Why is the gradient multiplied by grad_output in this

There is something I don’t quite understand in this tutorial:

There is example that implements a simple linear function in pytorch:

Inherit from Function

class LinearFunction(Function):

# Note that both forward and backward are @staticmethods
# bias is an optional argument
def forward(ctx, input, weight, bias=None):
    ctx.save_for_backward(input, weight, bias)
    output = input.mm(weight.t())
    if bias is not None:
        output += bias.unsqueeze(0).expand_as(output)
    return output

# This function has only a single output, so it gets only one gradient
def backward(ctx, grad_output):
    # This is a pattern that is very convenient - at the top of backward
    # unpack saved_tensors and initialize all gradients w.r.t. inputs to
    # None. Thanks to the fact that additional trailing Nones are
    # ignored, the return statement is simple even when the function has
    # optional inputs.
    input, weight, bias = ctx.saved_tensors
    grad_input = grad_weight = grad_bias = None

    # These needs_input_grad checks are optional and there only to
    # improve efficiency. If you want to make your code simpler, you can
    # skip them. Returning gradients for inputs that don't require it is
    # not an error.
    if ctx.needs_input_grad[0]:
        grad_input = grad_output.mm(weight)
    if ctx.needs_input_grad[1]:
        grad_weight = grad_output.t().mm(input)
    if bias is not None and ctx.needs_input_grad[2]:
        grad_bias = grad_output.sum(0)

    return grad_input, grad_weight, grad_bias

The question is probably simplistic, but if the backward is simply the “gradient formula”, I don’t understand why the gradient is multiplied by grad_output?

In my mind I have :

e.g. say x = input for clarity
w= weight
b = bias

f (x)= w* x + b

d f/ dx = w
d f/ dw = x
d f/ db = 1

What piece of the puzzle am I missing?

You are missing the use the gradient calculated by the loss (or the next layer). Backpropagation for a Linear Layer by Justin Johnson derives the formula for a simple case and was part of the CS231n lecture.

1 Like