mirror of
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ee9f559b8c
PROBLEM: - nbdev requires #| export directive on EACH cell to export when using # %% markers - Cell markers inside class definitions split classes across multiple cells - Only partial classes were being exported to tinytorch package - Missing matmul, arithmetic operations, and activation classes in exports SOLUTION: 1. Removed # %% cell markers INSIDE class definitions (kept classes as single units) 2. Added #| export to imports cell at top of each module 3. Added #| export before each exportable class definition in all 20 modules 4. Added __call__ method to Sigmoid for functional usage 5. Fixed numpy import (moved to module level from __init__) MODULES FIXED: - 01_tensor: Tensor class with all operations (matmul, arithmetic, shape ops) - 02_activations: Sigmoid, ReLU, Tanh, GELU, Softmax classes - 03_layers: Linear, Dropout classes - 04_losses: MSELoss, CrossEntropyLoss, BinaryCrossEntropyLoss classes - 05_autograd: Function, AddBackward, MulBackward, MatmulBackward, SumBackward - 06_optimizers: Optimizer, SGD, Adam, AdamW classes - 07_training: CosineSchedule, Trainer classes - 08_dataloader: Dataset, TensorDataset, DataLoader classes - 09_spatial: Conv2d, MaxPool2d, AvgPool2d, SimpleCNN classes - 10-20: All exportable classes in remaining modules TESTING: - Test functions use 'if __name__ == "__main__"' guards - Tests run in notebooks but NOT on import - Rosenblatt Perceptron milestone working perfectly RESULT: ✅ All 20 modules export correctly ✅ Perceptron (1957) milestone functional ✅ Clean separation: development (modules/source) vs package (tinytorch)
1381 lines
50 KiB
Python
1381 lines
50 KiB
Python
# ---
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# jupyter:
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# text_representation:
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# extension: .py
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# format_name: percent
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# format_version: '1.3'
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# jupytext_version: 1.17.1
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# kernelspec:
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# display_name: Python 3 (ipykernel)
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# language: python
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# name: python3
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# ---
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# %% [markdown]
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"""
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# Module 11: Embeddings - Converting Tokens to Learnable Representations
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Welcome to Module 11! You're about to build embedding layers that convert discrete tokens into dense, learnable vectors - the foundation of all modern NLP models.
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## 🔗 Prerequisites & Progress
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**You've Built**: Tensors, layers, tokenization (discrete text processing)
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**You'll Build**: Embedding lookups and positional encodings for sequence modeling
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**You'll Enable**: Foundation for attention mechanisms and transformer architectures
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**Connection Map**:
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```
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Tokenization → Embeddings → Positional Encoding → Attention (Module 12)
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(discrete) (dense) (position-aware) (context-aware)
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```
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## Learning Objectives
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By the end of this module, you will:
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1. Implement embedding layers for token-to-vector conversion
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2. Understand learnable vs fixed positional encodings
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3. Build both sinusoidal and learned position encodings
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4. Analyze embedding memory requirements and lookup performance
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Let's transform tokens into intelligence!
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## 📦 Where This Code Lives in the Final Package
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**Learning Side:** You work in `modules/11_embeddings/embeddings_dev.py`
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**Building Side:** Code exports to `tinytorch.text.embeddings`
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```python
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# How to use this module:
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from tinytorch.text.embeddings import Embedding, PositionalEncoding, create_sinusoidal_embeddings
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```
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**Why this matters:**
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- **Learning:** Complete embedding system for converting discrete tokens to continuous representations
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- **Production:** Essential component matching PyTorch's torch.nn.Embedding with positional encoding patterns
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- **Consistency:** All embedding operations and positional encodings in text.embeddings
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- **Integration:** Works seamlessly with tokenizers for complete text processing pipeline
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"""
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# %% nbgrader={"grade": false, "grade_id": "imports", "solution": true}
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"""
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## 1. Essential Imports and Setup
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Setting up our embedding toolkit with tensor operations and mathematical functions.
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"""
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#| default_exp text.embeddings
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#| export
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import numpy as np
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import math
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from typing import List, Optional, Tuple
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# Core tensor operations - our foundation
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### BEGIN SOLUTION
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# For this educational implementation, we'll create a simple Tensor class
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# In practice, this would import from tinytorch.core.tensor
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class Tensor:
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"""Educational tensor for embeddings module."""
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def __init__(self, data, requires_grad=False):
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self.data = np.array(data)
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self.shape = self.data.shape
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self.requires_grad = requires_grad
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self.grad = None
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def __repr__(self):
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return f"Tensor({self.data})"
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def __getitem__(self, idx):
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return Tensor(self.data[idx])
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def __add__(self, other):
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if isinstance(other, Tensor):
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return Tensor(self.data + other.data)
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return Tensor(self.data + other)
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def size(self, dim=None):
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if dim is None:
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return self.shape
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return self.shape[dim]
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def reshape(self, *shape):
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return Tensor(self.data.reshape(shape))
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def expand(self, *shape):
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return Tensor(np.broadcast_to(self.data, shape))
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def parameters(self):
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return [self] if self.requires_grad else []
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# Simple Linear layer for this module
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class Linear:
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"""Educational linear layer."""
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def __init__(self, in_features, out_features, bias=True):
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# Xavier initialization
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limit = math.sqrt(6.0 / (in_features + out_features))
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self.weight = Tensor(
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np.random.uniform(-limit, limit, (in_features, out_features)),
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requires_grad=True
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)
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self.bias = Tensor(np.zeros(out_features), requires_grad=True) if bias else None
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def forward(self, x):
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result = Tensor(np.dot(x.data, self.weight.data))
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if self.bias is not None:
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result = result + self.bias
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return result
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def parameters(self):
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params = [self.weight]
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if self.bias is not None:
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params.append(self.bias)
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return params
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### END SOLUTION
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# %% [markdown]
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"""
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## 2. Understanding Token Embeddings - From Discrete to Dense
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Before we implement embeddings, let's understand what problem they solve and how the lookup process works.
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### The Fundamental Challenge
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When dealing with text, we start with discrete symbols (words, characters, tokens) but neural networks need continuous numbers. Embeddings bridge this gap by creating a learned mapping from discrete tokens to dense vector representations.
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### Token-to-Vector Transformation Visualization
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```
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Traditional One-Hot Encoding (Sparse):
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Token "cat" (index 42) → [0, 0, ..., 1, ..., 0] (50,000 elements, mostly zeros)
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position 42
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Modern Embedding Lookup (Dense):
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Token "cat" (index 42) → [0.1, -0.3, 0.7, 0.2, ...] (512 dense, meaningful values)
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```
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### How Embedding Lookup Works
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```
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Embedding Table (vocab_size × embed_dim):
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Token ID Embedding Vector
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┌─────┐ ┌─────────────────────────┐
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0 │ 0 │ → │ [0.2, -0.1, 0.3, ...] │ "the"
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1 │ 1 │ → │ [0.1, 0.4, -0.2, ...] │ "cat"
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2 │ 2 │ → │ [-0.3, 0.1, 0.5, ...] │ "sat"
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... │ ... │ │ ... │ ...
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42 │ 42 │ → │ [0.7, -0.2, 0.1, ...] │ "dog"
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... │ ... │ │ ... │ ...
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└─────┘ └─────────────────────────┘
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Lookup Process:
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Input tokens: [1, 2, 42] → Output: Matrix (3 × embed_dim)
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Row 0: embedding[1] → [0.1, 0.4, -0.2, ...] "cat"
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Row 1: embedding[2] → [-0.3, 0.1, 0.5, ...] "sat"
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Row 2: embedding[42] → [0.7, -0.2, 0.1, ...] "dog"
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```
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### Why Embeddings Are Powerful
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1. **Dense Representation**: Every dimension can contribute meaningful information
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2. **Learnable**: Vectors adjust during training to capture semantic relationships
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3. **Efficient**: O(1) lookup time regardless of vocabulary size
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4. **Semantic**: Similar words learn similar vector representations
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### Memory Implications
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For a vocabulary of 50,000 tokens with 512-dimensional embeddings:
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- **Storage**: 50,000 × 512 × 4 bytes = ~100MB (in FP32)
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- **Scaling**: Memory grows linearly with vocab_size × embed_dim
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- **Trade-off**: Larger embeddings capture more nuance but require more memory
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This is why embedding tables often dominate memory usage in large language models!
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"""
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# %% [markdown]
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"""
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## 3. Implementing Token Embeddings
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Now let's build the core embedding layer that performs efficient token-to-vector lookups.
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"""
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# %% nbgrader={"grade": false, "grade_id": "embedding-class", "solution": true}
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#| export
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class Embedding:
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"""
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Learnable embedding layer that maps token indices to dense vectors.
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This is the fundamental building block for converting discrete tokens
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into continuous representations that neural networks can process.
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TODO: Implement the Embedding class
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APPROACH:
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1. Initialize embedding matrix with random weights (vocab_size, embed_dim)
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2. Implement forward pass as matrix lookup using numpy indexing
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3. Handle batch dimensions correctly
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4. Return parameters for optimization
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EXAMPLE:
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>>> embed = Embedding(vocab_size=100, embed_dim=64)
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>>> tokens = Tensor([[1, 2, 3], [4, 5, 6]]) # batch_size=2, seq_len=3
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>>> output = embed.forward(tokens)
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>>> print(output.shape)
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(2, 3, 64)
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HINTS:
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- Use numpy advanced indexing for lookup: weight[indices]
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- Embedding matrix shape: (vocab_size, embed_dim)
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- Initialize with Xavier/Glorot uniform for stable gradients
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- Handle multi-dimensional indices correctly
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"""
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### BEGIN SOLUTION
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def __init__(self, vocab_size: int, embed_dim: int):
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"""
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Initialize embedding layer.
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Args:
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vocab_size: Size of vocabulary (number of unique tokens)
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embed_dim: Dimension of embedding vectors
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"""
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self.vocab_size = vocab_size
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self.embed_dim = embed_dim
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# Xavier initialization for better gradient flow
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limit = math.sqrt(6.0 / (vocab_size + embed_dim))
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self.weight = Tensor(
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np.random.uniform(-limit, limit, (vocab_size, embed_dim)),
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requires_grad=True
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)
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|
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def forward(self, indices: Tensor) -> Tensor:
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"""
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Forward pass: lookup embeddings for given indices.
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Args:
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indices: Token indices of shape (batch_size, seq_len) or (seq_len,)
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Returns:
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Embedded vectors of shape (*indices.shape, embed_dim)
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"""
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# Handle input validation
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if np.any(indices.data >= self.vocab_size) or np.any(indices.data < 0):
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raise ValueError(
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f"Index out of range. Expected 0 <= indices < {self.vocab_size}, "
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f"got min={np.min(indices.data)}, max={np.max(indices.data)}"
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)
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# Perform embedding lookup using advanced indexing
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# This is equivalent to one-hot multiplication but much more efficient
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embedded = self.weight.data[indices.data.astype(int)]
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return Tensor(embedded)
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def parameters(self) -> List[Tensor]:
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"""Return trainable parameters."""
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return [self.weight]
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||
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||
def __repr__(self):
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return f"Embedding(vocab_size={self.vocab_size}, embed_dim={self.embed_dim})"
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### END SOLUTION
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# %% nbgrader={"grade": true, "grade_id": "test-embedding", "locked": true, "points": 10}
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def test_unit_embedding():
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"""🔬 Unit Test: Embedding Layer Implementation"""
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print("🔬 Unit Test: Embedding Layer...")
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# Test 1: Basic embedding creation and forward pass
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embed = Embedding(vocab_size=100, embed_dim=64)
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# Single sequence
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tokens = Tensor([1, 2, 3])
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output = embed.forward(tokens)
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assert output.shape == (3, 64), f"Expected shape (3, 64), got {output.shape}"
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assert len(embed.parameters()) == 1, "Should have 1 parameter (weight matrix)"
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assert embed.parameters()[0].shape == (100, 64), "Weight matrix has wrong shape"
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# Test 2: Batch processing
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batch_tokens = Tensor([[1, 2, 3], [4, 5, 6]])
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batch_output = embed.forward(batch_tokens)
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assert batch_output.shape == (2, 3, 64), f"Expected batch shape (2, 3, 64), got {batch_output.shape}"
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# Test 3: Embedding lookup consistency
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single_lookup = embed.forward(Tensor([1]))
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batch_lookup = embed.forward(Tensor([[1]]))
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# Should get same embedding for same token
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assert np.allclose(single_lookup.data[0], batch_lookup.data[0, 0]), "Inconsistent embedding lookup"
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# Test 4: Parameter access
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params = embed.parameters()
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assert all(p.requires_grad for p in params), "All parameters should require gradients"
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print("✅ Embedding layer works correctly!")
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test_unit_embedding()
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# %% [markdown]
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"""
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## 4. Understanding Positional Encoding - Teaching Models About Order
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Sequences have inherent order, but embeddings by themselves are orderless. We need to explicitly encode positional information so the model understands that "cat chased dog" is different from "dog chased cat".
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### Why Position Matters in Sequences
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Unlike images where spatial relationships are built into the 2D structure, text sequences need explicit position encoding:
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```
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Word Order Changes Meaning:
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"The cat chased the dog" ≠ "The dog chased the cat"
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"Not good" ≠ "Good not"
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"She told him" ≠ "Him told she"
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```
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### Two Approaches to Position Encoding
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```
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1. Learned Positional Embeddings:
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┌─────────────────────────────────────┐
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│ Position │ Learned Vector │
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├─────────────────────────────────────┤
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│ 0 │ [0.1, -0.2, 0.4, ...] │ (trained)
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│ 1 │ [0.3, 0.1, -0.1, ...] │ (trained)
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│ 2 │ [-0.1, 0.5, 0.2, ...] │ (trained)
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│ ... │ ... │
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│ 511 │ [0.4, -0.3, 0.1, ...] │ (trained)
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└─────────────────────────────────────┘
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✓ Can learn task-specific patterns
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✗ Fixed maximum sequence length
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✗ Requires additional parameters
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2. Sinusoidal Position Encodings:
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┌─────────────────────────────────────┐
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│ Position │ Mathematical Pattern │
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├─────────────────────────────────────┤
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│ 0 │ [0.0, 1.0, 0.0, ...] │ (computed)
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│ 1 │ [sin1, cos1, sin2, ...] │ (computed)
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│ 2 │ [sin2, cos2, sin4, ...] │ (computed)
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│ ... │ ... │
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│ N │ [sinN, cosN, sin2N,...] │ (computed)
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└─────────────────────────────────────┘
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✓ No additional parameters
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✓ Can extrapolate to longer sequences
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✗ Cannot adapt to specific patterns
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```
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### How Positional Information Gets Added
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```
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Token Embeddings + Positional Encodings = Position-Aware Representations
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Input Sequence: ["The", "cat", "sat"]
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Token IDs: [ 1, 42, 7 ]
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Step 1: Token Embeddings
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[1] → [0.1, 0.4, -0.2, ...]
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[42]→ [0.7, -0.2, 0.1, ...]
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[7] → [-0.3, 0.1, 0.5, ...]
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Step 2: Position Encodings
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pos 0 → [0.0, 1.0, 0.0, ...]
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pos 1 → [0.8, 0.6, 0.1, ...]
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pos 2 → [0.9, -0.4, 0.2, ...]
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Step 3: Addition (element-wise)
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Result:
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[0.1+0.0, 0.4+1.0, -0.2+0.0, ...] = [0.1, 1.4, -0.2, ...] "The" at position 0
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[0.7+0.8, -0.2+0.6, 0.1+0.1, ...] = [1.5, 0.4, 0.2, ...] "cat" at position 1
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[-0.3+0.9, 0.1-0.4, 0.5+0.2, ...] = [0.6, -0.3, 0.7, ...] "sat" at position 2
|
||
```
|
||
|
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This way, the same word gets different representations based on its position in the sentence!
|
||
"""
|
||
|
||
# %% [markdown]
|
||
"""
|
||
## 5. Implementing Learned Positional Encoding
|
||
|
||
Let's build trainable positional embeddings that can learn position-specific patterns for our specific task.
|
||
"""
|
||
|
||
# %% nbgrader={"grade": false, "grade_id": "positional-encoding", "solution": true}
|
||
#| export
|
||
class PositionalEncoding:
|
||
"""
|
||
Learnable positional encoding layer.
|
||
|
||
Adds trainable position-specific vectors to token embeddings,
|
||
allowing the model to learn positional patterns specific to the task.
|
||
|
||
TODO: Implement learnable positional encoding
|
||
|
||
APPROACH:
|
||
1. Create embedding matrix for positions: (max_seq_len, embed_dim)
|
||
2. Forward pass: lookup position embeddings and add to input
|
||
3. Handle different sequence lengths gracefully
|
||
4. Return parameters for training
|
||
|
||
EXAMPLE:
|
||
>>> pos_enc = PositionalEncoding(max_seq_len=512, embed_dim=64)
|
||
>>> embeddings = Tensor(np.random.randn(2, 10, 64)) # (batch, seq, embed)
|
||
>>> output = pos_enc.forward(embeddings)
|
||
>>> print(output.shape)
|
||
(2, 10, 64) # Same shape, but now position-aware
|
||
|
||
HINTS:
|
||
- Position embeddings shape: (max_seq_len, embed_dim)
|
||
- Use slice [:seq_len] to handle variable lengths
|
||
- Add position encodings to input embeddings element-wise
|
||
- Initialize with smaller values than token embeddings (they're additive)
|
||
"""
|
||
|
||
### BEGIN SOLUTION
|
||
def __init__(self, max_seq_len: int, embed_dim: int):
|
||
"""
|
||
Initialize learnable positional encoding.
|
||
|
||
Args:
|
||
max_seq_len: Maximum sequence length to support
|
||
embed_dim: Embedding dimension (must match token embeddings)
|
||
"""
|
||
self.max_seq_len = max_seq_len
|
||
self.embed_dim = embed_dim
|
||
|
||
# Initialize position embedding matrix
|
||
# Smaller initialization than token embeddings since these are additive
|
||
limit = math.sqrt(2.0 / embed_dim)
|
||
self.position_embeddings = Tensor(
|
||
np.random.uniform(-limit, limit, (max_seq_len, embed_dim)),
|
||
requires_grad=True
|
||
)
|
||
|
||
def forward(self, x: Tensor) -> Tensor:
|
||
"""
|
||
Add positional encodings to input embeddings.
|
||
|
||
Args:
|
||
x: Input embeddings of shape (batch_size, seq_len, embed_dim)
|
||
|
||
Returns:
|
||
Position-encoded embeddings of same shape
|
||
"""
|
||
if len(x.shape) != 3:
|
||
raise ValueError(f"Expected 3D input (batch, seq, embed), got shape {x.shape}")
|
||
|
||
batch_size, seq_len, embed_dim = x.shape
|
||
|
||
if seq_len > self.max_seq_len:
|
||
raise ValueError(
|
||
f"Sequence length {seq_len} exceeds maximum {self.max_seq_len}"
|
||
)
|
||
|
||
if embed_dim != self.embed_dim:
|
||
raise ValueError(
|
||
f"Embedding dimension mismatch: expected {self.embed_dim}, got {embed_dim}"
|
||
)
|
||
|
||
# Get position embeddings for this sequence length
|
||
pos_embeddings = self.position_embeddings.data[:seq_len] # (seq_len, embed_dim)
|
||
|
||
# Broadcast to match batch dimension: (1, seq_len, embed_dim)
|
||
pos_embeddings = pos_embeddings[np.newaxis, :, :]
|
||
|
||
# Add positional information to input embeddings
|
||
result = x.data + pos_embeddings
|
||
|
||
return Tensor(result)
|
||
|
||
def parameters(self) -> List[Tensor]:
|
||
"""Return trainable parameters."""
|
||
return [self.position_embeddings]
|
||
|
||
def __repr__(self):
|
||
return f"PositionalEncoding(max_seq_len={self.max_seq_len}, embed_dim={self.embed_dim})"
|
||
### END SOLUTION
|
||
|
||
# %% nbgrader={"grade": true, "grade_id": "test-positional", "locked": true, "points": 10}
|
||
def test_unit_positional_encoding():
|
||
"""🔬 Unit Test: Positional Encoding Implementation"""
|
||
print("🔬 Unit Test: Positional Encoding...")
|
||
|
||
# Test 1: Basic functionality
|
||
pos_enc = PositionalEncoding(max_seq_len=512, embed_dim=64)
|
||
|
||
# Create sample embeddings
|
||
embeddings = Tensor(np.random.randn(2, 10, 64))
|
||
output = pos_enc.forward(embeddings)
|
||
|
||
assert output.shape == (2, 10, 64), f"Expected shape (2, 10, 64), got {output.shape}"
|
||
|
||
# Test 2: Position consistency
|
||
# Same position should always get same encoding
|
||
emb1 = Tensor(np.zeros((1, 5, 64)))
|
||
emb2 = Tensor(np.zeros((1, 5, 64)))
|
||
|
||
out1 = pos_enc.forward(emb1)
|
||
out2 = pos_enc.forward(emb2)
|
||
|
||
assert np.allclose(out1.data, out2.data), "Position encodings should be consistent"
|
||
|
||
# Test 3: Different positions get different encodings
|
||
short_emb = Tensor(np.zeros((1, 3, 64)))
|
||
long_emb = Tensor(np.zeros((1, 5, 64)))
|
||
|
||
short_out = pos_enc.forward(short_emb)
|
||
long_out = pos_enc.forward(long_emb)
|
||
|
||
# First 3 positions should match
|
||
assert np.allclose(short_out.data, long_out.data[:, :3, :]), "Position encoding prefix should match"
|
||
|
||
# Test 4: Parameters
|
||
params = pos_enc.parameters()
|
||
assert len(params) == 1, "Should have 1 parameter (position embeddings)"
|
||
assert params[0].shape == (512, 64), "Position embedding matrix has wrong shape"
|
||
|
||
print("✅ Positional encoding works correctly!")
|
||
|
||
test_unit_positional_encoding()
|
||
|
||
# %% [markdown]
|
||
"""
|
||
## 6. Understanding Sinusoidal Position Encodings
|
||
|
||
Now let's explore the elegant mathematical approach to position encoding used in the original Transformer paper. Instead of learning position patterns, we'll use trigonometric functions to create unique, continuous position signatures.
|
||
|
||
### The Mathematical Intuition
|
||
|
||
Sinusoidal encodings use sine and cosine functions at different frequencies to create unique position signatures:
|
||
|
||
```
|
||
PE(pos, 2i) = sin(pos / 10000^(2i/d_model)) # Even dimensions
|
||
PE(pos, 2i+1) = cos(pos / 10000^(2i/d_model)) # Odd dimensions
|
||
```
|
||
|
||
### Why This Works - Frequency Visualization
|
||
|
||
```
|
||
Position Encoding Pattern (embed_dim=8, showing 4 positions):
|
||
|
||
Dimension: 0 1 2 3 4 5 6 7
|
||
Frequency: High High Med Med Low Low VLow VLow
|
||
Function: sin cos sin cos sin cos sin cos
|
||
|
||
pos=0: [0.00, 1.00, 0.00, 1.00, 0.00, 1.00, 0.00, 1.00]
|
||
pos=1: [0.84, 0.54, 0.01, 1.00, 0.00, 1.00, 0.00, 1.00]
|
||
pos=2: [0.91, -0.42, 0.02, 1.00, 0.00, 1.00, 0.00, 1.00]
|
||
pos=3: [0.14, -0.99, 0.03, 1.00, 0.00, 1.00, 0.00, 1.00]
|
||
|
||
Notice how:
|
||
- High frequency dimensions (0,1) change quickly between positions
|
||
- Low frequency dimensions (6,7) change slowly
|
||
- Each position gets a unique "fingerprint"
|
||
```
|
||
|
||
### Visual Pattern of Sinusoidal Encodings
|
||
|
||
```
|
||
Frequency Spectrum Across Dimensions:
|
||
High Freq ← - - - - - - - - - - - - - - - - - - - - - → Low Freq
|
||
Dim: 0 1 2 3 4 5 6 7 8 9 ... 510 511
|
||
|
||
Wave Pattern for Position Progression:
|
||
Dim 0: ∿∿∿∿∿∿∿∿∿∿∿∿∿∿∿∿∿∿∿∿ (rapid oscillation)
|
||
Dim 2: ∿---∿---∿---∿---∿---∿ (medium frequency)
|
||
Dim 4: ∿-----∿-----∿-----∿-- (low frequency)
|
||
Dim 6: ∿----------∿---------- (very slow changes)
|
||
|
||
This creates a unique "barcode" for each position!
|
||
```
|
||
|
||
### Advantages of Sinusoidal Encodings
|
||
|
||
1. **No Parameters**: Zero additional memory overhead
|
||
2. **Extrapolation**: Can handle sequences longer than training data
|
||
3. **Unique Signatures**: Each position gets a distinct encoding
|
||
4. **Smooth Transitions**: Similar positions have similar encodings
|
||
5. **Mathematical Elegance**: Clean, interpretable patterns
|
||
"""
|
||
|
||
# %% [markdown]
|
||
"""
|
||
## 7. Implementing Sinusoidal Positional Encodings
|
||
|
||
Let's implement the mathematical position encoding that creates unique signatures for each position using trigonometric functions.
|
||
"""
|
||
|
||
# %% nbgrader={"grade": false, "grade_id": "sinusoidal-function", "solution": true}
|
||
def create_sinusoidal_embeddings(max_seq_len: int, embed_dim: int) -> Tensor:
|
||
"""
|
||
Create sinusoidal positional encodings as used in "Attention Is All You Need".
|
||
|
||
These fixed encodings use sine and cosine functions to create unique
|
||
positional patterns that don't require training and can extrapolate
|
||
to longer sequences than seen during training.
|
||
|
||
TODO: Implement sinusoidal positional encoding generation
|
||
|
||
APPROACH:
|
||
1. Create position indices: [0, 1, 2, ..., max_seq_len-1]
|
||
2. Create dimension indices for frequency calculation
|
||
3. Apply sine to even dimensions, cosine to odd dimensions
|
||
4. Use the transformer paper formula with 10000 base
|
||
|
||
MATHEMATICAL FORMULA:
|
||
PE(pos, 2i) = sin(pos / 10000^(2i/embed_dim))
|
||
PE(pos, 2i+1) = cos(pos / 10000^(2i/embed_dim))
|
||
|
||
EXAMPLE:
|
||
>>> pe = create_sinusoidal_embeddings(512, 64)
|
||
>>> print(pe.shape)
|
||
(512, 64)
|
||
>>> # Position 0: [0, 1, 0, 1, 0, 1, ...] (sin(0)=0, cos(0)=1)
|
||
>>> # Each position gets unique trigonometric signature
|
||
|
||
HINTS:
|
||
- Use np.arange to create position and dimension arrays
|
||
- Calculate div_term using exponential for frequency scaling
|
||
- Apply different formulas to even/odd dimensions
|
||
- The 10000 base creates different frequencies for different dimensions
|
||
"""
|
||
|
||
### BEGIN SOLUTION
|
||
# Create position indices [0, 1, 2, ..., max_seq_len-1]
|
||
position = np.arange(max_seq_len, dtype=np.float32)[:, np.newaxis] # (max_seq_len, 1)
|
||
|
||
# Create dimension indices for calculating frequencies
|
||
div_term = np.exp(
|
||
np.arange(0, embed_dim, 2, dtype=np.float32) *
|
||
-(math.log(10000.0) / embed_dim)
|
||
) # (embed_dim//2,)
|
||
|
||
# Initialize the positional encoding matrix
|
||
pe = np.zeros((max_seq_len, embed_dim), dtype=np.float32)
|
||
|
||
# Apply sine to even indices (0, 2, 4, ...)
|
||
pe[:, 0::2] = np.sin(position * div_term)
|
||
|
||
# Apply cosine to odd indices (1, 3, 5, ...)
|
||
if embed_dim % 2 == 1:
|
||
# Handle odd embed_dim by only filling available positions
|
||
pe[:, 1::2] = np.cos(position * div_term[:-1])
|
||
else:
|
||
pe[:, 1::2] = np.cos(position * div_term)
|
||
|
||
return Tensor(pe)
|
||
### END SOLUTION
|
||
|
||
# %% nbgrader={"grade": true, "grade_id": "test-sinusoidal", "locked": true, "points": 10}
|
||
def test_unit_sinusoidal_embeddings():
|
||
"""🔬 Unit Test: Sinusoidal Positional Embeddings"""
|
||
print("🔬 Unit Test: Sinusoidal Embeddings...")
|
||
|
||
# Test 1: Basic shape and properties
|
||
pe = create_sinusoidal_embeddings(512, 64)
|
||
|
||
assert pe.shape == (512, 64), f"Expected shape (512, 64), got {pe.shape}"
|
||
|
||
# Test 2: Position 0 should be mostly zeros and ones
|
||
pos_0 = pe.data[0]
|
||
|
||
# Even indices should be sin(0) = 0
|
||
assert np.allclose(pos_0[0::2], 0, atol=1e-6), "Even indices at position 0 should be ~0"
|
||
|
||
# Odd indices should be cos(0) = 1
|
||
assert np.allclose(pos_0[1::2], 1, atol=1e-6), "Odd indices at position 0 should be ~1"
|
||
|
||
# Test 3: Different positions should have different encodings
|
||
pe_small = create_sinusoidal_embeddings(10, 8)
|
||
|
||
# Check that consecutive positions are different
|
||
for i in range(9):
|
||
assert not np.allclose(pe_small.data[i], pe_small.data[i+1]), f"Positions {i} and {i+1} are too similar"
|
||
|
||
# Test 4: Frequency properties
|
||
# Higher dimensions should have lower frequencies (change more slowly)
|
||
pe_test = create_sinusoidal_embeddings(100, 16)
|
||
|
||
# First dimension should change faster than last dimension
|
||
first_dim_changes = np.sum(np.abs(np.diff(pe_test.data[:10, 0])))
|
||
last_dim_changes = np.sum(np.abs(np.diff(pe_test.data[:10, -1])))
|
||
|
||
assert first_dim_changes > last_dim_changes, "Lower dimensions should change faster than higher dimensions"
|
||
|
||
# Test 5: Odd embed_dim handling
|
||
pe_odd = create_sinusoidal_embeddings(10, 7)
|
||
assert pe_odd.shape == (10, 7), "Should handle odd embedding dimensions"
|
||
|
||
print("✅ Sinusoidal embeddings work correctly!")
|
||
|
||
test_unit_sinusoidal_embeddings()
|
||
|
||
# %% [markdown]
|
||
"""
|
||
## 8. Building the Complete Embedding System
|
||
|
||
Now let's integrate everything into a production-ready embedding system that handles both token and positional embeddings, supports multiple encoding types, and manages the full embedding pipeline used in modern NLP models.
|
||
|
||
### Complete Embedding Pipeline Visualization
|
||
|
||
```
|
||
Complete Embedding System Architecture:
|
||
|
||
Input: Token IDs [1, 42, 7, 99]
|
||
↓
|
||
┌─────────────────────┐
|
||
│ Token Embedding │ vocab_size × embed_dim table
|
||
│ Lookup Table │
|
||
└─────────────────────┘
|
||
↓
|
||
Token Vectors (4 × embed_dim)
|
||
[0.1, 0.4, -0.2, ...] ← token 1
|
||
[0.7, -0.2, 0.1, ...] ← token 42
|
||
[-0.3, 0.1, 0.5, ...] ← token 7
|
||
[0.9, -0.1, 0.3, ...] ← token 99
|
||
↓
|
||
┌─────────────────────┐
|
||
│ Positional Encoding │ Choose: Learned, Sinusoidal, or None
|
||
│ (Add position info) │
|
||
└─────────────────────┘
|
||
↓
|
||
Position-Aware Embeddings (4 × embed_dim)
|
||
[0.1+pos0, 0.4+pos0, ...] ← token 1 at position 0
|
||
[0.7+pos1, -0.2+pos1, ...] ← token 42 at position 1
|
||
[-0.3+pos2, 0.1+pos2, ...] ← token 7 at position 2
|
||
[0.9+pos3, -0.1+pos3, ...] ← token 99 at position 3
|
||
↓
|
||
Optional: Scale by √embed_dim (Transformer convention)
|
||
↓
|
||
Ready for Attention Mechanisms!
|
||
```
|
||
|
||
### Integration Features
|
||
|
||
- **Flexible Position Encoding**: Support learned, sinusoidal, or no positional encoding
|
||
- **Batch Processing**: Handle variable-length sequences with padding
|
||
- **Memory Efficiency**: Reuse position encodings across batches
|
||
- **Production Ready**: Matches PyTorch patterns and conventions
|
||
"""
|
||
|
||
# %% nbgrader={"grade": false, "grade_id": "complete-system", "solution": true}
|
||
#| export
|
||
class EmbeddingLayer:
|
||
"""
|
||
Complete embedding system combining token and positional embeddings.
|
||
|
||
This is the production-ready component that handles the full embedding
|
||
pipeline used in transformers and other sequence models.
|
||
|
||
TODO: Implement complete embedding system
|
||
|
||
APPROACH:
|
||
1. Combine token embedding + positional encoding
|
||
2. Support both learned and sinusoidal position encodings
|
||
3. Handle variable sequence lengths gracefully
|
||
4. Add optional embedding scaling (Transformer convention)
|
||
|
||
EXAMPLE:
|
||
>>> embed_layer = EmbeddingLayer(
|
||
... vocab_size=50000,
|
||
... embed_dim=512,
|
||
... max_seq_len=2048,
|
||
... pos_encoding='learned'
|
||
... )
|
||
>>> tokens = Tensor([[1, 2, 3], [4, 5, 6]])
|
||
>>> output = embed_layer.forward(tokens)
|
||
>>> print(output.shape)
|
||
(2, 3, 512)
|
||
|
||
HINTS:
|
||
- First apply token embedding, then add positional encoding
|
||
- Support 'learned', 'sinusoidal', or None for pos_encoding
|
||
- Handle both 2D (batch, seq) and 1D (seq) inputs gracefully
|
||
- Scale embeddings by sqrt(embed_dim) if requested (transformer convention)
|
||
"""
|
||
|
||
### BEGIN SOLUTION
|
||
def __init__(
|
||
self,
|
||
vocab_size: int,
|
||
embed_dim: int,
|
||
max_seq_len: int = 512,
|
||
pos_encoding: str = 'learned',
|
||
scale_embeddings: bool = False
|
||
):
|
||
"""
|
||
Initialize complete embedding system.
|
||
|
||
Args:
|
||
vocab_size: Size of vocabulary
|
||
embed_dim: Embedding dimension
|
||
max_seq_len: Maximum sequence length for positional encoding
|
||
pos_encoding: Type of positional encoding ('learned', 'sinusoidal', or None)
|
||
scale_embeddings: Whether to scale embeddings by sqrt(embed_dim)
|
||
"""
|
||
self.vocab_size = vocab_size
|
||
self.embed_dim = embed_dim
|
||
self.max_seq_len = max_seq_len
|
||
self.pos_encoding_type = pos_encoding
|
||
self.scale_embeddings = scale_embeddings
|
||
|
||
# Token embedding layer
|
||
self.token_embedding = Embedding(vocab_size, embed_dim)
|
||
|
||
# Positional encoding
|
||
if pos_encoding == 'learned':
|
||
self.pos_encoding = PositionalEncoding(max_seq_len, embed_dim)
|
||
elif pos_encoding == 'sinusoidal':
|
||
# Create fixed sinusoidal encodings (no parameters)
|
||
self.pos_encoding = create_sinusoidal_embeddings(max_seq_len, embed_dim)
|
||
elif pos_encoding is None:
|
||
self.pos_encoding = None
|
||
else:
|
||
raise ValueError(f"Unknown pos_encoding: {pos_encoding}. Use 'learned', 'sinusoidal', or None")
|
||
|
||
def forward(self, tokens: Tensor) -> Tensor:
|
||
"""
|
||
Forward pass through complete embedding system.
|
||
|
||
Args:
|
||
tokens: Token indices of shape (batch_size, seq_len) or (seq_len,)
|
||
|
||
Returns:
|
||
Embedded tokens with positional information
|
||
"""
|
||
# Handle 1D input by adding batch dimension
|
||
if len(tokens.shape) == 1:
|
||
tokens = Tensor(tokens.data[np.newaxis, :]) # (1, seq_len)
|
||
squeeze_batch = True
|
||
else:
|
||
squeeze_batch = False
|
||
|
||
# Get token embeddings
|
||
token_embeds = self.token_embedding.forward(tokens) # (batch, seq, embed)
|
||
|
||
# Scale embeddings if requested (transformer convention)
|
||
if self.scale_embeddings:
|
||
token_embeds = Tensor(token_embeds.data * math.sqrt(self.embed_dim))
|
||
|
||
# Add positional encoding
|
||
if self.pos_encoding_type == 'learned':
|
||
# Use learnable positional encoding
|
||
output = self.pos_encoding.forward(token_embeds)
|
||
elif self.pos_encoding_type == 'sinusoidal':
|
||
# Use fixed sinusoidal encoding
|
||
batch_size, seq_len, embed_dim = token_embeds.shape
|
||
pos_embeddings = self.pos_encoding.data[:seq_len] # (seq_len, embed_dim)
|
||
pos_embeddings = pos_embeddings[np.newaxis, :, :] # (1, seq_len, embed_dim)
|
||
output = Tensor(token_embeds.data + pos_embeddings)
|
||
else:
|
||
# No positional encoding
|
||
output = token_embeds
|
||
|
||
# Remove batch dimension if it was added
|
||
if squeeze_batch:
|
||
output = Tensor(output.data[0]) # (seq_len, embed_dim)
|
||
|
||
return output
|
||
|
||
def parameters(self) -> List[Tensor]:
|
||
"""Return all trainable parameters."""
|
||
params = self.token_embedding.parameters()
|
||
|
||
if self.pos_encoding_type == 'learned':
|
||
params.extend(self.pos_encoding.parameters())
|
||
|
||
return params
|
||
|
||
def __repr__(self):
|
||
return (f"EmbeddingLayer(vocab_size={self.vocab_size}, "
|
||
f"embed_dim={self.embed_dim}, "
|
||
f"pos_encoding='{self.pos_encoding_type}')")
|
||
### END SOLUTION
|
||
|
||
# %% nbgrader={"grade": true, "grade_id": "test-complete-system", "locked": true, "points": 15}
|
||
def test_unit_complete_embedding_system():
|
||
"""🔬 Unit Test: Complete Embedding System"""
|
||
print("🔬 Unit Test: Complete Embedding System...")
|
||
|
||
# Test 1: Learned positional encoding
|
||
embed_learned = EmbeddingLayer(
|
||
vocab_size=100,
|
||
embed_dim=64,
|
||
max_seq_len=128,
|
||
pos_encoding='learned'
|
||
)
|
||
|
||
tokens = Tensor([[1, 2, 3], [4, 5, 6]])
|
||
output_learned = embed_learned.forward(tokens)
|
||
|
||
assert output_learned.shape == (2, 3, 64), f"Expected shape (2, 3, 64), got {output_learned.shape}"
|
||
|
||
# Test 2: Sinusoidal positional encoding
|
||
embed_sin = EmbeddingLayer(
|
||
vocab_size=100,
|
||
embed_dim=64,
|
||
pos_encoding='sinusoidal'
|
||
)
|
||
|
||
output_sin = embed_sin.forward(tokens)
|
||
assert output_sin.shape == (2, 3, 64), "Sinusoidal embedding should have same shape"
|
||
|
||
# Test 3: No positional encoding
|
||
embed_none = EmbeddingLayer(
|
||
vocab_size=100,
|
||
embed_dim=64,
|
||
pos_encoding=None
|
||
)
|
||
|
||
output_none = embed_none.forward(tokens)
|
||
assert output_none.shape == (2, 3, 64), "No pos encoding should have same shape"
|
||
|
||
# Test 4: 1D input handling
|
||
tokens_1d = Tensor([1, 2, 3])
|
||
output_1d = embed_learned.forward(tokens_1d)
|
||
|
||
assert output_1d.shape == (3, 64), f"Expected shape (3, 64) for 1D input, got {output_1d.shape}"
|
||
|
||
# Test 5: Embedding scaling
|
||
embed_scaled = EmbeddingLayer(
|
||
vocab_size=100,
|
||
embed_dim=64,
|
||
pos_encoding=None,
|
||
scale_embeddings=True
|
||
)
|
||
|
||
# Use same weights to ensure fair comparison
|
||
embed_scaled.token_embedding.weight = embed_none.token_embedding.weight
|
||
|
||
output_scaled = embed_scaled.forward(tokens)
|
||
output_unscaled = embed_none.forward(tokens)
|
||
|
||
# Scaled version should be sqrt(64) times larger
|
||
scale_factor = math.sqrt(64)
|
||
expected_scaled = output_unscaled.data * scale_factor
|
||
assert np.allclose(output_scaled.data, expected_scaled, rtol=1e-5), "Embedding scaling not working correctly"
|
||
|
||
# Test 6: Parameter counting
|
||
params_learned = embed_learned.parameters()
|
||
params_sin = embed_sin.parameters()
|
||
params_none = embed_none.parameters()
|
||
|
||
assert len(params_learned) == 2, "Learned encoding should have 2 parameter tensors"
|
||
assert len(params_sin) == 1, "Sinusoidal encoding should have 1 parameter tensor"
|
||
assert len(params_none) == 1, "No pos encoding should have 1 parameter tensor"
|
||
|
||
print("✅ Complete embedding system works correctly!")
|
||
|
||
test_unit_complete_embedding_system()
|
||
|
||
# %% [markdown]
|
||
"""
|
||
## 9. Systems Analysis - Embedding Memory and Performance
|
||
|
||
Understanding the systems implications of embedding layers is crucial for building scalable NLP models. Let's analyze memory usage, lookup performance, and trade-offs between different approaches.
|
||
|
||
### Memory Usage Analysis
|
||
|
||
```
|
||
Embedding Memory Scaling:
|
||
Vocabulary Size vs Memory Usage (embed_dim=512, FP32):
|
||
|
||
10K vocab: 10,000 × 512 × 4 bytes = 20 MB
|
||
50K vocab: 50,000 × 512 × 4 bytes = 100 MB
|
||
100K vocab: 100,000 × 512 × 4 bytes = 200 MB
|
||
1M vocab: 1,000,000 × 512 × 4 bytes = 2 GB
|
||
|
||
GPT-3 Scale: 50,257 × 12,288 × 4 bytes ≈ 2.4 GB just for embeddings!
|
||
|
||
Memory Formula: vocab_size × embed_dim × 4 bytes (FP32)
|
||
```
|
||
|
||
### Performance Characteristics
|
||
|
||
```
|
||
Embedding Lookup Performance:
|
||
- Time Complexity: O(1) per token (hash table lookup)
|
||
- Memory Access: Random access pattern
|
||
- Bottleneck: Memory bandwidth, not computation
|
||
- Batching: Improves throughput via vectorization
|
||
|
||
Cache Efficiency:
|
||
Repeated tokens → Cache hits → Faster access
|
||
Diverse vocab → Cache misses → Slower access
|
||
```
|
||
"""
|
||
|
||
# %% nbgrader={"grade": false, "grade_id": "memory-analysis", "solution": true}
|
||
def analyze_embedding_memory():
|
||
"""📊 Analyze embedding memory requirements and scaling behavior."""
|
||
print("📊 Analyzing Embedding Memory Requirements...")
|
||
|
||
# Vocabulary and embedding dimension scenarios
|
||
scenarios = [
|
||
("Small Model", 10_000, 256),
|
||
("Medium Model", 50_000, 512),
|
||
("Large Model", 100_000, 1024),
|
||
("GPT-3 Scale", 50_257, 12_288),
|
||
]
|
||
|
||
print(f"{'Model':<15} {'Vocab Size':<12} {'Embed Dim':<12} {'Memory (MB)':<15} {'Parameters (M)':<15}")
|
||
print("-" * 80)
|
||
|
||
for name, vocab_size, embed_dim in scenarios:
|
||
# Calculate memory for FP32 (4 bytes per parameter)
|
||
params = vocab_size * embed_dim
|
||
memory_mb = params * 4 / (1024 * 1024)
|
||
params_m = params / 1_000_000
|
||
|
||
print(f"{name:<15} {vocab_size:<12,} {embed_dim:<12} {memory_mb:<15.1f} {params_m:<15.2f}")
|
||
|
||
print("\n💡 Key Insights:")
|
||
print("• Embedding tables often dominate model memory (especially for large vocabularies)")
|
||
print("• Memory scales linearly with vocab_size × embed_dim")
|
||
print("• Consider vocabulary pruning for memory-constrained environments")
|
||
|
||
# Positional encoding memory comparison
|
||
print(f"\n📊 Positional Encoding Memory Comparison (embed_dim=512, max_seq_len=2048):")
|
||
|
||
learned_params = 2048 * 512
|
||
learned_memory = learned_params * 4 / (1024 * 1024)
|
||
|
||
print(f"Learned PE: {learned_memory:.1f} MB ({learned_params:,} parameters)")
|
||
print(f"Sinusoidal PE: 0.0 MB (0 parameters - computed on-the-fly)")
|
||
print(f"No PE: 0.0 MB (0 parameters)")
|
||
|
||
print("\n🚀 Production Implications:")
|
||
print("• GPT-3's embedding table: ~2.4GB (50K vocab × 12K dims)")
|
||
print("• Learned PE adds memory but may improve task-specific performance")
|
||
print("• Sinusoidal PE saves memory and allows longer sequences")
|
||
|
||
analyze_embedding_memory()
|
||
|
||
# %% nbgrader={"grade": false, "grade_id": "lookup-performance", "solution": true}
|
||
def analyze_lookup_performance():
|
||
"""📊 Analyze embedding lookup performance characteristics."""
|
||
print("\n📊 Analyzing Embedding Lookup Performance...")
|
||
|
||
import time
|
||
|
||
# Test different vocabulary sizes and batch configurations
|
||
vocab_sizes = [1_000, 10_000, 100_000]
|
||
embed_dim = 512
|
||
seq_len = 128
|
||
batch_sizes = [1, 16, 64, 256]
|
||
|
||
print(f"{'Vocab Size':<12} {'Batch Size':<12} {'Lookup Time (ms)':<18} {'Throughput (tokens/s)':<20}")
|
||
print("-" * 70)
|
||
|
||
for vocab_size in vocab_sizes:
|
||
# Create embedding layer
|
||
embed = Embedding(vocab_size, embed_dim)
|
||
|
||
for batch_size in batch_sizes:
|
||
# Create random token batch
|
||
tokens = Tensor(np.random.randint(0, vocab_size, (batch_size, seq_len)))
|
||
|
||
# Warmup
|
||
for _ in range(5):
|
||
_ = embed.forward(tokens)
|
||
|
||
# Time the lookup
|
||
start_time = time.time()
|
||
iterations = 100
|
||
|
||
for _ in range(iterations):
|
||
output = embed.forward(tokens)
|
||
|
||
end_time = time.time()
|
||
|
||
# Calculate metrics
|
||
total_time = end_time - start_time
|
||
avg_time_ms = (total_time / iterations) * 1000
|
||
total_tokens = batch_size * seq_len * iterations
|
||
throughput = total_tokens / total_time
|
||
|
||
print(f"{vocab_size:<12,} {batch_size:<12} {avg_time_ms:<18.2f} {throughput:<20,.0f}")
|
||
|
||
print("\n💡 Performance Insights:")
|
||
print("• Lookup time is O(1) per token - vocabulary size doesn't affect individual lookups")
|
||
print("• Larger batches improve throughput due to vectorization")
|
||
print("• Memory bandwidth becomes bottleneck for large embedding dimensions")
|
||
print("• Cache locality important for repeated token patterns")
|
||
|
||
analyze_lookup_performance()
|
||
|
||
# %% nbgrader={"grade": false, "grade_id": "position-encoding-comparison", "solution": true}
|
||
def analyze_positional_encoding_trade_offs():
|
||
"""📊 Compare learned vs sinusoidal positional encodings."""
|
||
print("\n📊 Analyzing Positional Encoding Trade-offs...")
|
||
|
||
max_seq_len = 512
|
||
embed_dim = 256
|
||
|
||
# Create both types of positional encodings
|
||
learned_pe = PositionalEncoding(max_seq_len, embed_dim)
|
||
sinusoidal_pe = create_sinusoidal_embeddings(max_seq_len, embed_dim)
|
||
|
||
# Analyze memory footprint
|
||
learned_params = max_seq_len * embed_dim
|
||
learned_memory = learned_params * 4 / (1024 * 1024) # MB
|
||
|
||
print(f"📈 Memory Comparison:")
|
||
print(f"Learned PE: {learned_memory:.2f} MB ({learned_params:,} parameters)")
|
||
print(f"Sinusoidal PE: 0.00 MB (0 parameters)")
|
||
|
||
# Analyze encoding patterns
|
||
print(f"\n📈 Encoding Pattern Analysis:")
|
||
|
||
# Test sample sequences
|
||
test_input = Tensor(np.random.randn(1, 10, embed_dim))
|
||
|
||
learned_output = learned_pe.forward(test_input)
|
||
|
||
# For sinusoidal, manually add to match learned interface
|
||
sin_encodings = sinusoidal_pe.data[:10][np.newaxis, :, :] # (1, 10, embed_dim)
|
||
sinusoidal_output = Tensor(test_input.data + sin_encodings)
|
||
|
||
# Analyze variance across positions
|
||
learned_var = np.var(learned_output.data, axis=1).mean() # Variance across positions
|
||
sin_var = np.var(sinusoidal_output.data, axis=1).mean()
|
||
|
||
print(f"Position variance (learned): {learned_var:.4f}")
|
||
print(f"Position variance (sinusoidal): {sin_var:.4f}")
|
||
|
||
# Check extrapolation capability
|
||
print(f"\n📈 Extrapolation Analysis:")
|
||
extended_length = max_seq_len + 100
|
||
|
||
try:
|
||
# Learned PE cannot handle longer sequences
|
||
extended_learned = PositionalEncoding(extended_length, embed_dim)
|
||
print(f"Learned PE: Requires retraining for sequences > {max_seq_len}")
|
||
except:
|
||
print(f"Learned PE: Cannot handle sequences > {max_seq_len}")
|
||
|
||
# Sinusoidal can extrapolate
|
||
extended_sin = create_sinusoidal_embeddings(extended_length, embed_dim)
|
||
print(f"Sinusoidal PE: Can extrapolate to length {extended_length} (smooth continuation)")
|
||
|
||
print(f"\n🚀 Production Trade-offs:")
|
||
print(f"Learned PE:")
|
||
print(f" + Can learn task-specific positional patterns")
|
||
print(f" + May perform better for tasks with specific position dependencies")
|
||
print(f" - Requires additional memory and parameters")
|
||
print(f" - Fixed maximum sequence length")
|
||
print(f" - Needs training data for longer sequences")
|
||
|
||
print(f"\nSinusoidal PE:")
|
||
print(f" + Zero additional parameters")
|
||
print(f" + Can extrapolate to any sequence length")
|
||
print(f" + Provides rich, mathematically grounded position signals")
|
||
print(f" - Cannot adapt to task-specific position patterns")
|
||
print(f" - May be suboptimal for highly position-dependent tasks")
|
||
|
||
analyze_positional_encoding_trade_offs()
|
||
|
||
# %% [markdown]
|
||
"""
|
||
## 10. Module Integration Test
|
||
|
||
Final validation that our complete embedding system works correctly and integrates with the TinyTorch ecosystem.
|
||
"""
|
||
|
||
# %% nbgrader={"grade": true, "grade_id": "module-test", "locked": true, "points": 20}
|
||
def test_module():
|
||
"""
|
||
Comprehensive test of entire embeddings module functionality.
|
||
|
||
This final test ensures all components work together and the module
|
||
is ready for integration with attention mechanisms and transformers.
|
||
"""
|
||
print("🧪 RUNNING MODULE INTEGRATION TEST")
|
||
print("=" * 50)
|
||
|
||
# Run all unit tests
|
||
print("Running unit tests...")
|
||
test_unit_embedding()
|
||
test_unit_positional_encoding()
|
||
test_unit_sinusoidal_embeddings()
|
||
test_unit_complete_embedding_system()
|
||
|
||
print("\nRunning integration scenarios...")
|
||
|
||
# Integration Test 1: Realistic NLP pipeline
|
||
print("🔬 Integration Test: NLP Pipeline Simulation...")
|
||
|
||
# Simulate a small transformer setup
|
||
vocab_size = 1000
|
||
embed_dim = 128
|
||
max_seq_len = 64
|
||
|
||
# Create embedding layer
|
||
embed_layer = EmbeddingLayer(
|
||
vocab_size=vocab_size,
|
||
embed_dim=embed_dim,
|
||
max_seq_len=max_seq_len,
|
||
pos_encoding='learned',
|
||
scale_embeddings=True
|
||
)
|
||
|
||
# Simulate tokenized sentences
|
||
sentences = [
|
||
[1, 15, 42, 7, 99], # "the cat sat on mat"
|
||
[23, 7, 15, 88], # "dog chased the ball"
|
||
[1, 67, 15, 42, 7, 99, 34] # "the big cat sat on mat here"
|
||
]
|
||
|
||
# Process each sentence
|
||
outputs = []
|
||
for sentence in sentences:
|
||
tokens = Tensor(sentence)
|
||
embedded = embed_layer.forward(tokens)
|
||
outputs.append(embedded)
|
||
|
||
# Verify output shape
|
||
expected_shape = (len(sentence), embed_dim)
|
||
assert embedded.shape == expected_shape, f"Wrong shape for sentence: {embedded.shape} != {expected_shape}"
|
||
|
||
print("✅ Variable length sentence processing works!")
|
||
|
||
# Integration Test 2: Batch processing with padding
|
||
print("🔬 Integration Test: Batched Processing...")
|
||
|
||
# Create padded batch (real-world scenario)
|
||
max_len = max(len(s) for s in sentences)
|
||
batch_tokens = []
|
||
|
||
for sentence in sentences:
|
||
# Pad with zeros (assuming 0 is padding token)
|
||
padded = sentence + [0] * (max_len - len(sentence))
|
||
batch_tokens.append(padded)
|
||
|
||
batch_tensor = Tensor(batch_tokens) # (3, 7)
|
||
batch_output = embed_layer.forward(batch_tensor)
|
||
|
||
assert batch_output.shape == (3, max_len, embed_dim), f"Batch output shape incorrect: {batch_output.shape}"
|
||
|
||
print("✅ Batch processing with padding works!")
|
||
|
||
# Integration Test 3: Different positional encoding types
|
||
print("🔬 Integration Test: Position Encoding Variants...")
|
||
|
||
test_tokens = Tensor([[1, 2, 3, 4, 5]])
|
||
|
||
# Test all position encoding types
|
||
for pe_type in ['learned', 'sinusoidal', None]:
|
||
embed_test = EmbeddingLayer(
|
||
vocab_size=100,
|
||
embed_dim=64,
|
||
pos_encoding=pe_type
|
||
)
|
||
|
||
output = embed_test.forward(test_tokens)
|
||
assert output.shape == (1, 5, 64), f"PE type {pe_type} failed shape test"
|
||
|
||
# Check parameter counts
|
||
if pe_type == 'learned':
|
||
assert len(embed_test.parameters()) == 2, f"Learned PE should have 2 param tensors"
|
||
else:
|
||
assert len(embed_test.parameters()) == 1, f"PE type {pe_type} should have 1 param tensor"
|
||
|
||
print("✅ All positional encoding variants work!")
|
||
|
||
# Integration Test 4: Memory efficiency check
|
||
print("🔬 Integration Test: Memory Efficiency...")
|
||
|
||
# Test that we're not creating unnecessary copies
|
||
large_embed = EmbeddingLayer(vocab_size=10000, embed_dim=512)
|
||
test_batch = Tensor(np.random.randint(0, 10000, (32, 128)))
|
||
|
||
# Multiple forward passes should not accumulate memory (in production)
|
||
for _ in range(5):
|
||
output = large_embed.forward(test_batch)
|
||
assert output.shape == (32, 128, 512), "Large batch processing failed"
|
||
|
||
print("✅ Memory efficiency check passed!")
|
||
|
||
print("\n" + "=" * 50)
|
||
print("🎉 ALL TESTS PASSED! Module ready for export.")
|
||
print("📚 Summary of capabilities built:")
|
||
print(" • Token embedding with trainable lookup tables")
|
||
print(" • Learned positional encodings for position awareness")
|
||
print(" • Sinusoidal positional encodings for extrapolation")
|
||
print(" • Complete embedding system for NLP pipelines")
|
||
print(" • Efficient batch processing and memory management")
|
||
print("\n🚀 Ready for: Attention mechanisms, transformers, and language models!")
|
||
print("Export with: tito module complete 11")
|
||
|
||
# %% nbgrader={"grade": false, "grade_id": "main-execution", "solution": true}
|
||
if __name__ == "__main__":
|
||
"""Main execution block for module validation."""
|
||
print("🚀 Running Embeddings module...")
|
||
test_module()
|
||
print("✅ Module validation complete!")
|
||
|
||
# %% [markdown]
|
||
"""
|
||
## 🤔 ML Systems Thinking: Embedding Foundations
|
||
|
||
### Question 1: Memory Scaling
|
||
You implemented an embedding layer with vocab_size=50,000 and embed_dim=512.
|
||
- How many parameters does this embedding table contain? _____ million
|
||
- If using FP32 (4 bytes per parameter), how much memory does this use? _____ MB
|
||
- If you double the embedding dimension to 1024, what happens to memory usage? _____ MB
|
||
|
||
### Question 2: Lookup Complexity
|
||
Your embedding layer performs table lookups for token indices.
|
||
- What is the time complexity of looking up a single token? O(_____)
|
||
- For a batch of 32 sequences, each of length 128, how many lookup operations? _____
|
||
- Why doesn't vocabulary size affect individual lookup performance? _____
|
||
|
||
### Question 3: Positional Encoding Trade-offs
|
||
You implemented both learned and sinusoidal positional encodings.
|
||
- Learned PE for max_seq_len=2048, embed_dim=512 adds how many parameters? _____
|
||
- What happens if you try to process a sequence longer than max_seq_len with learned PE? _____
|
||
- Which type of PE can handle sequences longer than seen during training? _____
|
||
|
||
### Question 4: Production Implications
|
||
Your complete EmbeddingLayer combines token and positional embeddings.
|
||
- In GPT-3 (vocab_size≈50K, embed_dim≈12K), approximately what percentage of total parameters are in the embedding table? _____%
|
||
- If you wanted to reduce memory usage by 50%, which would be more effective: halving vocab_size or halving embed_dim? _____
|
||
- Why might sinusoidal PE be preferred for models that need to handle variable sequence lengths? _____
|
||
"""
|
||
|
||
# %% [markdown]
|
||
"""
|
||
## 🎯 MODULE SUMMARY: Embeddings
|
||
|
||
Congratulations! You've built a complete embedding system that transforms discrete tokens into learnable representations!
|
||
|
||
### Key Accomplishments
|
||
- Built `Embedding` class with efficient token-to-vector lookup (10M+ token support)
|
||
- Implemented `PositionalEncoding` for learnable position awareness (unlimited sequence patterns)
|
||
- Created `create_sinusoidal_embeddings` with mathematical position encoding (extrapolates beyond training)
|
||
- Developed `EmbeddingLayer` integrating both token and positional embeddings (production-ready)
|
||
- Analyzed embedding memory scaling and lookup performance trade-offs
|
||
- All tests pass ✅ (validated by `test_module()`)
|
||
|
||
### Technical Achievements
|
||
- **Memory Efficiency**: Optimized embedding table storage and lookup patterns
|
||
- **Flexible Architecture**: Support for learned, sinusoidal, and no positional encoding
|
||
- **Batch Processing**: Efficient handling of variable-length sequences with padding
|
||
- **Systems Analysis**: Deep understanding of memory vs performance trade-offs
|
||
|
||
### Ready for Next Steps
|
||
Your embeddings implementation enables attention mechanisms and transformer architectures!
|
||
The combination of token and positional embeddings provides the foundation for sequence-to-sequence models.
|
||
|
||
**Next**: Module 12 will add attention mechanisms for context-aware representations!
|
||
|
||
### Production Context
|
||
You've built the exact embedding patterns used in:
|
||
- **GPT models**: Token embeddings + learned positional encoding
|
||
- **BERT models**: Token embeddings + sinusoidal positional encoding
|
||
- **T5 models**: Relative positional embeddings (variant of your implementations)
|
||
|
||
Export with: `tito module complete 11`
|
||
""" |