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To understand why this is hard, think about the numbers.
A small network might have a few thousand weights. A larger one has millions. Each weight is a number that can be set to anything: 0.3, 1.7, -0.4.
When you have a million weights and each one can take any value, the number of possible combinations is not just large. It's so large that if you tried one combination every second, you would not finish before the universe ended.
Trying them all is not an option.
What you need is a smarter approach. And here's the key insight: you don't need to find the perfect combination from scratch. You just need to keep moving in the right direction.
Imagine you're standing on a hillside in thick fog. You can't see the valley. You can't see the whole landscape. But you can feel which way the ground slopes under your feet.
So you take a step downhill. Then check again. Then another step.
You don't need to see the whole mountain. You just need to know which way is down from where you're standing.
Backpropagation tells you which direction is downhill. What's left is knowing how to walk.