Backpropagation in Gradient Descent for Neural Networks vs. Linear Regression

Question

I'm trying to understand "Back Propagation" as it is used in Neural Nets that are optimized using Gradient Descent. Reading through the literature it seems to do a few things.

1. Use random weights to start with and get error values
2. Perform Gradient Descent on the loss function using these weights to arrive at new weights.
3. Update the weights with these new weights until the loss function is minimized.

The steps above seem to be the EXACT process to solve for Linear Models (Regression for e.g.)? Andrew Ng's excellent course on Coursera for Machine Learning does exactly that for Linear Regression.

So, I'm trying to understand if BackPropagation does anything more than gradient descent on the loss function.. and if not, why is it only referenced in the case of Neural Nets and why not for GLMs (Generalized Linear Models). They all seem to be doing the same thing- what might I be missing?

Answer 1

The main division happens to be hiding in plain sight: linearity. In fact, extend to question to continuity of the first derivative, and you'll encapsulate most of the difference.

First of all, take note of one basic principle of neural nets (NN): a NN with linear weights and linear dependencies is a GLM. Also, having multiple hidden layers is equivalent to a single hidden layer: it's still linear combinations from input to output.

A "modern' NN has non-linear layers: ReLUs (change negative values to 0), pooling (max, min, or mean of several values), dropouts (randomly remove some values), and other methods destroy our ability to smoothly apply Gradient Descent (GD) to the model. Instead, we take many of the principles and work backward, applying limited corrections layer by layer, all the way back to the weights at layer 1.

Lather, rinse, repeat until convergence.

Does that clear up the problem for you?

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You got it!

A typical ReLU is

f(x) = x if x > 0,
       0 otherwise

A typical pooling layer reduces the input length and width by a factor of 2; in each 2x2 square, only the maximum value is passed through. Dropout simply kills off random values to make the model retrain those weights from "primary sources". Each of these is a headache for GD, so we have to do it layer by layer.

Answer 2

So, I'm trying to understand if BackPropagation does anything more than gradient descent on the loss function.. and if not, why is it only referenced in the case of Neural Nets

I think (at least originally) back propagation of errors meant less than what you describe: the term "backpropagation of errors" only refered to the method of calculating derivatives of the loss function, instead of e.g. automatic differentiation, symbolic differentiation, or numerical differentiation. No matter what the gradient was then used for (e.g. Gradient Descent, or maybe Levenberg/Marquardt).

They all seem to be doing the same thing- what might I be missing?

They're using different models. If your neural network used linear neurons, it would be equivalent to linear regression.