Capacity And Generalization - Learning Module

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Double Descent

Challenging the Classical Narrative

For decades, the classical bias-variance tradeoff taught us a simple story: as model complexity increases, first test error decreases (reducing bias), then it increases (variance dominates). The optimal model lies at the sweet spot between underfitting and overfitting.

This U-shaped curve became gospel in machine learning education. But it told only part of the story.

In recent years, researchers discovered something remarkable: if you keep increasing model complexity past the overfitting peak, test error eventually decreases again. The classical U-curve is actually just the first half of a double-descent pattern.

This discovery, formalized and extensively studied around 2019-2020, fundamentally changes how we think about model capacity. It explains puzzling empirical observations—like why training massive neural networks to zero training error often works well—and provides theoretical grounding for practices that previously seemed like folklore.

What You Will Master

By the end of this page, you will understand: (1) the double descent phenomenon and its key regimes, (2) the interpolation threshold and why it matters, (3) model-wise, epoch-wise, and sample-wise double descent, and (4) the theoretical explanations and practical implications.

The Classical U-Curve Revisited

Let's first establish what the classical theory predicted, and where it was observed to hold.

1.1 The Traditional View

The classical bias-variance decomposition for test error:

$$\text{Test Error} = \text{Bias}^2 + \text{Variance} + \text{Noise}$$

As model complexity increases:

Bias decreases — The model can fit more complex patterns, reducing systematic error
Variance increases — The model becomes more sensitive to the specific training set
The sum first decreases, then increases, forming a U-shape

This was observed reliably in many settings: polynomial regression, kernel methods with fixed kernels, small neural networks. The advice was clear: find the elbow of the U-curve using validation data.

classical_u_curve.py
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"""
The Classical U-Curve: Where it holds
 
Demonstrating the traditional bias-variance tradeoff in polynomial regression.
"""
 
import numpy as np
from sklearn.preprocessing import PolynomialFeatures
from sklearn.linear_model import LinearRegression
from sklearn.model_selection import cross_val_score
import warnings
warnings.filterwarnings('ignore')
 
def true_function(x):
    """True underlying function (cubic polynomial)."""
    return 1 + 0.5*x - 0.3*x**2 + 0.1*x**3
 
def generate_data(n_samples=100, noise_std=0.5, seed=None):
    if seed is not None:
        np.random.seed(seed)
    X = np.random.uniform(-3, 3, n_samples).reshape(-1, 1)
    y = true_function(X.flatten()) + np.random.normal(0, noise_std, n_samples)
    return X, y
 
# Generate training and test data
np.random.seed(42)
X_train, y_train = generate_data(n_samples=30, noise_std=0.5)
X_test, y_test = generate_data(n_samples=1000, noise_std=0.5, seed=123)
 
print("Classical U-Curve: Polynomial Regression")
print("=" * 60)
print(f"Training samples: {len(y_train)}")
print(f"True function: cubic polynomial")
print()
 
degrees = list(range(1, 21))
train_errors = []
test_errors = []
 
for degree in degrees:
    # Fit polynomial
    poly = PolynomialFeatures(degree)
    X_train_poly = poly.fit_transform(X_train)
    X_test_poly = poly.transform(X_test)
    
    model = LinearRegression()
    model.fit(X_train_poly, y_train)
    
    # Compute errors
    train_pred = model.predict(X_train_poly)
    test_pred = model.predict(X_test_poly)
    
    train_mse = np.mean((train_pred - y_train)**2)
    test_mse = np.mean((test_pred - y_test)**2)
    
    train_errors.append(train_mse)
    test_errors.append(test_mse)
 
# Print results
print(f"{'Degree':<8} {'Train MSE':<12} {'Test MSE':<12} {'Regime'}")
print("-" * 50)
 
for i, degree in enumerate(degrees):
    if degree <= 3:
        regime = "Underfitting"
    elif degree <= 10:
        regime = "Good fit"
    else:
        regime = "Overfitting"
    
    print(f"{degree:<8} {train_errors[i]:<12.4f} {test_errors[i]:<12.4f} {regime}")
 
# Find optimal
best_degree = degrees[np.argmin(test_errors)]
print()
print(f"Optimal degree: {best_degree} (Test MSE: {min(test_errors):.4f})")
print()
print("OBSERVATIONS:")
print("  - Test error decreases as we reduce underfitting (degrees 1-3)")
print("  - Test error minimum around true complexity (degree 3-4)")
print("  - Test error increases with overfitting (degrees 5+)")
print("  This is the classical U-curve!")
print()
print("BUT NOTE: We stopped at degree 20 (which is < 30 samples)")
print("What happens beyond the interpolation threshold?")

1.2 Where the Classical View Breaks Down

The classical U-curve assumes we stay in a regime where the model cannot perfectly fit the training data. But what happens when model capacity exceeds the number of training samples?

The Interpolation Threshold:

When the number of parameters $p$ approaches the number of samples $n$, something profound happens:

At $p < n$: Model is underdetermined, training loss > 0
At $p = n$: Model can exactly interpolate training data (training loss → 0)
At $p > n$: Model is overparameterized, infinitely many interpolating solutions exist

The classical U-curve describes behavior up to $p \approx n$. But modern deep learning operates far in the $p >> n$ regime. What happens there?

The Gap in Classical Theory

Classical statistical learning theory was developed when computational constraints kept models small relative to data. The regime where parameters vastly outnumber samples was considered impractical. Deep learning shattered this assumption, operating in a regime where p/n ratios can exceed 1000x. Classical theory simply didn't address this regime.

The Double Descent Discovery

2.1 Early Observations

Scattered observations of double descent appeared as early as the 1990s (Opper, 1995; Advani & Saxe, 2017), but the phenomenon wasn't widely recognized until Belkin et al. (2019) and Nakkiran et al. (2019) systematically documented it.

Key Finding: When you continue increasing model complexity past the interpolation threshold, test error doesn't stay high—it decreases again, often reaching values lower than the first minimum!

This creates a characteristic double-descent curve:

Under-parameterized regime ($p << n$): Classical U-curve behavior
Interpolation threshold ($p \approx n$): Peak in test error
Over-parameterized regime ($p >> n$): Test error decreases again

2.2 Visualizing Double Descent

The double descent curve has a distinctive shape:

Test Error
    │
    │     Peak at interpolation
    │           ╱╲
    │          ╱  ╲
    │         ╱    ╲
    │    ╱╲  ╱      ╲
    │   ╱  ╲╱        ╲_____ Second descent
    │  ╱
    │ ╱ First descent
    │╱
    └─────────────────────────────►
                                Model Complexity (p/n)
                    ↑
            Interpolation threshold (p ≈ n)

The Three Regimes:

Regime	Condition	Behavior
Under-parameterized	p < n	Classical bias-variance tradeoff
Interpolation threshold	p ≈ n	Worst generalization (variance explosion)
Over-parameterized	p >> n	Good generalization (benign overfitting)

double_descent_demo.py
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"""
Double Descent Phenomenon
 
Demonstrating how test error has two descent phases as model complexity increases.
"""
 
import numpy as np
from sklearn.kernel_ridge import KernelRidge
from sklearn.preprocessing import StandardScaler
import warnings
warnings.filterwarnings('ignore')
 
def generate_regression_data(n_samples, n_features, noise_std=0.1, seed=None):
    """
    Generate regression data where only a few features are relevant.
    """
    if seed is not None:
        np.random.seed(seed)
    
    X = np.random.randn(n_samples, n_features)
    
    # True function: depends on first 5 features only
    true_weights = np.zeros(n_features)
    true_weights[:5] = np.random.randn(5)
    
    y = X @ true_weights + np.random.randn(n_samples) * noise_std
    return X, y, true_weights
 
def random_features_regression(X_train, y_train, X_test, n_features):
    """
    Random features regression: projects data to n_features dimensions
    using random Gaussian weights, then fits linear regression.
    
    This model class shows clear double descent behavior.
    """
    input_dim = X_train.shape[1]
    
    # Random projection
    np.random.seed(42)  # Fixed for reproducibility
    W = np.random.randn(input_dim, n_features) / np.sqrt(input_dim)
    
    # Apply random features with ReLU activation
    X_train_rf = np.maximum(X_train @ W, 0)
    X_test_rf = np.maximum(X_test @ W, 0)
    
    # Add small regularization near interpolation threshold for stability
    n_samples = X_train.shape[0]
    ratio = n_features / n_samples
    
    if 0.8 < ratio < 1.2:
        # Near interpolation threshold: use small ridge for numerical stability
        ridge = 1e-4
    elif ratio > 1.2:
        # Overparameterized: minimum norm solution
        ridge = 1e-8
    else:
        # Underparameterized: small ridge
        ridge = 1e-6
    
    # Solve with ridge
    if n_features <= n_samples:
        # Standard solution
        XtX = X_train_rf.T @ X_train_rf + ridge * np.eye(n_features)
        theta = np.linalg.solve(XtX, X_train_rf.T @ y_train)
    else:
        # Minimum norm solution for overparameterized case
        XXt = X_train_rf @ X_train_rf.T + ridge * np.eye(n_samples)
        alpha = np.linalg.solve(XXt, y_train)
        theta = X_train_rf.T @ alpha
    
    train_pred = X_train_rf @ theta
    test_pred = X_test_rf @ theta
    
    train_mse = np.mean((train_pred - y_train)**2)
    test_mse = np.mean((test_pred - y_test)**2)
    
    return train_mse, test_mse
 
# Setup
n_train = 100
n_test = 500
input_dim = 50
noise_std = 0.3
 
X_train, y_train, _ = generate_regression_data(n_train, input_dim, noise_std, seed=42)
X_test, y_test, _ = generate_regression_data(n_test, input_dim, noise_std, seed=123)
 
# Standardize
scaler = StandardScaler()
X_train = scaler.fit_transform(X_train)
X_test = scaler.transform(X_test)
 
# Vary number of random features (model complexity)
# Key: sweep through under-parameterized, interpolation, and over-parameterized
feature_counts = [10, 20, 30, 50, 70, 80, 90, 95, 100, 105, 110, 120, 150, 200, 300, 500, 1000]
 
print("Double Descent in Random Features Regression")
print("=" * 70)
print(f"Training samples: {n_train}")
print(f"Interpolation threshold: ~{n_train} features")
print()
 
results = []
for n_features in feature_counts:
    train_mse, test_mse = random_features_regression(X_train, y_train, X_test, n_features)
    ratio = n_features / n_train
    
    if ratio < 0.9:
        regime = "Under-param"
    elif ratio < 1.1:
        regime = "THRESHOLD"
    else:
        regime = "Over-param"
    
    results.append((n_features, ratio, train_mse, test_mse, regime))
 
print(f"{'Features':<10} {'p/n Ratio':<12} {'Train MSE':<12} {'Test MSE':<12} {'Regime'}")
print("-" * 60)
 
for n_feat, ratio, train_err, test_err, regime in results:
    print(f"{n_feat:<10} {ratio:<12.2f} {train_err:<12.4f} {test_err:<12.4f} {regime}")
 
# Find minima
test_errors = [r[3] for r in results]
ratios = [r[1] for r in results]
 
# First minimum (before threshold)
under_param = [(r, e) for r, e in zip(ratios, test_errors) if r < 0.9]
if under_param:
    first_min_ratio = min(under_param, key=lambda x: x[1])
    print(f"\nFirst minimum: p/n ≈ {first_min_ratio[0]:.2f} (Test MSE: {first_min_ratio[1]:.4f})")
 
# Find peak
threshold = [(r, e) for r, e in zip(ratios, test_errors) if 0.9 <= r <= 1.1]
if threshold:
    peak = max(threshold, key=lambda x: x[1])
    print(f"Peak at threshold: p/n ≈ {peak[0]:.2f} (Test MSE: {peak[1]:.4f})")
 
# Second minimum (after threshold)
over_param = [(r, e) for r, e in zip(ratios, test_errors) if r > 1.5]
if over_param:
    second_min = min(over_param, key=lambda x: x[1])
    print(f"Second descent: p/n ≈ {second_min[0]:.2f} (Test MSE: {second_min[1]:.4f})")
 
print("\n" + "=" * 70)
print("KEY OBSERVATIONS:")
print("1. Test error initially decreases, then peaks near interpolation")
print("2. Past interpolation, test error DECREASES again (double descent!)")
print("3. Heavily overparameterized models can generalize BETTER than")
print("   optimally-tuned underparameterized models")

Why the Peak at Interpolation?

At the interpolation threshold (p ≈ n), the model has just enough capacity to fit the training data exactly, but only ONE interpolating solution exists. This solution is highly sensitive to noise—small changes in training data cause large changes in the fit. The variance is maximal. Past the threshold, infinitely many interpolating solutions exist, and the optimization algorithm can select the 'best' one (e.g., minimum norm), which tends to generalize well.

Model-Wise Double Descent

Model-wise double descent refers to the phenomenon as we increase model size (number of parameters, layers, units, etc.) while keeping the training setup fixed.

3.1 In Neural Networks

Nakkiran et al. (2019) demonstrated model-wise double descent in practical deep learning:

ResNets on CIFAR-10:

Small ResNets (few layers/channels): Test error decreases with size
Medium ResNets (near interpolation): Test error peaks
Large ResNets (heavily overparameterized): Test error decreases again

Transformers on Language Tasks:

Similar pattern observed: test perplexity has double-descent shape
Larger models eventually outperform medium-sized ones, even without early stopping

3.2 The Practical Implication

This has profound implications for model selection:

Classical Wisdom: "Don't overfit! Find the optimal complexity."

Double Descent Wisdom: "If your model is 'overfitting,' either make it smaller OR much larger. The interpolation threshold is the worst place to be."

3.3 Why This Matters for Deep Learning

Modern deep learning operates squarely in the overparameterized regime:

Setting	Parameters (p)	Samples (n)	Ratio (p/n)
ResNet-50 on ImageNet	25M	1.2M	~20x
BERT-Large	340M	~3.3B tokens	~0.1x (but effectively higher)
GPT-3	175B	~300B tokens	~0.5x
Vision Transformer-Large	632M	~14M images	~45x

All of these are in the "second descent" regime. They work because they're overparameterized, not despite it.

The key insight: Overparameterization is not a bug—it's a feature. When combined with appropriate training procedures (SGD, appropriate loss functions), more parameters provide:

Easier optimization (more paths to good solutions)
Better implicit regularization
Lower effective capacity for the actual solution found

Double Descent vs. Just Training More

Double descent is distinct from simply training longer or tuning hyperparameters better. Even with optimal early stopping and regularization, the double descent pattern persists. It's a fundamental property of the model class, not an artifact of training procedure.

Epoch-Wise Double Descent

The double descent phenomenon isn't just about model size—it also appears across training time. Epoch-wise double descent shows a similar pattern as you train longer.

4.1 The Phenomenon

For a fixed model architecture, as training progresses:

Early epochs: Fitting begins, training loss drops, test loss drops
Effective interpolation point: Model starts fitting training noise, test loss increases
Late epochs: Model continues refining, test loss decreases again

This means the classical advice to "stop early to avoid overfitting" can actually prevent you from reaching the second descent!

epoch_wise_double_descent.py
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"""
Epoch-Wise Double Descent
 
Demonstrating how test error can decrease, increase, then decrease again
as training progresses.
"""
 
import numpy as np
import torch
import torch.nn as nn
from torch.utils.data import DataLoader, TensorDataset
 
class MLP(nn.Module):
    """Overparameterized MLP for demonstrating epoch-wise double descent."""
    def __init__(self, input_dim, hidden_dim, output_dim):
        super().__init__()
        self.layers = nn.Sequential(
            nn.Linear(input_dim, hidden_dim),
            nn.ReLU(),
            nn.Linear(hidden_dim, hidden_dim),
            nn.ReLU(),
            nn.Linear(hidden_dim, output_dim)
        )
    
    def forward(self, x):
        return self.layers(x)
 
def generate_noisy_classification_data(n_samples, n_features, noise_rate=0.1, seed=None):
    """
    Generate classification data with label noise.
    Label noise is key for observing epoch-wise double descent.
    """
    if seed is not None:
        np.random.seed(seed)
        torch.manual_seed(seed)
    
    # Generate data from two Gaussians
    X1 = np.random.randn(n_samples // 2, n_features) + 1
    X2 = np.random.randn(n_samples // 2, n_features) - 1
    X = np.vstack([X1, X2]).astype(np.float32)
    
    y = np.array([0] * (n_samples // 2) + [1] * (n_samples // 2))
    
    # Add label noise
    n_flip = int(noise_rate * n_samples)
    flip_idx = np.random.choice(n_samples, n_flip, replace=False)
    y[flip_idx] = 1 - y[flip_idx]
    
    return torch.FloatTensor(X), torch.LongTensor(y)
 
# Settings
n_train, n_test = 200, 500
n_features = 20
hidden_dim = 500  # Overparameterized
noise_rate = 0.15  # Label noise is important for epoch-wise double descent
n_epochs = 4000
 
X_train, y_train = generate_noisy_classification_data(n_train, n_features, noise_rate, seed=42)
X_test, y_test = generate_noisy_classification_data(n_test, n_features, noise_rate, seed=123)
 
train_loader = DataLoader(TensorDataset(X_train, y_train), batch_size=n_train, shuffle=True)
test_loader = DataLoader(TensorDataset(X_test, y_test), batch_size=n_test)
 
model = MLP(n_features, hidden_dim, 2)
criterion = nn.CrossEntropyLoss()
optimizer = torch.optim.SGD(model.parameters(), lr=0.01)
 
print("Epoch-Wise Double Descent")
print("=" * 70)
print(f"Training samples: {n_train}")
print(f"Model parameters: {sum(p.numel() for p in model.parameters())}")
print(f"Label noise rate: {noise_rate:.0%}")
print()
 
# Track metrics
epochs_to_record = [1, 10, 50, 100, 200, 500, 1000, 2000, 3000, 4000]
results = []
 
for epoch in range(1, n_epochs + 1):
    model.train()
    for X_batch, y_batch in train_loader:
        optimizer.zero_grad()
        output = model(X_batch)
        loss = criterion(output, y_batch)
        loss.backward()
        optimizer.step()
    
    if epoch in epochs_to_record:
        model.eval()
        with torch.no_grad():
            # Training metrics
            train_output = model(X_train)
            train_loss = criterion(train_output, y_train).item()
            train_acc = (train_output.argmax(1) == y_train).float().mean().item()
            
            # Test metrics
            test_output = model(X_test)
            test_loss = criterion(test_output, y_test).item()
            test_acc = (test_output.argmax(1) == y_test).float().mean().item()
            
            results.append((epoch, train_loss, train_acc, test_loss, test_acc))
 
# Print results
print(f"{'Epoch':<10} {'Train Loss':<12} {'Train Acc':<12} {'Test Loss':<12} {'Test Acc':<12}")
print("-" * 60)
 
test_losses = []
for epoch, train_loss, train_acc, test_loss, test_acc in results:
    test_losses.append((epoch, test_loss))
    print(f"{epoch:<10} {train_loss:<12.4f} {train_acc:<12.2%} {test_loss:<12.4f} {test_acc:<12.2%}")
 
# Analyze the pattern
print("\n" + "=" * 70)
print("Analysis:")
 
# Find first minimum, peak, second minimum
early = [(e, l) for e, l in test_losses if e <= 100]
mid = [(e, l) for e, l in test_losses if 100 < e <= 1000]
late = [(e, l) for e, l in test_losses if e > 1000]
 
if early:
    first_min = min(early, key=lambda x: x[1])
    print(f"  First descent minimum: Epoch {first_min[0]} (Loss: {first_min[1]:.4f})")
 
if mid:
    peak = max(mid, key=lambda x: x[1])
    print(f"  Peak (worst generalization): Epoch {peak[0]} (Loss: {peak[1]:.4f})")
 
if late:
    second_min = min(late, key=lambda x: x[1])
    print(f"  Second descent minimum: Epoch {second_min[0]} (Loss: {second_min[1]:.4f})")
 
print("\nKEY INSIGHT: Training PAST the 'overfitting point' can improve")
print("generalization. The model first learns signal, then noise, then")
print("'unlearns' the noise while retaining the signal.")

4.2 Why Epoch-Wise Double Descent Occurs

The Ordering of Learning:

Neural networks tend to learn patterns in a specific order:

First: Simple, global patterns (main signal)
Then: Complex, local patterns (hard examples + noise)
Finally: The network 'stabilizes' and some noise patterns are forgotten

Label Noise is Key:

Epoch-wise double descent is most pronounced with label noise. When some labels are wrong:

Early: Model learns the correct patterns (test loss drops)
Middle: Model fits the wrong labels (test loss increases)
Late: Somehow, the noise fitting 'washes out' and generalization improves

The mechanism for the late improvement is still debated, but may relate to:

SGD noise causing 'forgetting' of spurious correlations
Weight regularization from continuous training
The geometry of the loss landscape favoring simple solutions

Practical Consideration

Epoch-wise double descent doesn't always occur, and when it does, the second descent can take a very long time. In practice, it's often more efficient to use explicit regularization or early stopping to stay in the first descent regime. However, for very large models with large compute budgets, training through the second descent may be worthwhile.

Sample-Wise Double Descent

The third manifestation of double descent: what happens when we fix the model and vary the amount of training data?

5.1 The Counter-Intuitive Observation

Classically, we expect: more data → better generalization. Always.

But double descent reveals a surprising pattern for overparameterized models:

As we add more training data:

Initially: Test error decreases (as expected)
Near interpolation threshold: Test error can increase (more data hurts!)
Past threshold: Test error decreases again (back to normal)

This means there's a regime where adding more data temporarily makes things worse.

5.2 Intuition

Why adding data can hurt (temporarily):

When a model is at the interpolation threshold:

Adding more data pushes us from overparameterized (good) to the threshold (bad)
The model now has exactly enough capacity to fit all points
This is the worst-generalization regime

The 'data cliff':

If your model has p parameters:

With n < p samples: Overparameterized, picks minimum-norm solution (good)
With n ≈ p samples: Interpolation threshold, variance explosion (bad)
With n > p samples: Underparameterized, classical regularization needed (okay)

So if you're in the overparameterized regime, adding just enough data to hit the threshold can hurt!

sample_wise_double_descent.py
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"""
Sample-Wise Double Descent
 
Demonstrating that adding more training data can temporarily hurt generalization.
"""
 
import numpy as np
from sklearn.preprocessing import PolynomialFeatures, StandardScaler
import warnings
warnings.filterwarnings('ignore')
 
def generate_all_data(n_total=500, noise_std=0.3, seed=42):
    """Generate a pool of data to draw from."""
    np.random.seed(seed)
    X = np.random.randn(n_total, 1)
    y = np.sin(2 * np.pi * X.flatten()) + np.random.normal(0, noise_std, n_total)
    return X, y
 
def fit_and_evaluate(X_train, y_train, X_test, y_test, n_features):
    """
    Fit polynomial features with minimum-norm / ridge solution.
    """
    n_samples = X_train.shape[0]
    
    # Create polynomial features
    poly = PolynomialFeatures(n_features - 1)  # n_features = degree + 1
    X_train_poly = poly.fit_transform(X_train)
    X_test_poly = poly.transform(X_test)
    
    # Scale features for numerical stability
    scaler = StandardScaler()
    X_train_scaled = scaler.fit_transform(X_train_poly)
    X_test_scaled = scaler.transform(X_test_poly)
    
    # Solve with appropriate method
    ridge = 1e-8
    
    if n_features <= n_samples:
        # Standard ridge solution
        XtX = X_train_scaled.T @ X_train_scaled + ridge * np.eye(n_features)
        theta = np.linalg.solve(XtX, X_train_scaled.T @ y_train)
    else:
        # Minimum norm solution
        XXt = X_train_scaled @ X_train_scaled.T + ridge * np.eye(n_samples)
        alpha = np.linalg.solve(XXt, y_train)
        theta = X_train_scaled.T @ alpha
    
    # Predictions
    test_pred = X_test_scaled @ theta
    test_mse = np.mean((test_pred - y_test)**2)
    
    return test_mse
 
# Setup: Fixed model complexity (number of features)
model_complexity = 50  # 50 polynomial features
 
# Generate data pools
X_all, y_all = generate_all_data(n_total=200, seed=42)
X_test, y_test = generate_all_data(n_total=500, seed=123)
 
# Vary training set size
sample_sizes = [10, 20, 30, 40, 45, 48, 50, 52, 55, 60, 70, 80, 100, 150, 200]
 
print("Sample-Wise Double Descent")
print("=" * 70)
print(f"Fixed model complexity: {model_complexity} features")
print(f"Interpolation threshold: {model_complexity} samples")
print()
 
results = []
for n_train in sample_sizes:
    if n_train > len(X_all):
        continue
    
    X_train = X_all[:n_train]
    y_train = y_all[:n_train]
    
    test_mse = fit_and_evaluate(X_train, y_train, X_test, y_test, model_complexity)
    ratio = n_train / model_complexity
    
    if ratio < 0.9:
        regime = "Over-param"
    elif ratio < 1.1:
        regime = "THRESHOLD"
    else:
        regime = "Under-param"
    
    results.append((n_train, ratio, test_mse, regime))
 
print(f"{'Samples':<10} {'n/p Ratio':<12} {'Test MSE':<15} {'Regime'}")
print("-" * 55)
 
for n, ratio, mse, regime in results:
    print(f"{n:<10} {ratio:<12.2f} {mse:<15.4f} {regime}")
 
# Analysis
print("\n" + "=" * 70)
print("Analysis:")
 
# Identify key points
test_losses = [r[2] for r in results]
sample_counts = [r[0] for r in results]
ratios = [r[1] for r in results]
 
# Best in overparameterized regime
over = [(n, mse) for n, r, mse, _ in results if r < 0.9]
if over:
    best_over = min(over, key=lambda x: x[1])
    print(f"  Best in over-parameterized: n={best_over[0]} (MSE: {best_over[1]:.4f})")
 
# Worst near threshold
threshold = [(n, mse) for n, r, mse, _ in results if 0.9 <= r <= 1.1]
if threshold:
    worst = max(threshold, key=lambda x: x[1])
    print(f"  Worst at threshold: n={worst[0]} (MSE: {worst[1]:.4f})")
 
# Best in underparameterized
under = [(n, mse) for n, r, mse, _ in results if r > 1.1]
if under:
    best_under = min(under, key=lambda x: x[1])
    print(f"  Best in under-parameterized: n={best_under[0]} (MSE: {best_under[1]:.4f})")
 
print("\n" + "=" * 70)
print("KEY INSIGHT: For a fixed model, adding data near the interpolation")
print("threshold HURTS generalization. This is why 'more data is always")
print("better' is not quite true in the age of overparameterized models.")
print()
print("PRACTICAL ADVICE:")
print("  - Stay safely overparameterized (n << p) with implicit regularization")
print("  - OR stay safely underparameterized (n >> p) with explicit regularization")
print("  - Avoid the interpolation threshold (n ≈ p)")

Practical Implications

Sample-wise double descent is less commonly encountered in practice because: (1) most deep learning settings are safely overparameterized, (2) adding data typically pushes you further into a good regime. However, it's important when comparing results across dataset sizes or when working near the interpolation threshold. If adding data hurts performance, consider also increasing model size.

Theoretical Explanations

Several theoretical frameworks help explain why double descent occurs.

6.1 Random Matrix Theory Perspective

The Marchenko-Pastur Law and Beyond:

For linear models (including linear networks and kernel regression), random matrix theory provides precise predictions:

The eigenvalue distribution of $X^T X$ changes qualitatively at $p/n = 1$
At the threshold, the smallest eigenvalue approaches 0 → condition number explodes
Variance of the solution scales as $\frac{1}{\lambda_{\min}}$ → variance explodes

For linear regression, the test error is:

$$\text{Test Error} = \sigma^2 \cdot \frac{p}{n - p - 1} \quad \text{(for } p < n \text{)}$$

This diverges as $p \to n$, explaining the interpolation peak.

6.2 The Min-Norm Interpolation View

Why Overparameterized Is Better Than Threshold:

At the interpolation threshold ($p = n$):

Exactly one interpolating solution exists
This solution has no degrees of freedom—it must fit all points exactly
Any noise in the labels is perfectly captured

In the overparameterized regime ($p >> n$):

Infinitely many interpolating solutions exist
SGD/gradient descent finds the minimum norm solution among these
This minimum-norm solution is effectively regularized
The excess capacity allows 'choosing' a well-behaved interpolant

Key Insight: Having too many solutions is better than having exactly one bad solution.

6.3 Bias-Variance Revisited

Double descent can be understood through a refined bias-variance analysis:

At interpolation threshold:

Bias is low (model can represent the target well)
Variance is maximal (unique solution is unstable)

Past interpolation:

Bias remains low
Variance decreases because the minimum-norm constraint effectively regularizes
The implicit regularization strength increases with overparameterization

double_descent_theory.py
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"""
Theoretical Analysis of Double Descent
 
Demonstrating the variance explosion at the interpolation threshold.
"""
 
import numpy as np
 
def theoretical_risk_linear(p, n, sigma_noise=1.0):
    """
    Theoretical test risk for least squares.
    
    Under-parameterized (p < n):
        Risk ≈ bias² + σ² * p / (n - p - 1)
    
    Over-parameterized (p > n): 
        Risk ≈ σ² * n / (p - n)  (for minimum norm solution)
    
    This is simplified; actual formulas involve additional terms.
    """
    if p < n - 1:
        # Under-parameterized regime
        # Assuming negligible bias for simplicity
        variance = sigma_noise**2 * p / (n - p - 1)
        return variance
    elif p > n + 1:
        # Over-parameterized regime
        variance = sigma_noise**2 * n / (p - n - 1)
        return variance
    else:
        # At threshold - divergent
        return np.nan
 
# Fixed sample size, varying model complexity
n = 100
sigma = 1.0
 
ps = np.arange(10, 200, 5)  # Model complexity from 10 to 200
risks = []
 
for p in ps:
    risk = theoretical_risk_linear(p, n, sigma)
    risks.append(risk)
 
print("Theoretical Risk Analysis (Linear Model)")
print("=" * 60)
print(f"Sample size n = {n}")
print(f"Noise σ = {sigma}")
print()
 
# Show key regimes
print(f"{'p (complexity)':<15} {'p/n':<10} {'Theoretical Risk':<20} {'Regime'}")
print("-" * 60)
 
for p, risk in zip(ps, risks):
    ratio = p / n
    if ratio < 0.9:
        regime = "Under-param"
    elif ratio < 1.1:
        regime = "THRESHOLD"
    else:
        regime = "Over-param"
    
    risk_str = f"{risk:.4f}" if not np.isnan(risk) else "DIVERGENT"
    print(f"{p:<15} {ratio:<10.2f} {risk_str:<20} {regime}")
 
print()
print("=" * 60)
print("THEORETICAL INSIGHTS:")
print()
print("1. Under-parameterized (p < n):")
print("   Risk grows as p → n because denominator (n - p) shrinks")
print()
print("2. Interpolation threshold (p ≈ n):")
print("   Risk DIVERGES - the unique interpolant is maximally unstable")
print()  
print("3. Over-parameterized (p > n):")
print("   Risk decreases as p grows because denominator (p - n) grows")
print("   More overparameterization = more implicit regularization")
print()
print("This explains the double descent shape!")
print("The peak is at p ≈ n, with descent on both sides.")
 
# Compute empirical verification
print("\n" + "=" * 60)
print("Empirical Verification (Monte Carlo)")
print("=" * 60)
 
def monte_carlo_risk(p, n, sigma_noise=1.0, n_trials=100):
    """Empirically estimate test risk via Monte Carlo."""
    risks = []
    
    for trial in range(n_trials):
        # Generate data
        X_train = np.random.randn(n, p)
        beta_true = np.zeros(p)  # No signal, just noise
        y_train = X_train @ beta_true + np.random.randn(n) * sigma_noise
        
        X_test = np.random.randn(n, p)
        y_test = X_test @ beta_true + np.random.randn(n) * sigma_noise
        
        # Fit
        try:
            if p <= n:
                # Standard least squares (or ridge with tiny lambda)
                XtX = X_train.T @ X_train + 1e-8 * np.eye(p)
                beta_hat = np.linalg.solve(XtX, X_train.T @ y_train)
            else:
                # Minimum norm solution
                XXt = X_train @ X_train.T + 1e-8 * np.eye(n)
                alpha = np.linalg.solve(XXt, y_train)
                beta_hat = X_train.T @ alpha
            
            # Test error
            test_pred = X_test @ beta_hat
            test_mse = np.mean((test_pred - y_test)**2)
            risks.append(test_mse)
        except:
            risks.append(np.nan)
    
    return np.nanmean(risks)
 
print(f"\n{'p':<10} {'Theoretical':<15} {'Empirical':<15}")
print("-" * 45)
 
for p in [20, 50, 80, 90, 95, 105, 110, 120, 150, 180]:
    theoretical = theoretical_risk_linear(p, n, sigma)
    empirical = monte_carlo_risk(p, n, sigma, n_trials=50)
    
    th_str = f"{theoretical:.4f}" if not np.isnan(theoretical) else "N/A"
    em_str = f"{empirical:.4f}" if not np.isnan(empirical) else "N/A"
    
    print(f"{p:<10} {th_str:<15} {em_str:<15}")

Theory vs. Practice

The theoretical analysis of double descent is most complete for linear models and kernel methods. For deep neural networks, the picture is more complex—nonlinearity, feature learning, and the specific optimization trajectory all play roles. However, the qualitative pattern of double descent is observed empirically even where theory is incomplete.

Practical Implications and Guidelines

Understanding double descent changes how we should approach model selection and training in practice.

7.1 Revised Model Selection Principles

Old Approach (Classical):

Start with a simple model
Gradually increase complexity
Stop when validation error starts increasing
The best model is at the 'elbow'

New Understanding (Post-Double-Descent):

If possible, go straight to overparameterized models
Use implicit regularization (SGD, initialization, architecture)
If you see overfitting, consider making the model larger, not smaller
Avoid the interpolation threshold (model size ≈ dataset size)

Practical Guidelines

•Go big or go small. The worst regime is near the interpolation threshold. Either stay safely underparameterized (with explicit regularization) or safely overparameterized (with implicit regularization).
•Consider label noise. Double descent effects are amplified by label noise. If your labels are noisy, overparameterized models with implicit regularization often work better than carefully tuned smaller models.
•Train longer for large models. Overparameterized models may need more epochs to reach the second descent in the epoch-wise sense. Don't give up too early.
•Match model scaling with data scaling. If you add more data, consider also increasing model size to stay in the 'good' regime relative to the interpolation threshold.
•Regularization shifts the curve. Explicit regularization (L2, dropout) can smooth out the interpolation peak. With strong regularization, you may see a single descent curve.

7.2 When Not to Overparameterize

Overparameterization isn't always the answer:

Computational constraints: Overparameterized models require more compute for training and inference. If you're deployment-constrained, you may need to stay in the underparameterized regime with careful regularization.

Few-shot or low-data regimes: With very little data, even overparameterized implicit regularization may not be enough. Explicit priors (through regularization, data augmentation, or pre-training) become more important.

Interpretability requirements: Larger models are harder to interpret. If explanations are needed, simpler models may be preferable even at the cost of some performance.

Well-understood problem structure: If you know the problem structure well, domain-appropriate inductive biases (embodied in smaller, more structured models) may outperform brute-force overparameterization.

The Modern Recipe

For most deep learning tasks today: (1) Use an architecture appropriate for your domain (CNNs for images, Transformers for text). (2) Make the model large relative to your data. (3) Use standard training (SGD/Adam with standard learning rates). (4) Use light explicit regularization (weight decay, dropout). (5) Use early stopping based on validation—but be aware you might be stopping before the second descent.

Summary and Key Insights

Double descent is one of the most important discoveries in modern machine learning theory. It reconciles decades of empirical observations with theoretical predictions and fundamentally changes how we think about the bias-variance tradeoff.

Key Takeaways

•The classical U-curve is incomplete. Past the interpolation threshold, test error descends again. The full picture is a double-descent curve.
•The interpolation threshold is the worst place. When model capacity exactly matches data size (p ≈ n), variance explodes and generalization is worst.
•Overparameterization can help generalization. When p >> n, the minimum-norm interpolant is implicitly regularized, leading to good generalization despite zero training error.
•Three manifestations: Model-wise (increasing parameters), epoch-wise (training longer), and sample-wise (varying data size) all show double descent patterns.
•Label noise amplifies the effect. Double descent is most pronounced when labels contain noise, making the distinction between fitting signal vs. noise more stark.
•Practical implication: Go big or go small. Avoid the interpolation threshold. Modern deep learning succeeds partly because models are safely overparameterized.

Looking Ahead

In the next page, we'll dive deeper into overparameterization—exploring why having more parameters than data points leads to better generalization when combined with appropriate training procedures. We'll see how overparameterization enables easier optimization, smoother loss landscapes, and more effective implicit regularization.

Page Complete

You now understand the double descent phenomenon—one of the most important revisions to classical statistical learning theory in recent years. This knowledge explains why modern deep learning can use massive models successfully and provides practical guidance for model selection in the overparameterized era.