NEURAL NETWORKS WITH TRANSFORMED ACTIVATION FUNCTION LAYERS

1 . A method performed by one or more computers, the method comprising:

receiving a network input; and

processing the network input using a neural network that comprises a plurality of neural network layers arranged as a directed graph to generate a network output for the network input, the plurality of neural network layers comprising a plurality of transformed activation function layers, and wherein processing the network input comprises, for each transformed activation function layer:

receiving a layer input for the transformed activation function layer;

generating, from the layer input, an activation input to an element-wise activation function for the transformed activation layer;

transforming the activation input using one or more input transformation constants to generate a transformed activation input;

applying the element-wise activation function to the transformed activation input to generate an initial activation output;

transforming the initial activation output using one or more output transformation constants to generate a transformed activation output, wherein the one or more input transformation constants and the one or more output transformation constants are based on properties of the neural network when the neural network is initialized prior to training the neural network; and

providing the transformed activation output as a layer output for the transformed activation function layer.

2 . The method of claim 1 , wherein generating, from the layer input, an activation input to an element-wise activation function for the transformed activation layer comprises:

applying an affine operation to the layer input.

3 . The method of claim 1 , wherein generating, from the layer input, an activation input to an element-wise activation function for the transformed activation layer comprises:

using the layer input as the activation input.

4 . The method of claim 1 , wherein transforming the activation input using one or more input transformation constants to generate a transformed activation input comprises:

generating an initial transformed activation input by multiplying the activation input by an input scale constant.

5 . The method of claim 4 , wherein transforming the activation input using one or more input transformation constants to generate a transformed activation input comprises:

generating the transformed activation input by adding an input shift constant to the initial transformed activation input.

6 . The method of claim 1 , wherein the activation function is a leaky RELU activation function that, for a given element, (i) is an identity operation when the given element is greater than or equal to zero and (ii) multiplies the given element by a slope value when the given element is less than zero, and wherein an output scale constant is defined by the slope value.

7 . The method of claim 6 , wherein the output scale constant is equal to a square root of a ratio between (i) 2 and (ii) a sum of one and a square of the slope value.

8 . The method of claim 1 , wherein transforming the initial activation output using one or more output transformation constants to generate a transformed activation output comprises:

generating a shifted initial activation output by adding an output shift constant to the activation output.

9 . The method of claim 8 , wherein transforming the initial activation output using one or more output transformation constants to generate a transformed activation output comprises:

generating the transformed activation output by multiplying the shifted initial activation output by an output scale constant.

10 . The method of claim 9 , wherein the activation function is a smooth activation function and wherein the output scale constant is based on a value of the shifted initial activation output for an element that has a value sampled from a noise distribution.

11 . The method of claim 1 , wherein the plurality of neural network layers comprise one or more normalized summation layers, and wherein processing the network input comprises, for each normalized summation layer:

receiving for each of a plurality of neural network layers that are connected to the normalized summation layer by an incoming edge in the directed graph, a respective layer output generated by the neural network layer during the processing of the network input;

applying a respective normalized weight to each of the respective layer outputs to generate a respective weighted layer output, wherein a sum of the squares of the respective normalized weights is equal to one; and

generating a layer output for the normalized summation layer by summing the respective weighted layer outputs.

12 . The method of claim 1 , wherein, for one or more of the plurality of transformed activation function layers, generating, from the layer input, an activation input to an element-wise activation function for the transformed activation layer comprises:

computing a convolution between a filter bank tensor for the layer and the layer input.

13 . The method of claim 12 , further comprising:

for each of the one or more transformed activation function layers, prior to training the neural network, initializing the filter bank tensor for the layer using Delta initialization.

14 . The method of claim 13 , wherein the Delta initialization uses an entry-wise Gaussian distribution.

15 . The method of claim 13 , wherein the Delta initialization uses a scaled-corrected uniform orthogonal (SUO) distribution.

16 . The method of claim 1 , wherein the one or more input transformation constants and the one or more output transformation constants are also based on a hyperparameter that represents a degree of nonlinearity of the operations performed by the neural network at initialization.

17 . The method of claim 16 , wherein the one or more input transformation constants and the one or more output transformation constants are based on the hyperparameter and an estimate of a maximal slope function of the neural network at initialization.

18 . The method of claim 1 , wherein the one or more input transformation constants and the one or more output transformation constants for each of the layers are selected such one or more constraints that are based on values of local C maps, local Q maps, or both for the plurality of layers in the neural network are satisfied.

19 . The method of claim 18 , wherein the one or more constraints are based on values of local C maps, wherein a local C map is a function that characterizes how well a cosine similarity function is preserved between the input and the output of a neural network layer, wherein a global C map comprises a composition local C maps for the plurality of neural network layers, and wherein the one or more input transformation constants and the one or more output transformation constants are selected based on a constraint on the global C map that represents preservation of the cosine similarity function by the neural network.

20 . The method of claim 18 , wherein the one or more constraints are based on values of local Q maps, wherein a local Q map is a function that characterizes a change in a squared magnitude of the element-wise activation function between the input and the output of a neural network layer, and wherein the one or more input transformation constants and the one or more output transformation constants are selected based on a constraint on the local Q map.

21 . The method of claim 18 , wherein at least one of the constraints is based on the hyperparameter and the estimate of the maximal slope function.

22 . The method of claim 1 , where processing the network input using the neural network comprises:

applying normalization to the network input to generate a normalized input; and

providing the normalized input as a layer input for one or more initial neural networks of the neural network.

23 . The method of claim 1 , wherein the network input is received during training of the neural network and wherein the method further comprises:

obtaining a target network output for the network input;

determining a gradient with respect to a set of parameters of the neural network of a loss function for the training of the neural network that measures a quality of the network output relative to the target network output; and

determining an update to the parameters of the neural network based at least on the gradient.

24 . A system comprising:

one or more computers; and

one or more storage devices storing instructions that, when executed by the one or more computers, cause the one or more computers to perform operations comprising:

receiving a network input; and

processing the network input using a neural network that comprises a plurality of neural network layers arranged as a directed graph to generate a network output for the network input, the plurality of neural network layers comprising a plurality of transformed activation function layers, and wherein processing the network input comprises, for each transformed activation function layer:

receiving a layer input for the transformed activation function layer;

generating, from the layer input, an activation input to an element-wise activation function for the transformed activation layer;

transforming the activation input using one or more input transformation constants to generate a transformed activation input;

applying the element-wise activation function to the transformed activation input to generate an initial activation output;

transforming the initial activation output using one or more output transformation constants to generate a transformed activation output, wherein the one or more input transformation constants and the one or more output transformation constants are based on properties of the neural network when the neural network is initialized prior to training the neural network; and

providing the transformed activation output as a layer output for the transformed activation function layer.

25 . One or more non-transitory computer-readable storage media storing instructions that when executed by one or more computers cause the one or more computers to perform operations comprising:

receiving a network input; and

processing the network input using a neural network that comprises a plurality of neural network layers arranged as a directed graph to generate a network output for the network input, the plurality of neural network layers comprising a plurality of transformed activation function layers, and wherein processing the network input comprises, for each transformed activation function layer:

receiving a layer input for the transformed activation function layer;

generating, from the layer input, an activation input to an element-wise activation function for the transformed activation layer;

transforming the activation input using one or more input transformation constants to generate a transformed activation input;

applying the element-wise activation function to the transformed activation input to generate an initial activation output;

transforming the initial activation output using one or more output transformation constants to generate a transformed activation output, wherein the one or more input transformation constants and the one or more output transformation constants are based on properties of the neural network when the neural network is initialized prior to training the neural network; and

providing the transformed activation output as a layer output for the transformed activation function layer.