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Autodiff

Automatic differentiation is a method by Seppo Linnainmaa for quickly computing the partial derivatives of a function defined by a straight line program. This method is very important in machine learning because it makes it easy to implement gradient-based optimization methods (which are, among other things, used to fit neural networks to data; did you know that neural networks are just straight line programs?).

Since I will teach this method in my class this semester, I thought I'd try my hand at implementing it.

The real "meat" of autodiff are the forward and backward methods. The rest is just a bunch of library functions that you can use in the straight line program.

Andrej Karpathy also has a nice implementation of autodiff for educational purposes.

Karpathy's micrograd

import numpy as np

class Var:
    def __init__(self, value=None, deriv=None, op=None, children=None, pds=None):
        self.value = value
        self.deriv = deriv
        self.op = op
        self.children = children
        self.pds = pds
        self.order = None

    def __repr__(self):
        return f'Var({self.value})'

    def forward(self):
        if self.order is None:
            self.order = topological_sort(self)
        for u in self.order:
            u.deriv = 0.
            if u.children is not None:
                u.value = u.op(*[ c.value for c in u.children ])

    def backward(self):
        for v in reversed(self.order):
            if v == self:
                v.deriv = 1.
            if v.children is not None:
                local_values = [ c.value for c in v.children ]
                for c, pd in zip(v.children, v.pds):
                    c.deriv += v.deriv * pd(*local_values)

def add(a, b):
    return Var(op=np.add, children=[a, b], pds=[__add_pd, __add_pd])

def subtract(a, b):
    return Var(op=np.subtract, children=[a, b], pds=[_add_pd, __subtract_pd_b])

def multiply(a, b):
    return Var(op=np.multiply, children=[a, b], pds=[__multiply_pd_a, __multiply_pd_b])

def divide(a, b):
    return Var(op=np.divide, children=[a, b], pds=[__divide_pd_a, __divide_pd_b])

def negative(a):
    return Var(op=np.negative, children=[a], pds=[__negative_pd])

def square(a):
    return Var(op=np.square, children=[a], pds=[__square_pd])

def exp(a):
    return Var(op=np.exp, children=[a], pds=[__exp_pd])

def sin(a):
    return Var(op=np.sin, children=[a], pds=[__sin_pd])

def cos(a):
    return Var(op=np.cos, children=[a], pds=[__cos_pd])

def __add_pd(a, b):
    return 1.

def __subtract_pd_b(a, b):
    return -1.

def __multiply_pd_a(a, b):
    return b

def __multiply_pd_b(a, b):
    return a

def __divide_pd_a(a, b):
    return 1. / b

def __divide_pd_b(a, b):
    return -a / (b * b)

def __negative_pd(a):
    return -1.

def __square_pd(a):
    return 2. * a

def __exp_pd(a):
    return np.exp(a)

def __sin_pd(a):
    return np.cos(a)

def __cos_pd(a):
    return -np.sin(a)

def topological_sort(v):
    visited = set()
    vertices = []
    def explore(v):
        if v not in visited:
            visited.add(v)
            if v.children is not None:
                for c in v.children:
                    explore(c)
            vertices.append(v)
    explore(v)
    return vertices

if __name__ == '__main__':
    x = Var(value=1)
    w = Var(value=4)
    v1 = multiply(x, w)
    v2 = sin(v1)
    v3 = add(v1, v2)
    v4 = square(v2)
    v5 = exp(v3)
    v6 = multiply(v4, w)
    v7 = add(v5, v6)
    for t in range(30):
        v7.forward()
        v7.backward()
        print(f'w={w.value}, v7={v7.value}, (dv7)/(dw)={w.deriv}')
        w.value -= 0.1 * w.deriv
    print(f'w={w.value}, v7={v7.value}, (dv7)/(dw)={w.deriv}')

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