Implement logistic 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. The core concept revolves around the logistic function (or sigmoid function), which maps any real-valued number into the range of (0, 1). Mathematically, the logistic function is defined as:

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

where $ z $ is a linear combination of the input features. The output can be interpreted as the probability that the output belongs to the positive class. The model parameters are optimized using maximum likelihood estimation, aiming to maximize the likelihood of observing the given data under the model's predictions.

In logistic regression, we define the hypothesis function as:

$$
H(X) = \sigma(W^TX + b)
$$

where $ W $ are the weights, $ b $ is the bias, and $ X $ is the input feature vector. The cost function for logistic regression is derived from the negative log-likelihood:

$$
J(W, b) = -\frac{1}{m} \sum_{i=1}^{m} [y^{(i)} \log(H(X^{(i)})) + (1 - y^{(i)}) \log(1 - H(X^{(i)}))]
$$

To optimize $ W $ and $ b $, we utilize gradient descent:

$$
W := W - \alpha \frac{\partial J}{\partial W}
$$

where $ \alpha $ is the learning rate. Regularization techniques (L1 and L2) can be added to the cost function to manage overfitting, which is crucial in high-dimensional feature spaces.

In a financial institution like Capital One, logistic regression can be used for credit scoring, predicting the likelihood of a customer defaulting on a loan. Implementing logistic regression from scratch allows for full control over the model's architecture and hyperparameters, such as regularization strength and learning rate. Additionally, handling class imbalance is critical; techniques such as oversampling the minority class or using class weights during optimization can significantly enhance model performance in imbalanced datasets typically seen in fraud detection.

Logistic regression is computationally efficient and interpretable, making it an excellent choice for binary classification tasks. However, it assumes a linear relationship between the features and the log-odds of the outcome, which can be a limitation in complex datasets. Regularization helps mitigate overfitting but requires careful tuning of hyperparameters. In cases of severe non-linearity or multi-class problems, other models like decision trees or neural networks might be more suitable. Thus, it's essential to analyze the dataset characteristics before choosing logistic regression.

A strong candidate would delve into advanced topics like the use of dropout as a regularization technique in more complex logistic regression frameworks (like neural networks). They could discuss evaluation metrics beyond accuracy, such as ROC-AUC and F1-score, especially in imbalanced datasets. Additionally, they could explore feature engineering techniques, such as polynomial feature expansion, to capture non-linear relationships while still using logistic regression. Finally, discussing the implications of interpretability in financial decisions can highlight the importance of model transparency in Capital One's context.

Implement logistic regression from scratch

Capital One

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Core Concept: Logistic Regression

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