ML Interview Q Series: Why is it not appropriate to use a linear regression model in place of logistic regression for classification tasks?

📚 Browse the full ML Interview series here.

Comprehensive Explanation

Logistic regression is fundamentally designed for binary (and by extension multi-class) classification, whereas linear regression is intended for continuous-valued outputs. Trying to use linear regression for a classification problem creates several practical and theoretical issues:

Output Range

A linear regression model produces any real-valued number from negative to positive infinity. In contrast, a classification task usually needs an output that can be interpreted as a probability (i.e., ranging strictly between 0 and 1). Logistic regression addresses this by applying the sigmoid or logistic function to map any real-valued input z into a [0, 1] range.

Here, z is w dot x + b, where w is the weight vector, x is the input features, and b is the bias. The value of z can be any real number, but after passing through the sigmoid function, the output is always between 0 and 1. This allows logistic regression to interpret the output directly as a class probability.

Decision Boundary and Interpretability

Logistic regression creates a linear decision boundary in the input feature space but uses the log-odds transformation to ensure that the predicted values are proper probabilities. With linear regression, even if you try to threshold the raw output (for example, at 0.5) to classify, the values can exceed the typical 0 to 1 probability range. This leads to difficulties in interpreting these values as valid probabilities.

Loss Function and Gradient Behavior

Linear regression typically uses a mean squared error loss. However, for a classification setting, this loss is neither appropriate nor efficient. In logistic regression, one usually employs cross-entropy (logistic) loss, which is more suitable for probability-based outputs because it penalizes confidently wrong predictions more severely. Cross-entropy loss also aligns well with the likelihood maximization perspective.

In the above expression, N is the number of training examples, y_{i} is the true label (0 or 1), and hat{y}_{i} is the predicted probability (output of the sigmoid function). This loss function has well-defined gradients and helps the model converge reliably to a better separating decision boundary.

Handling Outliers and Class Imbalance

Linear regression is more susceptible to outliers in its predictions for classification scenarios, as large negative or positive values can disproportionately influence the regression fit. Logistic regression, by producing outputs between 0 and 1, is less susceptible to extreme predictions. Moreover, logistic regression’s formulation allows better integration of class weights and thresholds when dealing with imbalanced datasets.

Probabilistic Interpretation

Logistic regression naturally lends itself to probability estimates for each class. This interpretation is helpful for tasks requiring calibrated probabilities, such as risk assessment, medical diagnostics, and other decision-making processes. Linear regression does not directly provide such a probabilistic interpretation for classification problems.

Follow-up Question 1

Could we just take the output of a linear regression, clamp it to [0, 1], and then interpret it as a probability?

A clamping approach does not solve the fundamental issues. Though you might forcibly squeeze the outputs into [0, 1], you lose the log-odds interpretation that underlies logistic regression. The linear regression objective is still not optimized for classification. You would not be directly maximizing the likelihood for the binary outcomes, and as a result, you could end up with a suboptimal decision boundary and worse predictive performance.

Follow-up Question 2

What if the prediction values of linear regression are generally within the [0, 1] range for a particular dataset? Would it be acceptable then?

Even if predictions sometimes happen to lie in [0, 1], the model is not guaranteed to output valid probabilities in all situations, especially for new test data or slightly different distributions. Moreover, the learning objective is not the same. The logistic regression objective function is specifically crafted to optimize classification performance and probabilistic interpretation. Relying on linear regression predictions to remain between [0, 1] could fail under a distribution shift or if the training data does not closely resemble future data.

Follow-up Question 3

Why does logistic regression typically use a log-loss (cross-entropy) instead of mean squared error?

The log-loss is more suitable for classification because it directly corresponds to maximizing the likelihood of the observed data under the Bernoulli distribution assumption. Mean squared error loss can lead to slower convergence and poorer decision boundaries for classification. Log-loss, on the other hand, penalizes misclassifications more sharply and encourages the model to assign high probabilities to the correct class, aligning well with the probability interpretation of logistic regression.

Follow-up Question 4

Are there other classification algorithms that use similar ideas to logistic regression but different link functions?

Yes, in generalized linear models (GLMs), different link functions can be chosen based on the nature of the target variable distribution. For binary classification, the logit link (used by logistic regression) is common, but alternatives exist (like the probit link for probit regression). Each link function has a slightly different interpretation for how it transforms linear combinations of features into probabilities, but the core idea of mapping real-valued inputs to [0, 1] remains consistent.

Follow-up Question 5

In practice, how do you implement logistic regression in a modern machine learning framework such as PyTorch or TensorFlow?

Below is a simple outline in Python with PyTorch. We assume you already have an input feature tensor X of size (N, d) and a label tensor y of size (N,) with values 0 or 1.

import torch
import torch.nn as nn

# Simple logistic regression model
class LogisticRegressionModel(nn.Module):
    def __init__(self, input_dim):
        super(LogisticRegressionModel, self).__init__()
        self.linear = nn.Linear(input_dim, 1)

    def forward(self, x):
        # Sigmoid activation for probability
        return torch.sigmoid(self.linear(x))

# Instantiate model
model = LogisticRegressionModel(input_dim=X.shape[1])
criterion = nn.BCELoss()  # Binary Cross Entropy Loss
optimizer = torch.optim.SGD(model.parameters(), lr=0.01)

# Training loop (simplified)
for epoch in range(100):
    optimizer.zero_grad()
    outputs = model(X)
    loss = criterion(outputs.squeeze(), y.float())
    loss.backward()
    optimizer.step()

In this example:

• We have a nn.Linear layer that computes the linear combination of inputs w^T x + b.

• We pass that result through a sigmoid function to ensure the outputs lie between 0 and 1.

• We use BCELoss, which implements the cross-entropy loss for binary classification.

This setup ensures the model’s output can be interpreted as probabilities and that the training optimizes for accurate classification in a probabilistic sense.