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autodiff

My implementation of an autodiff library that supports scalar and tensor-like (vector, matrix, and higher dimension) data types. Everything was done from scratch. The examples folder shows how to use it for some cases, including the implementation of a NeRF model that is able to learn an image, and a LeNet-5 network that can classify MNIST digits with 95% accuracy. This project was made just for fun.

Reference image Learning process Final learned image

There are many examples at the examples folder. You can check below some snippets for examples/nerf.cpp which implements a NeRF with positional encoding and 4 layers total.

Code walkthrough

Model

#include "autodiff/autodiff.h"

ad::Vector<3> forward(ad::Vector<2>& xy) {
    ad::Vector<32> input = ad::nn::positional_encoding<8>(xy);

    ad::Vector<128> l1 = ad::relu(w1 * input + b1);
    ad::Vector<128> l2 = ad::relu(w2 * l1 + b2);
    ad::Vector<128> l3 = ad::relu(w3 * l2 + b3);
    ad::Vector<3> output = ad::sigmoid(w4 * l3 + b4);

    return output;
}

The training chooses random pixels from an image and uses that as a loss so the network learns how to reproduce it.

Training

#include "autodiff/autodiff.h"

common::Bitmap3f image = common::load_bitmap<common::Color3f>(
    "sunflower.ppm");
auto [width, height] = y.size();

for (size_t step = 0; step < steps; ++step) {
    unsigned int px = rand() % width;
    unsigned int py = rand() % height;

    ad::Vector<2> xy({(float)px / width, (float)py / height});
    common::Tensor<float, 3> y_i = y(px, py);

    auto y_est = nerf.forward(xy);
    auto loss = ad::pow(y_est - y_i, 2);
    loss.backward();
    nerf.update(lr);
}

Usage

You only need to include autodiff.h:

#include "autodiff/autodiff.h"

The available classes and functions are listed below.

autodiff/value.h

Base types

  • ad::Value for scalar values (uses float)
  • ad::Vector<N> for vectors of size N (uses common::Tensor<float, N>)
  • ad::Matrix<N, M> for matrices of N rows and M columns (uses common::Tensor<float, N, M>)
  • ad::Tensor<Shape...> for general tensor data (uses common::Tensor<float, Shape...>)

Each of these forms a node that computes y = f(x). Each node has the following methods:

  • value(): Returns f(x)
  • backward(): Computes the derivatives of all children nodes in the graph (see grad()).
  • grad(): Returns dy/dx, where y is the variable you called backward() on, and x is the current node.
  • requires_grad(): Whether derivatives will be computed on its when backward() is called.
  • set_requires_grad(bool requires_grad): Sets the requires_grad attribute.
  • update(float lr): Short for value() -= grad() * lr

Basic operators

  • +, -, /
  • *: Either (1) at least one of the operands is a scalar (2) both are vectors, which results in element-wise multiplcation or (3) matrix-matrix/matrix-vector multiplication.

Extended operators

Those apply such functions in an element-wise manner if the input is a vector/matrix/tensor.

  • Exponential functions: ad::pow, ad::log, ad::exp.
  • Trigonometric functions: ad::sin, ad::cos.
  • Activation functions: ad::relu, ad::sigmoid.
  • ad::sum: Reduces input to a scalar.
  • ad::expand<N>: Works for scalar and vector data. Repeats it N times and returns this in a vector.
  • ad::flatten: Converts a Tensor to a flattened Vector

autodiff/nn.h

  • ad::nn::positional_encoding<N>(T& x): Converts x to {sin(x), cos(x), ... , sin(x*2^(N-1)), cos(x*2^(N-1))}. Accepts scalar or vector values. In the case of a vector with M values, the result is {sin(x_0), ... , sin(x_M), cos(x_0), ..., cos(x_M), ... }.
  • ad::nn::softmax(Vector<N>& x): Converts x to a probability distribution of N possible outcomes.
  • ad::nn::cross_entropy(Vector<N>& logits, Vector<N>& target): Computes the cross entropy loss.
  • ad::nn::conv_2d(Tensor<C_IN, H, W>& input, Tensor<C_OUT, C_IN, K, K>& kernel, Tensor<C_OUT>& bias): Applies a convolution (with stride 1, adding the necessary zero padding).
  • ad::nn::avg_pool_2d<N>: Does average pooling with stride N and adding the necessary zero padding (like ceil_mode=True in PyTorch).