Generating an output for a neural network output layer

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, 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, the method comprising:

precomputing a respective exponentiation measure for each possible output value of a predetermined finite set of possible output values that can be included in layer outputs generated by processing network inputs through the one or more initial neural network layers using the processing system;

storing the respective precomputed exponentiation measures;

obtaining a first layer output generated by the processing system by processing a first network input through the one or more initial neural network layers,

the first layer output having a plurality of first layer output values, and

each first layer output value being a respective one of the predetermined finite set of possible output values; and

processing the layer output through the softmax output layer to generate a first 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 first layer output values that are in the first layer output;

for each possible output value that occurs at least once in the in the plurality of layer output values, accessing the stored respective precomputed exponentiation measure for the possible output value, instead of re-computing the respective exponentiation measure;

determining a normalization factor for the layer output by combining the stored precomputed exponentiation measures for the possible output values that occur in the first layer output at least once in accordance with the counts of occurrences of the possible output values; and

determining, for each of the plurality of first layer output values, a softmax probability value from the respective precomputed exponentiation measure for the first layer output value and the normalization factor.

2. The method of claim 1 , wherein obtaining the first 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 corresponding first 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. 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.

5. The method of claim 1 , further comprising:

generating the first network input by converting one or more floating point values to fixed point values.

6. The method of claim 1 ,

wherein determining the respective precomputed exponentiation measure for each possible output value comprises exponentiating Euler's number by a multiplication of the possible output value.

7. The method of claim 1 , wherein each softmax probability value is determined by dividing each respective precomputed exponentiation measure by the normalization factor.

8. The method of claim 1 , wherein each of the finite set of possible output values is an output of a mapping function.

9. The method of claim 1 , wherein each of the finite set of possible output values is an output of a compression function.

10. 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, 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, the operations comprising:

precomputing a respective exponentiation measure for each possible output value of a predetermined finite set of possible output values that can be included in layer outputs generated by processing network inputs through the one or more initial neural network layers using the processing system;

storing the respective precomputed exponentiation measures;

obtaining a first layer output generated by the processing system processing a first network input through the one or more initial neural network layers,

the first layer output having a plurality of first layer output values, and

each first layer output value being a respective one of the predetermined finite set of possible output values; and

processing the layer output through the softmax output layer to generate a first 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 first layer output values that are in the first layer output;

for each possible output value that occurs at least once in the in the plurality of layer output values, accessing the stored precomputed exponentiation measure for the possible output value, instead of re-computing the respective exponentiation measure;

determining a normalization factor for the layer output by combining the stored precomputed exponentiation measures for the possible output values that occur in the first layer output at least once in accordance with the counts of occurrences of the possible output values; and

determining, for each of the plurality of first layer output values, a softmax probability value from the respective precomputed exponentiation measure for the first layer output value and the normalization factor.

11. The system of claim 10 , wherein obtaining the first 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 corresponding first layer output value based on the mapping data.

12. The system of claim 11 , 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.

13. The system of claim 10 , wherein each of the finite set of possible output values map to a respective value of an integer data type.

14. The system of claim 10 , further comprising:

generating the first network input by converting one or more floating point values to fixed point values.

15. The system of claim 10 ,

wherein determining the respective precomputed exponentiation measure for each possible output value comprises exponentiating Euler's number by a multiplication of the possible output value.

16. 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, 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, the operations comprising:

precomputing a respective exponentiation measure for each possible output value of a predetermined finite set of possible output values that can be included in layer outputs generated by processing network inputs through the one or more initial neural network layers using the processing system;

storing the respective precomputed exponentiation measures;

obtaining a first layer output generated by the processing system processing a first network input through the one or more initial neural network layers,

the first layer output having a plurality of first layer output values, and

each first layer output value being a respective one of the predetermined finite set of possible output values; and

processing the layer output through the softmax output layer to generate a first 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 first layer output values that are in the first layer output;

for each possible output value that occurs at least once in the in the plurality of layer output values, accessing the stored precomputed exponentiation measure for the possible output value, instead of re-computing the respective exponentiation measure;

determining a normalization factor for the layer output by combining the stored precomputed exponentiation measures for the possible output values that occur in the first layer output at least once in accordance with the counts of occurrences of the possible output values; and

determining, for each of the plurality of first layer output values, a softmax probability value from the respective precomputed exponentiation measure for the first layer output value and the normalization factor.

17. The non-transitory computer storage medium of claim 16 , wherein obtaining the first 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 corresponding first layer output value based on the mapping data.

18. The non-transitory computer storage medium of claim 17 , 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.

19. The non-transitory computer storage medium of claim 16 , wherein each of the finite set of possible output values map to a respective value of an integer data type.

20. The non-transitory computer storage medium of claim 16 , wherein determining the respective precomputed exponentiation measure for each possible output value comprises exponentiating Euler's number by a multiplication of the possible output value.