Backpropagation vs. Gradient Descent in Linear and Logistic Regression

Note that the term "backpropagation" is technically not used for standard linear or logistic regression. Instead, we simply call the optimization process Gradient Descent.

Backpropagation is also related to vanishing gradient.  Vanishing gradient can occur only across multiple layers. Thus both backpropagation and vanishing gradient are related to multiple layers only, then are never used for single layer models.

Important Note

The word propagation implies a sequence of layers. Because linear regression and logistic regression contain only a single layer of weights, there is no sequence of layers through which errors can propagate backward.

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What is Backpropagation?

Backpropagation is short for "backward propagation of errors." It is an algorithm specifically designed for multi-layer neural networks.

The Sequence

It uses the mathematical Chain Rule to calculate gradients. It starts at the final output layer, computes the error, and then passes (propagates) that error backward, layer by layer, to update the weights at the very beginning of the network.

During backpropagation, the error is propagated backward through multiple layers using the Chain Rule.

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Why the Confusion in Linear and Logistic Regression?

You might hear people use the term loosely in linear or logistic regression for two main reasons:

1. Neural Network Equivalent

A logistic regression model is mathematically identical to a single-neuron neural network with a sigmoid activation function.

2. The "One-Step" Chain Rule

Even in logistic regression, you use a small version of the Chain Rule. To find how the loss changes with respect to the weights,

∂Loss / ∂W = (∂Loss / ∂Prediction) × (∂Prediction / ∂Score) × (∂Score / ∂W)

Because this calculation proceeds backward from the Loss to the Weights, some instructors casually refer to it as "backpropagation", even though it only spans a single layer.

Logistic regression applies the Chain Rule over a single layer rather than propagating errors through multiple layers.

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Summary of Terms

Term	Definition	Used In
Gradient Descent	The overall optimization algorithm used to update weights to minimize error.	Linear regression, logistic regression, and neural networks.
Backpropagation	The specific method used inside gradient descent to calculate gradients across multiple sequential layers.	Multi-layer neural networks.

Key Takeaway

Gradient Descent is the optimization algorithm that updates model parameters to minimize the loss function. Backpropagation is the gradient-computation algorithm used within gradient descent for multi-layer neural networks. In standard linear regression and logistic regression, we typically refer only to Gradient Descent, since there are no multiple layers through which errors must propagate.