long context processing extrapolation

# Long Context Processing & Extrapolation

## Introduction & Motivation

Long context: extend context window beyond training length. Enable processing very long sequences. Applications: long documents, memory systems.

Motivation: Process documents longer than training context.

Applications: Legal documents, research papers, long conversations.

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## Core Concepts & Theory

### Context Extrapolation

Generalize beyond training length.

### Sparse Attention

Reduce quadratic complexity.

### Memory Mechanisms

Store long-term information.

### Compression

Summarize context.

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## Mathematical Formulation

Extrapolation Factor:
$$ ext{extrapolation} = \frac{ ext{test length}}{ ext{training length}}$$

Sparse Attention Complexity:
$$O(L \cdot k \log L) ext{ instead of } O(L^2)$$

Context Compression:
$$ ext{compressed} = ext{Summarize}( ext{context})$$

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## Advanced Theory & Extensions

### Hierarchical Context

Multi-level context representation.

### Retrieval-Based Context

Retrieve relevant passages.

### Segment-Level Recurrence

Process segments incrementally.

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## Computational Considerations

Standard attention: O(L²).

Sparse: O(L log L) or O(L).

Memory: Reduced with compression.

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## Practical Implementation Strategies

### Window Attention

Local attention windows.

### Strided Attention

Regular pattern access.

### Compression Strategy

Select important tokens.

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## Benchmark Datasets & Evaluation

Long Range Arena: Benchmark.

Wikitext: Long-form text.

PG-19: Books dataset.

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## Key Challenges & Limitations

### Capability Loss

Sparse attention hurts performance.

### Extrapolation Generalization

Limited to certain factors.

### Inference Efficiency

Sparse patterns overhead.

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## Hyperparameter Tuning

Window size: 512-2048.

Stride: 256-1024.

Memory size: 1k-100k tokens.

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## Real-World Applications & Case Studies

Legal Documents: Process contracts.

Research Papers: Analyze full papers.

Conversations: Long dialogue history.

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## Integration with Other Methods

Long context + sparse attention; + retrieval augmentation.

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## Summary & Key Takeaways

Long context processing extends model capabilities.

Principles:
1. Extrapolation: Generalize beyond training.
2. Sparse attention: Reduce complexity.
3. Compression: Summarize context.
4. Memory: Store long-term info.
5. Efficiency: Balance capability and speed.

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## Appendix: Practical Labs

### Lab 1: Strided Attention

import numpy as np

def strided_attention_mask(seq_len, stride=4):
 """Create strided attention pattern"""
 mask = np.zeros((seq_len, seq_len))
 
 for i in range(seq_len):
 # Attend to stride positions
 for j in range(i % stride, seq_len, stride):
 mask[i, j] = 1
 
 return mask

mask = strided_attention_mask(100, 4)
assert mask.sum() < 100 * 100
print(f"✓ Strided attention: {int(mask.sum())} active connections")

### Lab 2: Local Window Attention

import numpy as np

def local_window_attention(seq_len, window_size=16):
 """Local window attention pattern"""
 mask = np.zeros((seq_len, seq_len))
 
 for i in range(seq_len):
 start = max(0, i - window_size // 2)
 end = min(seq_len, i + window_size // 2)
 mask[i, start:end] = 1
 
 return mask

mask = local_window_attention(100, 16)
assert mask.sum() <= 100 * 16
print(f"✓ Local window attention: {int(mask.sum())} active")

### Lab 3: Context Compression

import numpy as np

def compress_context(context, compression_ratio=0.1):
 """Compress context to important tokens"""
 seq_len = len(context)
 num_keep = max(1, int(seq_len * compression_ratio))
 
 # Keep highest magnitude tokens
 importance = np.abs(context).mean(axis=1)
 keep_indices = np.argsort(importance)[-num_keep:]
 
 compressed = context[sorted(keep_indices)]
 return compressed

np.random.seed(42)
ctx = np.random.randn(1000, 768)
compressed = compress_context(ctx, 0.1)
assert len(compressed) <= 100
print(f"✓ Compression: {len(ctx)} → {len(compressed)}")

### Lab 4: Extrapolation Evaluation

import numpy as np

def evaluate_extrapolation(train_len, test_len, model_fn):
 """Evaluate extrapolation capability"""
 extrapolation_factor = test_len / train_len
 
 # Test on sequence longer than training
 test_seq = np.random.randn(test_len, 768)
 loss = model_fn(test_seq)
 
 return loss, extrapolation_factor

model = lambda x: np.random.rand()
loss, factor = evaluate_extrapolation(512, 2048, model)
assert factor == 4.0
print(f"✓ Extrapolation factor: {factor}x")

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