• Regularization is an essential concept in machine learning that aims to prevent overfitting - a common problem where a model performs well on training data but poorly on unseen data.
  • Regularization addresses this issue by introducing additional information to penalize extreme parameter weights, thus constraining the model. By adding this complexity penalty, the model is encouraged to be as simple as possible and to not fit too closely to the training data, enhancing its generalizability. Regularization techniques include L1 and L2 regularization, Dropout, and others.

L1 Regularization (Lasso)

  • L1 Regularization, also known as Lasso Regression, introduces a penalty equal to the absolute value of the magnitude of the model’s weights. The L1 term is computed as the sum of the absolute value of the weights. This tends to make weights exactly zero, especially for less important or irrelevant features, effectively reducing the model’s complexity.
  • L1 regularization has the beneficial side effect of performing feature selection. By driving certain weights to zero, L1 regularization effectively removes the corresponding features from the model. This can be helpful when dealing with high-dimensional data where some features may be irrelevant or redundant.
  • However, one potential drawback is that L1 regularization can lead to model underfitting if the penalty factor is set too high.

L2 Regularization (Ridge)

  • L2 Regularization, also known as Ridge Regression, adds a penalty equal to the square of the magnitude of the model’s weights. The L2 term is computed as the sum of the squared value of the weights. Unlike L1 regularization, L2 regularization does not result in zero weights but instead tends to distribute weights evenly across features.
  • L2 regularization can prevent overfitting by discouraging large weights, leading to a more balanced and generalized model. It is generally more robust to outliers than L1 regularization.
  • A potential drawback is that it may not perform well when the data has many irrelevant features, as it does not perform feature selection like L1 regularization does.

What are the differences between L1 and L2?

  • Regularization is a technique used to avoid overfitting by trying to make the model more simple. One way to apply regularization is by adding the weights to the loss function. This is done in order to consider minimizing unimportant weights. In L1 regularization we add the sum of the absolute of the weights to the loss function. In L2 regularization we add the sum of the squares of the weights to the loss function.
  • So both L1 and L2 regularization are ways to reduce overfitting, but to understand the difference it’s better to know how they are calculated:
    • Loss (L2) : Cost function + \(L\) * \(weights^2\)
    • Loss (L1) : Cost function + \(L\) * \(\|weights\|\)
      • Where \(L\) is the regularization parameter
  • L2 regularization penalizes huge parameters preventing any of the single parameters to get too large. But weights never become zeros. It adds parameters square to the loss. Preventing the model from overfitting on any single feature.
  • L1 regularization penalizes weights by adding a term to the loss function which is the absolute value of the loss. This leads to it removing small values of the parameters leading in the end to the parameter hitting zero and staying there for the rest of the epochs. Removing this specific variable completely from our calculation. So, It helps in simplifying our model. It is also helpful for feature selection as it shrinks the coefficient to zero which is not significant in the model.


  • Dropout is a different type of regularization technique commonly used in neural networks. It works by randomly setting a fraction of the input units for a given layer to zero during training, which can prevent overfitting.
  • By doing so, dropout effectively creates a form of ensemble learning, where during each training iteration, a different “thinned” network is trained. The resulting model can be thought of as an ensemble of these thinned networks.
  • Dropout helps to prevent complex co-adaptations of features during training, leading to a more robust and generalized model. It can significantly improve the performance of neural networks on supervised learning tasks.
  • However, a potential disadvantage of dropout is that it can increase the time required to train the model, as it essentially requires training multiple different networks.

Sparsity and Regularization

  • Sparsity refers to having a high proportion of zero coefficients in the solution. L1 regularization can lead to a sparse model, where only a subset of the features contribute to the model’s predictions. This is a direct result of L1 regularization’s ability to shrink coefficients to zero, effectively eliminating the least important features. This property makes L1 regularization a useful tool for feature selection, especially in high-dimensional data.
  • In contrast, L2 regularization does not lead to sparse solutions and does not perform feature selection. Instead, it tends to distribute weights evenly and keeps all features, but with smaller coefficients.
  • Regularization is a crucial tool in a machine learning practitioner’s toolbox, helping to manage overfitting, improve model generalization, and in some cases, even perform feature selection. The choice of regularization (L1, L2, Dropout) depends on the problem at hand, the nature of the data, and the specific model being used.


  • “In AI inference and machine learning, sparsity refers to a matrix of numbers that includes many zeros or values that will not significantly impact a calculation.”(source)
  • Improved model efficiency: Sparsity reduces the number of non-zero elements in the model, leading to more efficient computations and memory usage. By eliminating unnecessary parameters or features, sparse models can be faster and require less storage, making them more practical for deployment on resource-constrained devices or in large-scale systems.
  • Feature selection and interpretability: Sparsity can help identify the most relevant features or inputs for a given task. By encouraging sparsity in the model’s weights or feature representation, less important or redundant features can be effectively ignored, leading to a more compact and interpretable model. This can facilitate better understanding and insights into the underlying patterns and relationships in the data.
  • Regularization and generalization: Sparsity acts as a form of regularization, preventing overfitting by reducing model complexity. By encouraging sparsity, the model becomes more robust and less prone to fitting noise or irrelevant details in the training data. This regularization effect helps improve generalization, allowing the model to perform better on unseen data.
  • Compressed model representation: Sparsity can be leveraged to compress models and reduce storage requirements. Sparse representations enable more efficient model storage and transmission, which is particularly valuable in scenarios where bandwidth or storage capacity is limited, such as mobile applications or distributed systems.
  • Energy efficiency: Sparse models can consume less energy during both training and inference. The reduced number of operations required for sparse computations leads to lower power consumption, making sparse models more energy-efficient, especially in resource-constrained environments.
  • Link to more information on sparsity on Aman.ai.

Why add sparsity

  • Sparse vectors often have a large number of dimensions, and creating a feature cross can further increase the dimensionality. This can lead to a significant increase in model size and memory requirements.
  • In high-dimensional sparse vectors, it would be beneficial to promote weights to become exactly zero whenever possible. A weight of zero essentially removes the corresponding feature from the model, resulting in memory savings and potentially reducing noise in the model

How to add Sparsity

  • L1 Regularization: L1 regularization, also known as Lasso regularization, can encourage sparsity by adding a penalty term to the model’s loss function. This penalty term is proportional to the absolute values of the model’s weights. By minimizing the combined loss and penalty, L1 regularization tends to shrink less important weights to zero, effectively selecting a subset of features or parameters.
  • Group Lasso: Group Lasso extends L1 regularization to encourage sparsity at the group level. It is particularly useful when dealing with structured data, such as images or text, where the features can be organized into groups. Group Lasso promotes sparsity by shrinking entire groups of features together, effectively selecting only a subset of groups.
  • Elastic Net Regularization: Elastic Net combines both L1 and L2 regularization. It adds a penalty term that is a linear combination of the L1 and L2 norms of the model’s weights. This allows for both sparsity (through the L1 term) and shrinkage (through the L2 term), providing a balance between feature selection and model stability.
  • Dropout: Dropout is a regularization technique commonly used in neural networks. It randomly sets a fraction of the neurons or connections to zero during each training iteration. By doing so, dropout encourages individual neurons to be less reliant on the presence of specific input features, promoting a more robust and sparse representation.
  • Pruning: Pruning involves iteratively removing or setting small-weight connections to zero after training a model. It can be applied to various types of models, including neural networks. Pruning techniques identify and eliminate connections or weights that contribute less to the model’s performance, resulting in a sparser and more efficient model.
  • Quantization: Quantization reduces the precision of the model’s weights or activations, typically from floating-point to fixed-point representation. This reduction in precision can lead to increased sparsity by setting many of the less significant bits to zero, resulting in a more compact and sparse model representation.
  • Keep in mind, an excessively sparse models may sacrifice performance or miss important features.

When to remove Sparsity and how?

  • There may be situations where removing sparsity is desired in order to improve model performance or address specific requirements. Here are a few scenarios:
    1. Dense Representations: Sparse vectors with many dimensions can be computationally expensive and memory-intensive, especially when dealing with large-scale datasets. In some cases, transforming sparse vectors into dense representations can be beneficial for efficient storage and faster computations.
    2. Dense Neural Networks: Sparsity can pose challenges when training deep neural networks that require dense connections. Some network architectures, such as fully connected layers, may require dense representations to propagate information effectively across all dimensions.
    3. Domain-specific Considerations: Certain domains or applications may have specific requirements that necessitate dense representations. For example, in computer vision tasks, dense feature maps may be required for precise spatial information or detailed visual analysis.
  • To remove sparsity and convert sparse vectors into dense representations, you can employ techniques such as:
    1. Embeddings: Utilize embedding layers to transform high-dimensional sparse input into lower-dimensional dense representations. These embeddings can capture meaningful relationships between features and provide dense vector outputs.
    2. Dimensionality Reduction: Apply dimensionality reduction techniques like Principal Component Analysis (PCA) or t-SNE to reduce the dimensionality of sparse vectors while preserving important information. This can result in denser representations with reduced computational complexity.
    3. Feature Engineering: Identify and engineer new features that capture important patterns or interactions within the data. By combining or transforming sparse features, you can create denser feature representations that capture relevant information for the task at hand.
  • It’s important to note that the decision to remove sparsity and the techniques used to achieve it should be based on the specific problem, data characteristics, and performance goals of the model.