Logistic Regression Cannot Perfectly Model Well-separated Classes

I have made several observations when using logistic regression, but one of them is a bit peculiar:

Confused?

Let me explain my thought process.

Background

We all know that logistic regression outputs the probability of a class, which is given by:

What’s more, its loss function is the binary cross-entropy loss (or log loss), which is written as:

• When the true label yᵢ = 1, the loss value is → -log(ŷᵢ).

• When the true label yᵢ = 0, the loss value is → -log(1-ŷᵢ).

And as we all know, the model attempts to determine the parameters (θ₀, θ₁) by minimizing the loss function.

Proof

The above output probability can be rewritten as follows:

Simply put, we have represented the output probability function in terms of two other parameters.

All good?

Now consider the following 1D dataset with well-separated classes:

Modeling this with a logistic regression model from sklearn, we get the following:

Printing the (m,c) values from the below formulation, we get m=2.21, c=-2.33.

Let’s see if we can obtain a better regression curve now.

More specifically, we shall try fitting a logistic regression model with different different values of m.

The results are shown below:

From the above visual, it is clear that increasing the m parameter consistently leads to:

• A smaller (yet non-zero) loss value.

• A better regression fit.

And to obtain the best regression fit, the sigmoid curve must be entirely vertical in the middle, which is never possible.

Thus, the abovementioned point: “Logistic regression can never perfectly fit well-separated classes” is entirely valid.

That is why many open-source implementations (sklearn, for instance) stop after a few iterations.

So it is important to note that they still leave a little scope for improvement if needed.

I would love to know your thoughts on this little experiment.

On a side note, have you ever wondered the following:

1. Why do we use Sigmoid in logistic regression?

2. Why do we ‘log loss’ in logistic regression?

3. Why sklearn’s logistic regression has no learning rate hyperparamter?

Check out these two deep dives to learn this:

• Why Do We Use Sigmoid in Logistic Regression?

• Why Do We Use ‘log loss’ to Train Logistic Regression?

• Why Sklearn’s Logistic Regression Has No Learning Rate Hyperparameter?

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