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@timvieira
Last active May 14, 2022 04:32
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Simple example of manually performing "automatic" differentiation.
"""
Simple example of manually performing "automatic" differentiation
"""
import numpy as np
from numpy import exp, sin, cos
def f(x, with_grad=False):
# Need to cache intermediates from forward pass (might not use all of them).
a = exp(x)
b = a**2
c = a + b
d = exp(c)
e = sin(c)
f = d + e
if not with_grad:
return f
# initialize all adjoints to zero. "Adjoints" are specifically the gradients
# of intermediates wrt to the output variable. Often they are abbreviated
# since they all are df_*.
df_da = df_db = df_dc = df_dd = df_de = df_dx = 0
# Loop thru rules in reverse order
# f = d + e ==> two inputs -> two ajoint rules. (At the final output the
# adjoints are just the local gradients.)
df_dd = 1
df_de = 1
# e = sin(c)
de_dc = cos(c) # local gradient: how `e` changes with its input `c`
df_dc += df_de * de_dc # backprop: adjoint from `e` is "converted" into
# adjoint for `c` where `de/dc` is the "conversion
# factor" Note how the units match like in
# dimensional analysis: df/de * de/dc => df/dc. This
# is a useful sanity check.
# d = exp(c)
dd_dc = exp(c)
df_dc += df_dd * dd_dc # Note that because `c` is used twice in the program, its ajoint is a sum!
# c = a + b ==> two inputs -> two ajoint rules
dc_da = 1
dc_db = 1
df_da += df_dc * dc_da
df_db += df_dc * dc_db
# b = a**2
db_da = 2*a
df_da += df_db * db_da # variable `a` is used twice so its adjoints sum, just like `c`
# a = exp(x)
da_dx = exp(x)
df_dx += df_da * da_dx
# Output the gradient of the input variable
return f, df_dx
def main():
from arsenal.math.checkgrad import fdcheck
for x in np.random.uniform(-10, 3, size=10):
x = np.array([x])
fdcheck(lambda: f(x), x, f(x, with_grad=1)[1])
if __name__ == '__main__':
main()
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