Weight Regularization - Learning Module

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Max-Norm Constraints

Hard Constraints on Weight Magnitudes

L2 and L1 regularization apply "soft" constraints—they penalize large weights but don't prevent them. A sufficiently strong gradient signal can still push weights to extreme values, particularly early in training when gradients may be large and varied.

Max-norm constraints take a different approach: they impose hard limits on weight magnitudes. After each gradient update, weights are projected back onto a constraint set if they exceed the specified bound. This guarantees that weight norms never exceed a maximum value $c$, regardless of gradient magnitudes.

This technique is particularly valuable for:

Training stability with large learning rates
Preventing weight explosion during early training
Regularizing when combined with dropout
Scenarios where bounded weights are required by domain constraints

What You Will Learn

This page covers max-norm constraints comprehensively: the mathematical formulation, projection operations, relationship to constrained optimization, implementation with different norm types, interaction with dropout, and practical usage guidelines.

Mathematical Formulation

Max-norm regularization constrains the training problem:

$$\min_{\boldsymbol{\theta}} \mathcal{L}_{\text{data}}(\boldsymbol{\theta}) \quad \text{subject to} \quad |\mathbf{w}_j|_2 \leq c \quad \forall j$$

where $\mathbf{w}_j$ is the weight vector of the $j$-th unit (e.g., incoming weights to a neuron), and $c$ is the maximum allowed norm.

Types of Max-Norm:

Incoming weight norm: Constrain the L2 norm of weights entering each neuron $$|\mathbf{W}_{:,j}|_2 \leq c$$
Outgoing weight norm: Constrain weights leaving each neuron $$|\mathbf{W}_{i,:}|_2 \leq c$$
Full matrix norm: Constrain the Frobenius or spectral norm of the weight matrix (covered separately in spectral normalization)

Why Per-Unit Constraints?

Constraining per-unit rather than the full weight matrix provides finer control. Each neuron's capacity is bounded independently, preventing any single neuron from dominating while allowing the network as a whole to have high capacity.

The Projection Operation

Max-norm is enforced by projecting weights onto the constraint set after each gradient update.

Projection onto L2 Ball:

For a weight vector $\mathbf{w}$ with constraint $|\mathbf{w}|_2 \leq c$:

$$\mathbf{w}_{\text{proj}} = \begin{cases} \mathbf{w} & \text{if } |\mathbf{w}|_2 \leq c \ c \cdot \frac{\mathbf{w}}{|\mathbf{w}|_2} & \text{otherwise} \end{cases}$$

This is called rescaling or clipping: if the norm exceeds $c$, we scale the vector down to have norm exactly $c$, preserving its direction.

Algorithm:

Compute gradient and perform standard update
For each unit's weight vector:
- Compute L2 norm
- If norm > c, rescale to norm = c

max_norm_projection.py
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import numpy as np
import torch
import torch.nn as nn
 
def project_to_max_norm(weights, max_norm):
    """
    Project weight vectors onto L2 ball of radius max_norm.
    
    Args:
        weights: Weight matrix of shape (fan_out, fan_in)
                 Each row is incoming weights to one neuron
        max_norm: Maximum L2 norm per neuron
    
    Returns:
        Projected weights
    """
    # Compute norm of each row (incoming weights to each neuron)
    norms = np.linalg.norm(weights, axis=1, keepdims=True)
    
    # Compute scaling factor: min(1, max_norm / norm)
    # This equals 1 if norm <= max_norm, else max_norm/norm
    scale = np.clip(max_norm / (norms + 1e-8), 0, 1)
    
    return weights * scale
 
def apply_max_norm_constraint(model, max_norm):
    """
    Apply max-norm constraint to all Linear layers in a model.
    
    Should be called AFTER optimizer.step()
    """
    with torch.no_grad():
        for module in model.modules():
            if isinstance(module, nn.Linear):
                # Weight shape: (out_features, in_features)
                # Each row = incoming weights to one output neuron
                norms = module.weight.norm(p=2, dim=1, keepdim=True)
                scale = torch.clamp(max_norm / norms, max=1.0)
                module.weight.mul_(scale)
 
# Example usage in training loop
model = nn.Sequential(
    nn.Linear(784, 256),
    nn.ReLU(),
    nn.Linear(256, 10)
)
 
optimizer = torch.optim.SGD(model.parameters(), lr=0.01)
max_norm = 3.0  # Typical values: 1-5
 
for inputs, targets in dataloader:
    optimizer.zero_grad()
    loss = criterion(model(inputs), targets)
    loss.backward()
    optimizer.step()
    
    # Apply max-norm constraint AFTER the update
    apply_max_norm_constraint(model, max_norm)

Relationship to Soft Regularization

Max-norm constraints and L2 regularization are related through constrained optimization theory.

KKT Conditions:

For the constrained problem with L2 ball constraint, the Karush-Kuhn-Tucker (KKT) conditions show that at optimality, there exists a Lagrange multiplier $\mu \geq 0$ such that:

$$\nabla \mathcal{L}_{\text{data}} + \mu \mathbf{w} = 0$$

This looks exactly like the gradient of L2 regularized loss with $\lambda = \mu$. When the constraint is active (norm = c), $\mu > 0$; when inactive, $\mu = 0$.

Key Difference:

L2: Same $\lambda$ for all weights, regardless of how close to the "boundary"
Max-norm: Effective $\mu$ is large for weights at the boundary, zero for weights inside

Max-norm is "smarter"—it only penalizes weights that are actually constrained, leaving unconstrained weights free.

Soft (L2) vs Hard (Max-Norm) Constraints
Aspect	L2 Regularization	Max-Norm Constraint
Type	Soft penalty	Hard constraint
Weights > bound	Penalized but allowed	Impossible (projected out)
Weights < bound	Still penalized (toward zero)	No penalty
Effect	Shrinks all weights	Only clips outliers
Hyperparameter	λ (penalty strength)	c (max norm)
Stability	Can still explode if λ small	Guaranteed bounded

Interaction with Dropout

Max-norm constraints are particularly effective when combined with dropout. This combination was emphasized in the original dropout paper (Srivastava et al., 2014).

Why They Work Well Together:

Dropout creates noise: Random unit dropping injects high variance into gradients
Large gradients can explode weights: Without constraints, dropout noise can push weights to extreme values
Max-norm prevents explosion: Hard bounds ensure weights stay reasonable despite noisy gradients
Complementary regularization: Dropout provides implicit ensemble; max-norm provides capacity control

Recommended Usage:

When using dropout rates > 0.3:

Apply max-norm with $c \in [3, 5]$
This prevents the "weight explosion" phenomenon
Allows using higher dropout rates that would otherwise destabilize training

dropout_maxnorm_combination.py
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import torch
import torch.nn as nn
 
class DropoutWithMaxNorm(nn.Module):
    """
    Network combining dropout with max-norm constraints.
    
    Recommended when using aggressive dropout rates.
    """
    def __init__(self, input_dim, hidden_dim, output_dim, 
                 dropout_rate=0.5, max_norm=4.0):
        super().__init__()
        self.max_norm = max_norm
        
        self.layers = nn.Sequential(
            nn.Linear(input_dim, hidden_dim),
            nn.ReLU(),
            nn.Dropout(dropout_rate),
            nn.Linear(hidden_dim, hidden_dim),
            nn.ReLU(),
            nn.Dropout(dropout_rate),
            nn.Linear(hidden_dim, output_dim)
        )
    
    def forward(self, x):
        return self.layers(x)
    
    def apply_max_norm(self):
        """Apply max-norm constraint to all Linear layers."""
        with torch.no_grad():
            for module in self.modules():
                if isinstance(module, nn.Linear):
                    norms = module.weight.norm(p=2, dim=1, keepdim=True)
                    scale = torch.clamp(self.max_norm / norms, max=1.0)
                    module.weight.mul_(scale)
 
# Training loop
model = DropoutWithMaxNorm(784, 1024, 10, dropout_rate=0.5, max_norm=4.0)
optimizer = torch.optim.SGD(model.parameters(), lr=0.1, momentum=0.9)
 
for epoch in range(num_epochs):
    for inputs, targets in train_loader:
        optimizer.zero_grad()
        outputs = model(inputs)
        loss = criterion(outputs, targets)
        loss.backward()
        optimizer.step()
        
        # Apply max-norm after each update
        model.apply_max_norm()

The Dropout + Max-Norm Recipe

The original dropout paper found that max-norm (c=4) combined with dropout (p=0.5) and high momentum (0.9-0.99) allowed training with very large learning rates, achieving faster convergence and better results than either technique alone.

Variants and Extensions

Max-norm constraints can be applied in various ways beyond simple per-neuron L2 norms.

Max-Norm Variants

•Column-wise: Constrain incoming weights to each neuron (most common)
•Row-wise: Constrain outgoing weights from each neuron
•Full Frobenius: Constrain entire weight matrix norm
•Spectral: Constrain largest singular value (see spectral normalization)
•Group max-norm: Constrain groups of neurons together
•Layerwise varying: Different max-norm values per layer

max_norm_variants.py
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import torch
import torch.nn as nn
 
def apply_column_max_norm(weight, max_norm):
    """Constrain incoming weights to each neuron (columns of W^T)."""
    # For Linear layer: weight shape is (out, in)
    # Each ROW is incoming weights to one output neuron
    norms = weight.norm(p=2, dim=1, keepdim=True)
    scale = torch.clamp(max_norm / norms, max=1.0)
    return weight * scale
 
def apply_row_max_norm(weight, max_norm):
    """Constrain outgoing weights from each neuron (columns of W)."""
    # Each COLUMN is outgoing weights from one input neuron
    norms = weight.norm(p=2, dim=0, keepdim=True)
    scale = torch.clamp(max_norm / norms, max=1.0)
    return weight * scale
 
def apply_frobenius_max_norm(weight, max_norm):
    """Constrain entire matrix Frobenius norm."""
    norm = weight.norm(p='fro')
    if norm > max_norm:
        return weight * (max_norm / norm)
    return weight
 
def apply_layerwise_max_norm(model, max_norms):
    """
    Apply different max-norm to each layer.
    
    Args:
        model: Neural network
        max_norms: Dict mapping layer index to max_norm value
                   e.g., {0: 3.0, 1: 4.0, 2: 5.0}
    """
    with torch.no_grad():
        for i, module in enumerate(model.modules()):
            if isinstance(module, nn.Linear) and i in max_norms:
                norms = module.weight.norm(p=2, dim=1, keepdim=True)
                scale = torch.clamp(max_norms[i] / norms, max=1.0)
                module.weight.mul_(scale)

Practical Considerations

When to Use Max-Norm

•With dropout: Especially at high dropout rates (> 0.3)
•Stability concerns: When training with large learning rates
•Early stopping: When you want to train longer without overfitting
•Guaranteed bounds: When domain requires bounded activations
•Simple networks: MLPs benefit more than modern architectures

When Max-Norm May Not Help

•With batch normalization: BatchNorm already controls scales
•Modern architectures: ResNets, Transformers have other normalization
•Well-tuned L2: Properly tuned weight decay often suffices
•AdamW training: Decoupled weight decay provides good control

Typical Max-Norm Values
Scenario	Max-Norm c	Notes
Standard MLP	3-5	Original dropout paper recommendation
High dropout (0.5+)	3-4	Tighter constraint for stability
Low dropout (0.1-0.3)	4-5	Looser constraint acceptable
Large learning rate	2-3	Tighter to prevent explosion
Fine-tuning	5-10	Allow larger deviations from init

Summary: Max-Norm Constraints

Key Takeaways

•Max-norm constrains weight norms to hard upper bound $c$
•Projection rescales vectors exceeding the bound, preserving direction
•Hard vs soft: Unlike L2, max-norm guarantees bounded weights
•Dropout synergy: Particularly effective with high dropout rates
•Per-neuron: Typically applied to incoming weights of each neuron
•Less common now: Modern architectures use normalization layers instead

Page Complete

You now understand max-norm constraints as hard bounds on weight magnitudes. While less common in modern architectures with built-in normalization, max-norm remains valuable for MLPs with dropout and situations requiring guaranteed weight bounds. Next, we explore spectral normalization—a more sophisticated technique that constrains the spectral norm (largest singular value) of weight matrices.