---
title: "Backpropagation: Assigning Blame for an Error"
description: "Training a network means knowing how much each weight contributed to the error, which the chain rule gives by passing derivatives backwards through the layers. Every weight is then nudged by gradient "
canonical: https://lightmysky.com/learn/computing/backpropagation-assigning-blame-for-an-error-mt_5bwaSWfF96
source: https://lightmysky.com/learn/computing/backpropagation-assigning-blame-for-an-error-mt_5bwaSWfF96.md
retrieved: 2026-09-12
---

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# Backpropagation: Assigning Blame for an Error

Training a network means knowing how much each weight contributed to the error, which the chain rule gives by passing derivatives backwards through the layers. Every weight is then nudged by gradient descent, and the whole procedure is one pass forward and one pass back.

Subject: Computing · Area: Machine Learning · Ages 21 to 22
Page: https://lightmysky.com/learn/computing/backpropagation-assigning-blame-for-an-error-mt_5bwaSWfF96

## Ready when they can

- Propagate an error backwards through a two-layer network by hand
- Say what the chain rule contributes at each layer boundary
- Explain why a squashing function with a flat tail slows learning

## Lesson: Blame travels backward

Training a stack that can fence the opposite yes corners runs one pass forward and one pass back. Forward, inputs flow layer by layer into a prediction. Then a loss function scores the gap between the prediction and the target.

**Example.** Take y as the square of two x plus one, at x equals one. The inner layer gives three with a local rate of two. The outer square gives a local rate of six, and six times two makes twelve. A second stack, the square of x squared plus one at one, combines four times two into eight.

The backward pass walks the error from the output toward the input. At each layer boundary the chain rule multiplies the local rates together. Static backprop changes only weights and biases, never the wiring. Each weight then takes a small gradient descent nudge downhill against the loss.

A squashing function with a flat tail has a tiny local rate out there. Multiplying by near zero shrinks the signal each layer back, so early layers learn very slowly.

**Recap.** Push forward to score the error, multiply local rates backward, and nudge each weight downhill against the loss.

## Practice

17 questions on this page, each with its working shown.

## Needs first

- [The Multivariable Chain Rule](https://lightmysky.com/learn/mathematics/the-multivariable-chain-rule-mt_CHSHdNxPmJ)
- [Gradient Descent: Following the Slope Downhill](https://lightmysky.com/learn/computing/gradient-descent-following-the-slope-downhill-mt_EjfIT1WgMX)
- [From Perceptron to Multilayer Network](https://lightmysky.com/learn/computing/from-perceptron-to-multilayer-network-mt_TTa7KN1-IO)

## Opens up

- [Deep Learning: Convolution, Sequence and Scale](https://lightmysky.com/learn/computing/deep-learning-convolution-sequence-and-scale-mt_karzGN5JnI)
- [Attention as a Learned Lookup](https://lightmysky.com/learn/computing/attention-as-a-learned-lookup-mt_rrdtS1lHvC)