Regularization

Overview & Motivation

Regularization adds a penalty term $\Omega(\theta)$ to the loss function that discourages overly complex models:

$\mathcal{L}_{\text{total}} = \mathcal{L}(\hat{y}, y) + \lambda \, \Omega(\theta)$

where $\lambda > 0$ controls the strength of the penalty. Without regularization, a neural network with enough parameters can memorize the training data perfectly yet generalize poorly to new inputs — a phenomenon called overfitting. Regularization biases the optimizer toward simpler solutions that tend to generalize better.

Mathematical Theory

L2 Regularization (Ridge / Weight Decay)

$$\Omega_{L2}(\theta) = \frac{1}{2}\sum_{i=1}^{P} \theta_i^2 = \frac{1}{2}|\theta|_$2^{2}$$$

$\frac{\partial \Omega_{L2}}{\partial \theta_i} = \theta_i$

Effect on update rule:

$\theta_{t+1} = \theta_t - \eta(\nabla_\theta \mathcal{L} + \lambda \theta_t) = (1 - \eta\lambda)\theta_t - \eta \nabla_\theta \mathcal{L}$

The factor $(1 - \eta\lambda)$ shrinks weights toward zero each step — hence the name weight decay. Large weights are penalized quadratically, keeping the model smooth.

L1 Regularization (Lasso)

$\Omega_{L1}(\theta) = \sum_{i=1}^{P} |\theta_i| = \|\theta\|_1$

$\frac{\partial \Omega_{L1}}{\partial \theta_i} = \operatorname{sign}(\theta_i)$

Effect: L1 drives small weights exactly to zero, producing a sparse model. This is useful for feature selection — irrelevant connections are pruned automatically.

Comparison

Property	L1	L2
Penalty shape	Diamond (corners at axes)	Sphere
Sparsity	Promotes exact zeros	Shrinks toward zero but rarely reaches it
Gradient at $\theta_i = 0$	Undefined (sub-gradient)	Zero
Best for	Feature selection, sparse models	General-purpose weight control

Geometric Interpretation

The regularized loss can be viewed as constrained optimization:

$\min_\theta \mathcal{L}(\theta) \quad \text{subject to} \quad \Omega(\theta) \le c$

where $c$ is determined by $\lambda$. L1 constrains $\theta$ to a diamond, so the optimal point tends to lie at a corner (sparse). L2 constrains to a sphere, so the optimal point balances all dimensions.

Complexity Analysis

Operation	Time	Space
$\Omega_{L2}(\theta)$	$O(P)$	$O(1)$
$\nabla \Omega_{L2}$	$O(P)$	$O(P)$
$\Omega_{L1}(\theta)$	$O(P)$	$O(1)$
$\nabla \Omega_{L1}$	$O(P)$	$O(P)$

Regularization adds negligible computational cost — one pass over the parameter vector per training iteration.

Step-by-Step Walkthrough

Scenario: 3 parameters, $\theta = [0.8, -0.3, 0.5]$, $\lambda = 0.1$.

L2 Regularization:

Step	Computation	Result
Penalty	$\frac{1}{2}(0.64 + 0.09 + 0.25) = 0.49$	$\Omega_{L2} = 0.49$
Gradient	$[0.8, -0.3, 0.5]$	Added to $\nabla\mathcal{L}$
Contribution to loss	$0.1 \times 0.49 = 0.049$

L1 Regularization:

Step	Computation	Result
Penalty	$0.8 + 0.3 + 0.5 = 1.6$	$\Omega_{L1} = 1.6$
Gradient	$[1, -1, 1]$	Added to $\nabla\mathcal{L}$
Contribution to loss	$0.1 \times 1.6 = 0.16$

After several L1 updates ($\eta = 0.1$, $\lambda = 0.1$): the $\theta_2 = -0.3$ component, already small, is driven to exactly zero. The model effectively prunes that connection.

Pitfalls & Edge Cases

• $\lambda$ too large. The model underfits — weights are driven so close to zero that the network cannot represent the function. Cross-validate $\lambda$.
• $\lambda$ too small. Negligible effect; overfitting persists.
• L1 non-differentiability. At $\theta_i = 0$, the L1 gradient is undefined. Use sub-gradient $\operatorname{sign}(0) = 0$ or proximal operators for exact handling.
• Regularizing biases. Conventionally, bias parameters are excluded from regularization because they do not contribute to model complexity. This library regularizes all parameters in the flat vector — be aware of this if bias control matters.
• Fixed-point precision. The regularization term can be much smaller than the main loss when $\lambda$ is small. In low-precision fixed-point, it may round to zero. Scale $\lambda$ or use a wider accumulator.

Variants & Generalizations

Variant	Key Difference
Elastic Net	\Omega = \alpha \|\theta\|_1 + (1-\alpha)\|\theta\|_2^{2}$$; combines L1 sparsity with L2 smoothness
Dropout	Randomly zeroes activations during training; implicit ensemble regularization
Early stopping	Halts training before overfitting; regularization without modifying the loss
Data augmentation	Expands the training set with transformed copies; reduces overfitting by increasing data diversity
Spectral normalization	Constrains the spectral norm of weight matrices; stabilizes GAN training
Weight clipping	Hard constraint: $\left\|\theta_i\right\| \le c$; used in Wasserstein GANs

Applications

• Preventing overfitting — The primary use case for any neural network trained on limited data (common in embedded scenarios).
• Feature selection — L1 regularization identifies and prunes irrelevant input connections.
• Model compression — Sparse models (via L1) require less storage and computation for deployment on MCUs.
• Transfer learning — L2 regularization keeps fine-tuned weights close to pre-trained values.

Connections to Other Algorithms

graph TD
    Reg["Regularization"]
    Loss["Loss Functions"]
    Opt["Optimizer"]
    Model["Model"]
    LR["Linear Regression"]

    Reg -->|"λ Ω(θ) added to ℒ"| Loss
    Loss --> Opt
    Opt --> Model
    Reg -.->|"L2 + MSE = Ridge regression"| LR

Component	Relationship
Loss Functions	Regularization is a penalty added to the loss: $\mathcal{L}_{\text{total}} = \mathcal{L} + \lambda\Omega$
Optimizer	Receives the combined gradient $\nabla\mathcal{L} + \lambda\nabla\Omega$
Linear Regression	L2-regularized MSE with a linear model is Ridge regression; L1 is Lasso

References & Further Reading

• Goodfellow, I., Bengio, Y., and Courville, A., Deep Learning, MIT Press, 2016 — Chapter 7 (regularization).
• Tibshirani, R., "Regression shrinkage and selection via the lasso", JRSS-B, 58(1), 1996.
• Krogh, A. and Hertz, J.A., "A simple weight decay can improve generalization", NeurIPS, 1991.