The L1 Loss Gradient, Explained From Scratch
Last Updated on April 10, 2026 by Editorial Team
Author(s): Utkarsh Mittal
Originally published on Towards AI.
A complete, step-by-step walkthrough of how gradient descent works with absolute-value loss — with diagrams you can actually follow.
If you’ve ever read a deep learning tutorial and hit a derivative that seems to appear from nowhere, this article is for you. We’re going to break down one of the simplest — yet most instructive — gradient calculations in machine learning: the gradient of L1 (absolute-value) loss with respect to a single weight.
The article explains the gradient calculation of L1 loss through a structured approach, starting with a simple regression model and discussing its components, the loss function, and how to derive the gradient with respect to a weight. It emphasizes clarity by using concrete examples and progressively builds the understanding through the chain rule in calculus. The synopsis concludes by contrasting L1 loss’s insensitivity to outliers with L2 loss’s responsiveness to error magnitude, ultimately guiding on when to use each loss function effectively.
Read the full blog for free on Medium.
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