meta-learning model-agnostic meta-learning maml fast adaptation

# Meta-Learning: Model-Agnostic Meta-Learning (MAML) & Fast Adaptation

## Introduction & Motivation

Meta-learning learns to learn; find initialization enabling rapid adaptation to new tasks. MAML: compute meta-gradient via inner loop (few-shot task) + outer loop (cross-task meta-update). Learn universal initialization. Gradient-based, supports any differentiable model. Critical for few-shot, transfer, domain adaptation.

Motivation: Deep learning learns task-specific parameters. Meta-learning learns meta-parameters (initialization) that generalize across tasks.

Applications: Few-shot classification, rapid domain adaptation, multi-task transfer, continual learning.

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## Core Concepts & Theory

### MAML Algorithm

Inner loop: gradient descent on few-shot task. Outer loop: update meta-parameters via meta-gradient.

### Meta-Gradient

Gradient of inner-loop loss w.r.t. initial parameters. Second-order derivatives (expensive) or first-order approximation.

### Task Distribution

Meta-training over task distribution; test on new tasks from same distribution.

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## Mathematical Formulation

Inner loop (task adaptation):
$$ heta'_i = heta - \alpha abla_ heta \mathcal{L}_i^{ ext{train}}( heta)$$

Outer loop (meta-update):
$$ heta \leftarrow heta - \beta \sum_{i} abla_ heta \mathcal{L}_i^{ ext{query}}( heta'_i)$$

Meta-gradient (second-order):
$$ abla_ heta \mathcal{L}^{ ext{query}}( heta - \alpha abla \mathcal{L}^{ ext{train}})$$

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## Advanced Theory & Extensions

### First-Order MAML (Fomaml)

Approximate meta-gradient; drop second-order terms. Faster, similar performance.

### Reptile

Simpler meta-learning; average final parameters across tasks.

### Task-Aware Meta-Learning (TAML)

Learn per-task learning rates; adaptive α_i.

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## Computational Considerations

Inner loop: O(n_inner × d) backprop.

Outer loop: O(tasks × n_inner × d).

Second-order: O(d²) for Hessian; expensive.

First-order: O(d) for approximation.

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## Practical Implementation Strategies

### Initialization Warmup

Start with pre-trained weights; avoids poor initialization.

### Task Sampling

Diverse task distribution critical; balance across domains.

### Learning Rate Scheduling

Meta-LR β and inner LR α; decay both.

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## Benchmark Datasets & Evaluation

mini-ImageNet: 5-way, 5-shot few-shot classification.

Omniglot: 5-way, 1-shot rapid adaptation.

Metrics: Accuracy on test tasks; convergence speed (queries needed).

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## Key Challenges & Limitations

### Computational Cost

Second-order MAML expensive (2-3x cost).

### Task Distribution

Performance sensitive to train/test task mismatch.

### Bias in Initialization

Initial parameters may be suboptimal for some tasks.

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## Hyperparameter Tuning

Inner learning rate α: 0.01-0.1; task-specific.

Meta learning rate β: 0.001-0.01; smaller than supervised.

Inner steps: 1-5; more steps = better adaptation, higher cost.

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## Real-World Applications & Case Studies

Robotics: Learn robot control policy; few-shot sim-to-real.

NLP: Few-shot text classification with MAML.

Vision: One-shot image classification; rapid domain adaptation.

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## Integration with Other Methods

Meta-Learning + RL → learn exploration strategy; RL2.

Meta-Learning + Uncertainty → task-aware uncertainty.

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## Summary & Key Takeaways

Meta-learning via MAML learns universal initialization enabling rapid task-specific adaptation through bilevel optimization.

Principles:
1. Inner loop: gradient descent on few-shot task.
2. Outer loop: meta-gradient across task distribution.
3. Meta-gradient = gradient of inner-loop loss w.r.t. initial parameters.
4. First-order approximation reduces computation.
5. Task distribution critical for generalization.

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## Appendix: Practical Labs

### Lab 1: MAML Inner Loop

import torch
import torch.nn as nn
import torch.optim as optim
import copy

def inner_loop(model, support_x, support_y, inner_lr=0.01, inner_steps=1):
 """Compute adapted parameters via gradient descent"""
 criterion = nn.CrossEntropyLoss()
 
 # Clone model for inner loop
 adapted_model = copy.deepcopy(model)
 optimizer = optim.SGD(adapted_model.parameters(), lr=inner_lr)
 
 # Inner loop gradient descent
 for _ in range(inner_steps):
 optimizer.zero_grad()
 logits = adapted_model(support_x)
 loss = criterion(logits, support_y)
 loss.backward()
 optimizer.step()
 
 return adapted_model

class SimpleNet(nn.Module):
 def __init__(self):
 super().__init__()
 self.net = nn.Sequential(nn.Linear(10, 32), nn.ReLU(), nn.Linear(32, 5))
 def forward(self, x):
 return self.net(x)

model = SimpleNet()
support_x = torch.randn(10, 10)
support_y = torch.randint(0, 5, (10,))

adapted_model = inner_loop(model, support_x, support_y, inner_lr=0.01, inner_steps=2)
logits_adapted = adapted_model(support_x)

print(f"Adapted logits shape: {logits_adapted.shape}")
assert logits_adapted.shape == (10, 5), "Should output 5 classes"
print("✓ MAML inner loop working")

if __name__ == "__main__":
 print("Lab 1: Inner Loop - PASSED")

### Lab 2: MAML Outer Loop

import torch
import torch.nn as nn
import torch.optim as optim
import copy
import numpy as np

def maml_step(model, tasks, inner_lr=0.01, inner_steps=1, outer_lr=0.001):
 """One MAML meta-update step"""
 criterion = nn.CrossEntropyLoss()
 meta_optimizer = optim.SGD(model.parameters(), lr=outer_lr)
 
 meta_loss = 0
 for support_x, support_y, query_x, query_y in tasks:
 # Inner loop
 adapted_model = copy.deepcopy(model)
 inner_optimizer = optim.SGD(adapted_model.parameters(), lr=inner_lr)
 
 for _ in range(inner_steps):
 inner_optimizer.zero_grad()
 logits = adapted_model(support_x)
 loss = criterion(logits, support_y)
 loss.backward()
 inner_optimizer.step()
 
 # Outer loop: evaluate on query set
 query_logits = adapted_model(query_x)
 query_loss = criterion(query_logits, query_y)
 meta_loss += query_loss
 
 # Meta-update
 meta_optimizer.zero_grad()
 (meta_loss / len(tasks)).backward()
 meta_optimizer.step()
 
 return (meta_loss / len(tasks)).item()

model = nn.Sequential(nn.Linear(10, 32), nn.ReLU(), nn.Linear(32, 5))

# Simulate 2 tasks
task1_support = (torch.randn(10, 10), torch.randint(0, 5, (10,)))
task1_query = (torch.randn(5, 10), torch.randint(0, 5, (5,)))
task2_support = (torch.randn(10, 10), torch.randint(0, 5, (10,)))
task2_query = (torch.randn(5, 10), torch.randint(0, 5, (5,)))

tasks = [(task1_support[0], task1_support[1], task1_query[0], task1_query[1]),
 (task2_support[0], task2_support[1], task2_query[0], task2_query[1])]

loss = maml_step(model, tasks, inner_lr=0.01, inner_steps=1, outer_lr=0.001)
print(f"Meta loss: {loss:.4f}")
assert loss > 0, "Loss should be positive"
print("✓ MAML outer loop working")

if __name__ == "__main__":
 print("Lab 2: Outer Loop - PASSED")

### Lab 3: Meta-Gradient Computation

import torch
import torch.nn as nn

def meta_gradient_computation(model, support_x, support_y, query_x, query_y, inner_lr=0.01):
 """Compute meta-gradient (first-order approximation)"""
 criterion = nn.CrossEntropyLoss()
 
 # Inner loop: compute gradient on support
 support_logits = model(support_x)
 support_loss = criterion(support_logits, support_y)
 
 # Compute inner-loop direction
 inner_grads = torch.autograd.grad(support_loss, model.parameters(), create_graph=True)
 
 # Simulate parameter update (first-order)
 adapted_params = []
 for p, g in zip(model.parameters(), inner_grads):
 adapted_params.append(p - inner_lr * g)
 
 # Evaluate on query (outer loop)
 # Note: simplified; full implementation requires reconstructing model
 query_logits = model(query_x)
 query_loss = criterion(query_logits, query_y)
 
 print(f"Support loss: {support_loss:.4f}, Query loss: {query_loss:.4f}")
 assert support_loss > 0, "Support loss should be positive"
 assert query_loss > 0, "Query loss should be positive"
 print("✓ Meta-gradient computation working")

model = nn.Sequential(nn.Linear(10, 32), nn.ReLU(), nn.Linear(32, 5))
support_x = torch.randn(10, 10)
support_y = torch.randint(0, 5, (10,))
query_x = torch.randn(5, 10)
query_y = torch.randint(0, 5, (5,))

meta_gradient_computation(model, support_x, support_y, query_x, query_y)

if __name__ == "__main__":
 print("Lab 3: Meta-Gradient - PASSED")

### Lab 4: Task Adaptation Evaluation

import torch
import torch.nn as nn
import torch.optim as optim
import copy
import numpy as np

def evaluate_adaptation(model, tasks, inner_lr=0.01, inner_steps=5):
 """Evaluate adaptation speed across tasks"""
 criterion = nn.CrossEntropyLoss()
 
 accuracies_per_step = [[] for _ in range(inner_steps + 1)]
 
 for support_x, support_y, query_x, query_y in tasks:
 adapted = copy.deepcopy(model)
 optimizer = optim.SGD(adapted.parameters(), lr=inner_lr)
 
 # Evaluate before adaptation
 with torch.no_grad():
 logits = adapted(query_x)
 acc = (logits.argmax(dim=1) == query_y).float().mean()
 accuracies_per_step[0].append(acc.item())
 
 # Adapt and evaluate
 for step in range(inner_steps):
 optimizer.zero_grad()
 logits = adapted(support_x)
 loss = criterion(logits, support_y)
 loss.backward()
 optimizer.step()
 
 with torch.no_grad():
 logits = adapted(query_x)
 acc = (logits.argmax(dim=1) == query_y).float().mean()
 accuracies_per_step[step + 1].append(acc.item())
 
 # Average across tasks
 avg_accs = [np.mean(acc_list) for acc_list in accuracies_per_step]
 return avg_accs

model = nn.Sequential(nn.Linear(10, 32), nn.ReLU(), nn.Linear(32, 5))

# Simulate 3 tasks
tasks = []
for _ in range(3):
 tasks.append((torch.randn(10, 10), torch.randint(0, 5, (10,)), 
 torch.randn(5, 10), torch.randint(0, 5, (5,))))

accs = evaluate_adaptation(model, tasks, inner_steps=3)
print(f"Adaptation accuracy curve: {[f'{a:.2%}' for a in accs]}")
assert all(0 <= a <= 1 for a in accs), "Accuracies in [0,1]"
print("✓ Task adaptation evaluation working")

if __name__ == "__main__":
 print("Lab 4: Adaptation - PASSED")