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Note: Autograd is still being maintained but is no longer actively developed. The main developers (Dougal Maclaurin, David Duvenaud, Matt Johnson, and Jamie Townsend) are now working on JAX, with Dougal and Matt working on it full-time. JAX combines a new version of Autograd with extra features such as jit compilation.

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Autograd can automatically differentiate native Python and Numpy code. It can handle a large subset of Python's features, including loops, ifs, recursion and closures, and it can even take derivatives of derivatives of derivatives. It supports reverse-mode differentiation (a.k.a. backpropagation), which means it can efficiently take gradients of scalar-valued functions with respect to array-valued arguments, as well as forward-mode differentiation, and the two can be composed arbitrarily. The main intended application of Autograd is gradient-based optimization. For more information, check out the tutorial and the examples directory.

Example use:

>>> import autograd.numpy as np  # Thinly-wrapped numpy
>>> from autograd import grad    # The only autograd function you may ever need
>>> def tanh(x):                 # Define a function
...     y = np.exp(-2.0 * x)
...     return (1.0 - y) / (1.0 + y)
>>> grad_tanh = grad(tanh)       # Obtain its gradient function
>>> grad_tanh(1.0)               # Evaluate the gradient at x = 1.0
>>> (tanh(1.0001) - tanh(0.9999)) / 0.0002  # Compare to finite differences

We can continue to differentiate as many times as we like, and use numpy's vectorization of scalar-valued functions across many different input values:

>>> from autograd import elementwise_grad as egrad  # for functions that vectorize over inputs
>>> import matplotlib.pyplot as plt
>>> x = np.linspace(-7, 7, 200)
>>> plt.plot(x, tanh(x),
...          x, egrad(tanh)(x),                                     # first  derivative
...          x, egrad(egrad(tanh))(x),                              # second derivative
...          x, egrad(egrad(egrad(tanh)))(x),                       # third  derivative
...          x, egrad(egrad(egrad(egrad(tanh))))(x),                # fourth derivative
...          x, egrad(egrad(egrad(egrad(egrad(tanh)))))(x),         # fifth  derivative
...          x, egrad(egrad(egrad(egrad(egrad(egrad(tanh))))))(x))  # sixth  derivative

See the tanh example file for the code.


You can find a tutorial here.

End-to-end examples

How to install

Install Autograd using Pip:

pip install autograd

Some features require SciPy, which you can install separately or as an optional dependency along with Autograd:

pip install "autograd[scipy]"


Autograd was written by Dougal Maclaurin, David Duvenaud, Matt Johnson, Jamie Townsend and many other contributors. The package is currently still being maintained, but is no longer actively developed. Please feel free to submit any bugs or feature requests. We'd also love to hear about your experiences with autograd in general. Drop us an email!

We want to thank Jasper Snoek and the rest of the HIPS group (led by Prof. Ryan P. Adams) for helpful contributions and advice; Barak Pearlmutter for foundational work on automatic differentiation and for guidance on our implementation; and Analog Devices Inc. (Lyric Labs) and Samsung Advanced Institute of Technology for their generous support.