operations for processing a network input through a neural network having one or more initial neural network layers followed by a softmax output layer to generate a neural network output for the network input, the operations comprising: obtaining a layer output generated by processing the network input through the one or more initial neural network layers, wherein the initial neural network layers are implemented by a processing system that performs computations specified by the one or more initial neural network layers using quantized arithmetic such that output values generated by the processing system can take only values from a predetermined finite set of values, and wherein: the layer output has a plurality of layer output values, and each layer output value is a respective one of a predetermined finite set of possible output values; and processing the layer output through the softmax output layer to generate the neural network output for the network input, comprising: determining, for each possible output value in the predetermined finite set of possible output values, a count of occurrences of the possible output value among the plurality of layer output values that are in the layer output; for each possible output value that occurs at least once in the plurality of layer output values, determining a respective exponentiation measure of the possible output value; determining a normalization factor for the layer output by combining the exponentiation measures in accordance with the counts of occurrences of the possible output values; and determining, for each of the plurality of layer output values, a softmax probability value from the respective exponentiation measure for the layer output value and the normalization factor for the layer output. 19. The computer storage medium of claim 18, wherein obtaining the layer output comprises: receiving a plurality of initial layer output values from the processing system, the plurality of initial layer output values being unmapped output values of the one or more initial neural network layers that are each a respective one of the predetermined finite set of values; obtaining mapping data defining a mapping from the predetermined finite set of values to the predetermined finite set of possible layer output values; and determining, for each initial layer output value, a layer output value based on the mapping data. 20. (canceled) 21. The method of claim 1, further comprising, prior to obtaining the layer output: determining, for each possible output value in the predetermined finite set of possible output values, a precomputed exponentiation measure; and storing each precomputed exponentiation measure; and wherein determining the respective exponentiation measure for each possible output value occurring in the plurality of layer output values comprises accessing the stored precomputed exponentiation measure for the possible output value.