Understanding Gradients in Machine Learning. Taking derivatives of tensor-valued functions, with examples.
When I first began studying neural networks, I was immediately confronted with formulas for backpropagating gradients, starting from the loss function computed at the end of the network and working back layer-by-layer. These formulas already looked quite complex even despite making several basic assumptions — such as a fully-connected network and sigmoid activation functions after each neuron. I wondered how popular packages like TensorFlow and Pytorch perform the same operations but for arbitrary mathematical functions.
Modern machine learning packages use “automatic differentiation” (or “autograd”) to handle this, and with that name it sounds like everything just happens; as if you write the operations and the computer just figures out all the derivatives. But it can’t be this simple. In an operation such as convolution, a multiplicative sum is computed using a weight filter of some chosen size positioned at various locations over an input image, and the result is a new image in which each value corresponds to one such sum. How does “autograd” handle this?
A 2x2 convolution: How do you take the derivative of that? (Figure by author)
Here we will:
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