Learning processes for single hidden layer neural networks with linear output units
Learning processes for a single hidden layer neural network, including linear input units, nonlinear hidden units, and linear output units, calculate the lower-layer network parameter gradients by taking into consideration a solution for the upper-layer network parameters. The upper-layer network parameters are calculated by a closed form formula given the lower-layer network parameters. An accelerated gradient algorithm can be used to update the lower-layer network parameters. A weighted gradient also can be used. With the combination of these techniques, accelerated training with faster convergence, to a point with a lower error rate, can be obtained.
1. A computer-implemented process comprising:
receiving into memory data defining a single hidden layer neural network which includes a linear input layer, a nonlinear hidden layer, and a linear output layer, having lower layer weights applied to the input layer to provide an output of the hidden layer and upper layer weights applied to the output of the hidden layer to provide an output of the output layer;
computing a gradient of a square error with respect to the lower layer weights, wherein the gradient of the square error is a first function derived by a. specifying a second function defining the gradient of the square error using the lower layer weights and the upper layer weights, and b. substituting, in the second function, the upper layer weights with a third function defining the upper layer weights using the lower layer weights; and
within each learning stage, updating the lower layer weights according to the computed gradient using the first function and then updating the upper layer weights according to the third function defining the upper layer weights using the updated lower layer weights.
2. The computer-implemented process of claim 1 , further comprising updating the lower layer weights using a gradient history.
3. The computer implemented process of claim 2 , wherein computing the gradient uses a weight for each sample, wherein the weights are positively correlated to errors introduced by each sample.
4. The computer-implemented process of claim 1 , further comprising:
applying input data to the input layer;
computing error data from output data from the output layer and desired output data corresponding to the input data.
5. The computer-implemented process of claim 4 , wherein the input data is two dimensional images.
6. The computer-implemented process of claim 5 , wherein the two-dimensional images include handwriting from an input device and the output layer provides outputs classifying the handwriting into text.
7. The computer-implemented process of claim 4 , wherein the input data is audio data.
8. The computer-implemented process of claim 7 , wherein the audio data includes speech and the output layer provides outputs classifying contents of the audio data into components of speech.
9. An article of manufacture comprising:
a computer storage medium;
computer program instructions stored on the computer storage medium which, when processed by a processing device, instruct the processing device to perform a process comprising:
receiving into memory data defining a single hidden layer neural network which includes a linear input layer, a nonlinear hidden layer, and a linear output layer, having lower layer weights applied to the input layer to provide an output of the hidden layer and upper layer weights applied to the output of the hidden layer to provide and output of the output layer;
computing a gradient of a square error with respect to the lower layer weights, wherein the gradient of the square error is a first function derived by a. specifying a second function defining the gradient of the square error using the lower layer weights and the upper layer weights, and b. substituting, in the second function, the upper layer weights with a third function defining the upper layer weights using the lower layer weights; and
within each learning stage, updating the lower layer weights according to the computed gradient using the first function and then updating the upper layer weights according to the third function defining the upper layer weights using the updated lower layer weights.
10. The article of manufacture of claim 9 , wherein the process performed by the processing device further comprises updating the lower layer weights using a gradient history.
11. The article of manufacture of claim 10 , wherein the gradient uses a weight for each sample, wherein the weights are positively correlated to errors introduced by each sample.
12. The article of manufacture of claim 9 , wherein the process performed by the processing unit further comprises:
applying input data to the input layer; and
computing error data from output data from the output layer and desired output data corresponding to the input data.
13. The article of manufacture of claim 12 , wherein the input data is two dimensional images, and the output layer provides outputs classifying contents of the images.
14. The article of manufacture of claim 12 , wherein the input data is audio data including speech and the output layer provides outputs classifying contents of the audio data into components of speech.
15. A computing machine comprising:
a memory storing data defining a single hidden layer neural network which includes a linear layer, a nonlinear hidden layer, and a linear output layer, having lower layer weights applied to the input layer to provide an output of the hidden layer and upper layer weights applied to the output of the hidden layer to provide an output of the output layer;
a processing unit configured to compute a gradient of a square error with respect to the lower layer weights, wherein the gradient of the square error is a first function derived by a. specifying a second function defining the gradient of the square error using the lower layer weights and the upper layer weights, and b. substituting, in the second function, the upper layer weights with a third function defining the upper layer weights using the lower layer weights and, in each learning stage, to update the lower layer weights according to the computed gradient using the first function and then to update the upper layer weights according to the third function defining the upper layer weights using the updated lower layer weights.
16. The computing machine of claim 15 , wherein the processing unit is further configured to update the lower layer weights using a gradient history.
17. The computing machine of claim 16 , wherein the gradient uses a weight for each sample, wherein the weights are positively correlated to errors introduced by each sample.
18. The computing machine of claim 15 , wherein the processing unit is further configured to:
apply input data to the input layer; and
compute error data from output data from the output layer and desired output data corresponding to the input data.
19. The computing machine of claim 18 , wherein the input data is two dimensional images, and the output layer provides outputs classifying contents of the images.
20. The computing machine of claim 18 , wherein the input data is audio data including speech and the output layer provides outputs classifying contents of the audio data into components of speech.