Generating output sequences from input sequences using neural networks

1. A method comprising:

obtaining an input sequence having a first number of inputs arranged according to an input order;

processing each input in the input sequence using an encoder recurrent neural network to generate a respective encoder hidden state for each input in the input sequence; and

generating an output sequence having a second number of outputs arranged according to an output order, each output in the output sequence being selected from the inputs in the input sequence, comprising, for each position in the output order and beginning at an initial position in the output order:

generating a softmax output for the position using the encoder hidden states that is a pointer into the input sequence; and

selecting an input from the input sequence as the output at the position using the softmax output.

2. The method of claim 1 , wherein generating the softmax output for the position comprises, for the initial position in the output order:

processing a predetermined initial output using a decoder recurrent neural network to generate an initial decoder hidden state;

generating, from the initial decoder hidden state and the hidden states for the inputs in the input sequence, the softmax output for the initial position, wherein the softmax output comprises a respective output score for each position in the input order; and

selecting an input at the highest-scoring position in the input order according to the output scores as the output at the initial position in the output order.

3. The method of claim 2 , wherein processing the predetermined initial output using the decoder recurrent neural network to generate an initial decoder hidden state comprises:

initializing an internal state of the decoder recurrent neural network to the encoder hidden state for the last input in the input sequence; and

processing the predetermined initial output using the decoder recurrent neural network to update the initialized internal state to the initial decoder hidden state.

4. The method of claim 1 , wherein generating the softmax output for the position comprises, for each position after the initial position in the output order:

processing an output at the preceding position in the output order using a decoder recurrent neural network to generate a decoder hidden state for the output at the preceding position;

generating, from the decoder hidden state for the output at the preceding position and the hidden states for the inputs in the input sequence, the softmax output for the position, wherein the softmax output comprises a respective output score for each position in the input order; and

selecting an input at the highest-scoring position in the input order according to the output scores as the output at the position in the output order.

5. The method of claim 4 , wherein generating the softmax output for the position comprises:

generating an attention vector having a dimensionality equal to the first number of inputs in the input sequence, comprising, for each position in the input order:

combining the encoder hidden state for the input at the position in the input order and the decoder hidden state for the output at the preceding position in the output order to generate a value at a corresponding position in the vector; and

generating the respective output scores by processing the attention vector through a softmax output layer.

6. The method of claim 1 , further comprising, for each position in the output order:

determining whether or not the selected output at the position is a designated input;

when the selected output at the position is not the designated input, determining that an output should be generated for a subsequent position in the output order; and

when the selected output at the position is the designated input, determining that the position is the last position in the output order.

7. The method of claim 1 , further comprising:

generating one or more additional output sequences using a beam search technique;

determining a respective sequence score for the output sequence and for each additional output sequence; and

selecting an output sequence having a highest sequence score as a final output sequence for the input sequence.

8. The method of claim 1 , wherein:

obtaining the input sequence comprises obtaining respective coordinates of the first number of input points; and

generating the output sequence comprises generating a corresponding solution to a geometric problem with respect to the input points.

9. The method of claim 8 , wherein the geometric problem is a convex hull problem, a Delaunay triangulation problem, or a Travelling Salesman Problem.

10. 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:

obtaining an input sequence having a first number of inputs arranged according to an input order;

processing each input in the input sequence using an encoder recurrent neural network to generate a respective encoder hidden state for each input in the input sequence; and

generating an output sequence having a second number of outputs arranged according to an output order, each output in the output sequence being selected from the inputs in the input sequence, comprising, for each position in the output order and beginning at an initial position in the output order:

generating a softmax output for the position using the encoder hidden states that is a pointer into the input sequence; and

selecting an input from the input sequence as the output at the position using the softmax output.

11. The system of claim 10 , wherein generating the softmax output for the position comprises, for the initial position in the output order:

processing a predetermined initial output using a decoder recurrent neural network to generate an initial decoder hidden state;

generating, from the initial decoder hidden state and the hidden states for the inputs in the input sequence, the softmax output for the initial position, wherein the softmax output comprises a respective output score for each position in the input order; and

selecting an input at the highest-scoring position in the input order according to the output scores as the output at the initial position in the output order.

12. The system of claim 11 , wherein processing the predetermined initial output using the decoder recurrent neural network to generate an initial decoder hidden state comprises:

initializing an internal state of the decoder recurrent neural network to the encoder hidden state for the last input in the input sequence; and

processing the predetermined initial output using the decoder recurrent neural network to update the initialized internal state to the initial decoder hidden state.

13. The system of claim 10 , wherein generating the softmax output for the position comprises, for each position after the initial position in the output order:

processing an output at the preceding position in the output order using a decoder recurrent neural network to generate a decoder hidden state for the output at the preceding position;

generating, from the decoder hidden state for the output at the preceding position and the hidden states for the inputs in the input sequence, the softmax output for the position, wherein the softmax output comprises a respective output score for each position in the input order; and

selecting an input at the highest-scoring position in the input order according to the output scores as the output at the position in the output order.

14. The system of claim 13 , wherein generating the softmax output for the position comprises:

generating an attention vector having a dimensionality equal to the first number of inputs in the input sequence, comprising, for each position in the input order:

combining the encoder hidden state for the input at the position in the input order and the decoder hidden state for the output at the preceding position in the output order to generate a value at a corresponding position in the vector; and

generating the respective output scores by processing the attention vector through a softmax output layer.

15. The system of claim 10 , the operations further comprising, for each position in the output order:

determining whether or not the selected output at the position is a designated input;

when the selected output at the position is not the designated input, determining that an output should be generated for a subsequent position in the output order; and

when the selected output at the position is the designated input, determining that the position is the last position in the output order.

16. The system of claim 10 , the operations further comprising:

generating one or more additional output sequences using a beam search technique;

determining a respective sequence score for the output sequence and for each additional output sequence; and

selecting an output sequence having a highest sequence score as a final output sequence for the input sequence.

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

obtaining an input sequence having a first number of inputs arranged according to an input order;

processing each input in the input sequence using an encoder recurrent neural network to generate a respective encoder hidden state for each input in the input sequence; and

generating an output sequence having a second number of outputs arranged according to an output order, each output in the output sequence being selected from the inputs in the input sequence, comprising, for each position in the output order and beginning at an initial position in the output order:

generating a softmax output for the position using the encoder hidden states that is a pointer into the input sequence; and

selecting an input from the input sequence as the output at the position using the softmax output.

18. The computer storage media of claim 17 , wherein generating the softmax output for the position comprises, for the initial position in the output order:

processing a predetermined initial output using a decoder recurrent neural network to generate an initial decoder hidden state;

generating, from the initial decoder hidden state and the hidden states for the inputs in the input sequence, the softmax output for the initial position, wherein the softmax output comprises a respective output score for each position in the input order; and

selecting an input at the highest-scoring position in the input order according to the output scores as the output at the initial position in the output order.

19. The computer storage media of claim 18 , wherein processing the predetermined initial output using the decoder recurrent neural network to generate an initial decoder hidden state comprises:

initializing an internal state of the decoder recurrent neural network to the encoder hidden state for the last input in the input sequence; and

processing the predetermined initial output using the decoder recurrent neural network to update the initialized internal state to the initial decoder hidden state.

20. The computer storage media of claim 17 , the operations further comprising, for each position in the output order:

determining whether or not the selected output at the position is a designated input;

when the selected output at the position is not the designated input, determining that an output should be generated for a subsequent position in the output order; and

when the selected output at the position is the designated input, determining that the position is the last position in the output order.