Code Architecture — Pointer Networks Notebook

Overview

The notebook implements a Pointer Network (Ptr-Net) from scratch in PyTorch, trains it to solve small convex-hull problems, and visualizes the pointer attention path over input points. The model uses an LSTM encoder-decoder architecture where the attention mechanism acts as a pointer — at each decoding step, the attention distribution over encoder hidden states directly produces the probability of selecting each input point as the next output. No fixed output vocabulary is used; the “vocabulary” is dynamically the set of input points.

Section-by-Section Breakdown

1. Setup & Imports

• PyTorch, NumPy, Matplotlib
• Device detection (GPU if available)
• Reproducibility seeding

2. Synthetic Data Generation (generate_convex_hull_example)

Generates random 2D point sets and computes their convex hull using the monotone-chain algorithm. The output is the sequence of indices (into the input array) that form the hull boundary in clockwise/counterclockwise order.

Key functions:

• generate_convex_hull_example(n_points, coord_range=10): Returns (points, hull_indices) where points is an (n_points, 2) array of random floats and hull_indices is a list of indices forming the convex hull.
• generate_dataset(n_samples, min_points, max_points): Creates a dataset of varying-length point sets with their hull solutions.

Data shapes:

• Input points: (n_points, 2) — float coordinates
• Target output: (hull_length,) — integer indices into the input array
• Note: hull length varies per example (depends on point distribution)

3. Input Representation (PointSequence dataset)

• Each point (x, y) is represented as a 2D vector fed to the encoder LSTM
• Special start-of-sequence (SOS) and end-of-sequence (EOS) tokens are added to the decoder input/output
• Sequences are padded to the maximum length in each batch for efficient batching
• A separate index n_points is reserved as the EOS token position

Key class: PointSequenceDataset(torch.utils.data.Dataset)

• __getitem__(i): returns (points_tensor, hull_indices_tensor, n_points)
• Padding masks are used to handle variable-length sequences

4. Pointer Network Model (PointerNet)

The core model: an LSTM encoder + LSTM decoder with pointer attention.

Key class: PointerNet(nn.Module)

Components:

• Encoder: nn.LSTM(input_size=2, hidden_size=hidden_dim, batch_first=True) — processes the input point sequence. Outputs encoder hidden states e_1, ..., e_n of shape (batch, n_points, hidden_dim).
• Decoder: nn.LSTM(input_size=2, hidden_size=hidden_dim, batch_first=True) — generates output positions. At each step, receives the previously selected point (or SOS token for the first step).
• Pointer attention: Two linear projections W_s (decoder state) and W_e (encoder state) map to a shared attention space. The score for encoder state j at decoder step t is: u_j = v^T tanh(W_s s_t + W_e e_j), normalized via softmax over input positions. A mask excludes padding positions.

Key methods:

• forward(points, target_hull, n_points, teacher_forcing_ratio=1.0):
  1. Encode input points → encoder hidden states (batch, max_n, hidden_dim)
  2. Initialize decoder with SOS token
  3. For each decoder step (up to max output length):
     • Compute attention scores over all encoder positions
     • Apply padding mask
     • softmax → pointer probability distribution (batch, max_n)
     • If teacher forcing: next decoder input = points[batch, target[step]]
     • Else: next decoder input = points[batch, argmax(pointer_probs)]
  4. Return pointer_logits (list of (batch, max_n) tensors) and predicted indices

Architecture parameters:

• Hidden dimension: 128 (paper uses 512)
• Input dimension: 2 (x, y coordinates)
• LSTM layers: 1 (paper uses 1 for encoder, 1 for decoder)
• Maximum input length: 50 points (same as paper’s training range)

5. Training Loop

• Loss: Cross-entropy loss over the pointer distribution at each decoder step. The target is the index of the correct hull point. EOS is handled with a special index.
• Optimizer: Adam, learning rate 1e-3 (paper uses Adam with lr scheduling)
• Teacher forcing: Always on during training (paper uses teacher forcing)
• Batch size: 128 (paper uses 128)
• Epochs: 50 (paper trains for ~1000 epochs; reduced for GPU runtime)
• Curriculum: Training data includes point sets of sizes 5–50 (paper uses 5–50)

Training data shapes:

• Batch of points: (batch, max_n, 2)
• Batch of hull indices: (batch, max_hull_len) — padded
• Lengths tensor: (batch,) — actual number of points per example

6. Evaluation

• Generate test point sets of various sizes (including sizes larger than training max)
• Measure accuracy: percentage of correctly predicted hull sequences (exact match)
• Measure point-level accuracy: percentage of individual hull indices correctly predicted
• Report generalization to unseen lengths (e.g., train on 5–30, test on 35–50)

7. Pointer Path Visualization

• For sample test cases, plot the 2D point cloud with the predicted hull and true hull overlaid
• Visualize the attention distribution at each decoder step as a heatmap over input points
• Show the pointer “path”: which points were selected in order, with arrows connecting them
• Highlight mismatches between predicted and true hull

8. Sorting Task (Bonus)

• As a secondary task, apply the same Pointer Network to sort sequences of numbers
• Input: a sequence of random numbers; output: the indices that would sort them
• Demonstrates the generality of the pointer mechanism beyond geometric problems

Data Flow

Input points (n, 2)  →  [LSTM Encoder]  →  Encoder hidden states (n, hidden_dim)
                                                        ↓
SOS token (1, 2)  →  [LSTM Decoder]  →  Decoder state s_t (1, hidden_dim)
                                                        ↓
                              [Pointer Attention: u_j = v^T tanh(W_s s_t + W_e e_j)]
                                                        ↓
                              softmax(u_j) → pointer distribution (n,)
                                                        ↓
                              argmax → predicted index → points[index] → next decoder input
                                                        ↓
                              Repeat until EOS predicted
                                                        ↓
                              Output: sequence of indices [i_1, i_2, ..., i_k]

Deliberate Simplifications vs Full Paper

Aspect	Paper	This Notebook
Tasks	Convex hull, Delaunay triangulation, TSP	Convex hull + sorting
Hidden dimension	512	128
Training range	5–50 points	5–50 points (same)
Test generalization	Up to 500–1000 points	Up to 80 points
LSTM layers	1 (encoder + decoder)	1 (same)
Epochs	~1000	50
Beam search	Beam size 5 at inference	Greedy decoding (argmax)
Attention mechanism	Bahdanau additive attention	Bahdanau additive (same)
Data size	10M training examples	~10K training examples
TSP	Full TSP with near-optimal comparison	Not included (focus on convex hull)
Delaunay	Triangle prediction with 3-index output	Not included
Curriculum learning	Not used	Not used (same)
Evaluation	Exact sequence match + point accuracy	Same metrics

The simplifications keep the notebook self-contained and runnable within a Kaggle GPU session (~15 min) while preserving the core architecture: LSTM encoder-decoder with pointer attention, variable-length output dictionary, teacher-forced training, and generalization to longer sequences.