Implement linear regression from scratch

Question

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

Codemia · Accepted Answer

Logistic Regression is a statistical method used for binary classification problems. Unlike linear regression, which predicts continuous outcomes, logistic regression models the probability that a given input belongs to a particular category. The core concept lies in the logistic function (or sigmoid function), defined as:

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

where $ z = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + ... + \beta_n x_n $ is the linear combination of the input features. The output of the logistic function ranges between 0 and 1, making it suitable for binary classification where we interpret the output as the probability of the positive class.

In logistic regression, we optimize the weights $ \beta $ using the maximum likelihood estimation (MLE). The likelihood function for binary outcomes is given by:

$$ L(\beta) = \prod_{i=1}^{n} (y_i \sigma(z_i) + (1 - y_i)(1 - \sigma(z_i))) $$

To find the optimal $ \beta $, we take the natural logarithm of the likelihood to form the log-likelihood:

$$ \ell(\beta) = \sum_{i=1}^{n} (y_i \log(\sigma(z_i)) + (1 - y_i)\log(1 - \sigma(z_i))) $$

We then use gradient ascent to maximize this log-likelihood function. The gradient is computed with respect to the weights, updating them iteratively until convergence. The learning rate $ \alpha $ is a crucial hyperparameter that controls the step size during optimization.

In a company like Dropbox, logistic regression can be applied to various binary classification tasks, such as spam detection in file sharing, predicting user churn, or classifying documents based on their content. For instance, in user behavior analysis, logistic regression can help predict whether a user will upgrade to a paid plan based on their usage patterns. The implementation from scratch allows for fine-tuning and understanding of the model's behavior, which is crucial for building trust in automated systems.

Logistic regression is simple and interpretable, making it a great starting point for binary classification. However, it assumes a linear relationship between the features and the log-odds of the outcome, which can lead to poor performance on non-linear datasets. Additionally, it can suffer from overfitting if the number of features is high relative to the number of observations. Regularization techniques like L1 (Lasso) and L2 (Ridge) can help mitigate overfitting by penalizing large coefficients. In contrast, for very complex datasets, ensemble methods like random forests or gradient boosting may be more appropriate.

A strong candidate would explore enhancements like implementing regularization to prevent overfitting, discussing techniques for handling class imbalance (e.g., using weighted loss functions), and considering feature engineering strategies. Additionally, they would delve into model evaluation metrics appropriate for classification, such as ROC-AUC, precision-recall curves, and F1-score, especially in the context of imbalanced datasets. Understanding how to interpret coefficients can also be crucial for providing insights into feature importance, which can enhance model explainability and trustworthiness.

Implement linear regression from scratch

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