Pointer Networks (1506.03134v2)

Abstract: We introduce a new neural architecture to learn the conditional probability of an output sequence with elements that are discrete tokens corresponding to positions in an input sequence. Such problems cannot be trivially addressed by existent approaches such as sequence-to-sequence and Neural Turing Machines, because the number of target classes in each step of the output depends on the length of the input, which is variable. Problems such as sorting variable sized sequences, and various combinatorial optimization problems belong to this class. Our model solves the problem of variable size output dictionaries using a recently proposed mechanism of neural attention. It differs from the previous attention attempts in that, instead of using attention to blend hidden units of an encoder to a context vector at each decoder step, it uses attention as a pointer to select a member of the input sequence as the output. We call this architecture a Pointer Net (Ptr-Net). We show Ptr-Nets can be used to learn approximate solutions to three challenging geometric problems -- finding planar convex hulls, computing Delaunay triangulations, and the planar Travelling Salesman Problem -- using training examples alone. Ptr-Nets not only improve over sequence-to-sequence with input attention, but also allow us to generalize to variable size output dictionaries. We show that the learnt models generalize beyond the maximum lengths they were trained on. We hope our results on these tasks will encourage a broader exploration of neural learning for discrete problems.

• The paper introduces a novel architecture that uses attention as a pointer to dynamically select variable-length output indices.
• It applies the model to geometric problems, demonstrating high accuracy and strong generalization in tasks like convex hull, Delaunay triangulation, and TSP.
• Empirical evaluations show that the Pointer Network outperforms traditional seq2seq models, offering significant potential for combinatorial optimization.

Overview of "Pointer Networks"

The paper "Pointer Networks" by Oriol Vinyals, Meire Fortunato, and Navdeep Jaitly, introduces a novel neural architecture specifically designed to address problems where the output sequence consists of positions within the input sequence. Traditional neural network models such as sequence-to-sequence (seq2seq) and Neural Turing Machines (NTMs) are limited by their predefined output dictionary sizes, which constrains their applicability to problems with variable-length output sequences. The proposed Pointer Network (Ptr-Net) leverages a unique application of the attention mechanism to dynamically select elements from the input sequence as outputs, thus overcoming this limitation.

Key Contributions

The authors present the following main contributions:

1. Novel Architecture - The Pointer Network architecture uses a softmax probability distribution as a pointer to select input elements, effectively representing variable-length output dictionaries.
2. Application to Geometric Problems - The Ptr-Net is applied to three distinct combinatorial optimization problems: computing planar convex hulls, Delaunay triangulations, and solving the planar Traveling Salesman Problem (TSP). The model demonstrates the capability to generalize beyond the maximum lengths seen during training, showing promising performance on test problems with a greater number of points.
3. Empirical Evaluation - The paper provides an exhaustive empirical evaluation of the Ptr-Net model, demonstrating its superior performance over the baseline seq2seq models, particularly with input attention mechanisms.

Model Architecture

Sequence-to-Sequence Model with Attention

In common seq2seq models, a recurrent neural network (RNN) processes input sequences and a distinct RNN decodes the embeddings into output sequences. However, this approach is limited by the fixed size of the output dictionary. Bahdanau et al.'s attention mechanism improves this by allowing the decoder to attend to different parts of the input sequence. Nonetheless, it still assumes fixed output sizes.

Pointer Network (Ptr-Net)

Ptr-Net advances the use of attention by employing it as a pointer mechanism to select items from the input sequence. This refines the attention framework to directly output indices of the input elements, thus elegantly handling variable-length output dictionaries. The decoder RNN in Ptr-Net generates a context vector that the attention mechanism uses to produce a probability distribution over input indices at each decoding step.

Datasets and Problems

The paper details the application of Ptr-Net in tackling three geometric problems, each characterized by different output structure complexities:

1. Convex Hull - Given a set of planar points, the task is to identify the outermost points forming the convex polygon. Ptr-Net achieves high accuracy and area coverage, outperforming traditional seq2seq models.
2. Delaunay Triangulation - This task involves triangulating a set of points such that no point lies inside the circumcircle of any triangle. Ptr-Net exhibits robust performance in generating valid triangulations.
3. Traveling Salesman Problem (TSP) - The objective here is to find the shortest tour that visits each city exactly once. Ptr-Net not only handles variable input sizes but also provides competitive solutions even for larger problem instances.

Results

The empirical results highlight the efficacy of Ptr-Net:

• Convex Hull: Ptr-Net achieves 72.6% accuracy and 99.9% area coverage for $n=50$. Impressively, it generalizes to $n=500$ with a coverage of 99.2%.
• Delaunay Triangulation: The model maintains high triangle coverage, proving its utility in generating approximate triangulations.
• Traveling Salesman Problem: Ptr-Net trained on smaller instances (up to $n=20$) generalizes to larger instances, striking a balance between exact and approximate methods in solving the TSP.

Implications and Future Work

The Pointer Network's ability to model problems naturally involving variable output dictionaries has significant implications for the field of neural network applications in combinatorial optimization. Beyond geometric problems, Ptr-Nets are potentially applicable to diverse tasks requiring variable-length outputs, such as sorting and other optimization problems. Future research could explore extending Ptr-Nets to integrate with other advanced neural architectures like Neural Turing Machines and Memory Networks, further broadening their applicability and enhancing their problem-solving capabilities.

In conclusion, Ptr-Net represents an important step towards solving discrete problems with variable-size outputs. It stands as a promising tool for complex and computationally intensive tasks that traditionally posed challenges to existing neural network models.