~/writing/math-foundations-for-reasoning-models $ ls
Backpropagation and Training
Bridge to Deep Representations
Backpropagation explains how layered networks can be trained, setting up the question of what deeper layers learn.
~/writing/math-foundations-for-reasoning-models $ cat topic.md
Bridge To Deep Representations
Where We Are
This chapter explained how neural networks are trained.
The loop was:
forward pass -> loss computation -> backward pass -> parameter update
Backpropagation made the backward part precise.
loss -> gradients -> parameter updates
Training changes a network by using loss to update parameters.
What Backpropagation Added
The previous chapter showed that a neural network is a composition of functions.
This chapter showed how to train that composition.
composed functions -> forward output
loss -> backward gradients
gradients -> updates
Backpropagation tells each parameter how it contributed to loss.
Backpropagation makes layered function composition trainable.
Why Depth Comes Next
Once we can train layers, the next question is:
what happens when we stack many layers?
More layers create more stages of representation.
input
-> early representation
-> later representation
-> output
This leads to deep representations.
Depth means repeated transformation through many layers.
Representation Changes Across Layers
A layer does not just pass data forward unchanged.
It transforms the representation.
In a deeper network, later layers receive values already transformed by earlier layers.
raw input -> transformed signal -> more abstract signal -> output
The next chapter focuses on how depth and representation learning work.
Deep networks build representations through repeated learned transformations.
Bridge Question
The next chapter begins with:
what does depth add to a neural network?
That leads to:
depth
width
multi-layer networks
representation learning
feature hierarchy
vanishing gradients
exploding gradients
skip connections
After training mechanics, the next step is understanding learned representations.
Final Summary
- Training uses forward pass, loss, backward pass, and updates.
- Backpropagation computes gradients through composed functions.
- Gradients allow parameters in every layer to be updated.
- Once many layers can be trained, depth becomes important.
- Deep networks transform representations across layers.
- The next chapter focuses on deep representations.
~/writing/math-foundations-for-reasoning-models $ open
Continue the series.
Depth