Implement logistic regression from scratch

Question

Write a clean implementation of logistic regression without using ML libraries.

Codemia · Accepted Answer

Logistic Regression is a supervised learning algorithm used for binary classification problems. Unlike linear regression, which predicts continuous outcomes, logistic regression predicts the probability that a given input point belongs to a particular category. The core concept involves the logistic function (also known as the sigmoid function), which maps any real-valued number into the (0, 1) interval. Mathematically, the logistic function is defined as:

$$ \sigma(z) = \frac{1}{1 + e^{-z}} $$

where $ z $ is the linear combination of input features, $ z = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + ... + \beta_n x_n $. Here, $ \beta $ represents the weights that the model learns during training.

To train a logistic regression model, we utilize the Maximum Likelihood Estimation (MLE) method to optimize the weights. The loss function for logistic regression is the negative log-likelihood, defined as:

$$ L(\beta) = -\sum_{i=1}^{m} [y^{(i)} \log(h(x^{(i)})) + (1 - y^{(i)}) \log(1 - h(x^{(i)}))] $$

where $ h(x^{(i)}) = \sigma(z^{(i)}) $ is the predicted probability of the positive class. The optimization process is carried out using Gradient Descent, which updates the weights iteratively:

$$ \beta := \beta - \alpha \nabla L(\beta) $$

where $ \alpha $ is the learning rate and $ \nabla L(\beta) $ is the gradient of the loss function with respect to the parameters.

At Plaid, logistic regression can be instrumental in various applications such as fraud detection, risk assessment, and customer segmentation. For instance, when analyzing transaction data, logistic regression can help predict whether a transaction is fraudulent based on features like transaction amount, location, and frequency. Implementing logistic regression from scratch allows for greater control and customization, such as adjusting the learning rate, regularization techniques, or feature selection methods, to tailor the model to specific datasets and business needs.

The strengths of logistic regression include its simplicity, interpretability, and efficiency with linearly separable data. However, it has limitations, such as its inability to model complex relationships unless transformed features are used. Regularization techniques (like L1 and L2) can mitigate overfitting, especially in high-dimensional datasets, but they can also lead to underfitting if applied too aggressively. Additionally, logistic regression may not perform well with imbalanced datasets, where the positive class is rare. In such cases, alternative methods like ensemble techniques (e.g., Random Forests or Gradient Boosting) may be more suitable.

A strong candidate would discuss the importance of feature scaling (using techniques like min-max normalization or standardization) to improve convergence during training. They could also elaborate on model evaluation metrics, such as Precision, Recall, F1-score, and ROC-AUC, specifically in the context of imbalanced classes. Furthermore, discussing hyperparameter tuning strategies, such as cross-validation and grid search, would demonstrate an understanding of how to optimize model performance effectively. Lastly, exploring advanced topics like multi-class logistic regression or extensions to other classification algorithms could provide insight into the candidate's depth of knowledge.

Implement logistic regression from scratch

Plaid

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Key Topics to Cover

How to Approach This

Possible Follow-up Questions

Sharpen Your Skills on Codemia

Sample Answer

Core Concept: Logistic Regression

How It Works: Optimization via Gradient Descent

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