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"""
Demonstration of the Automatic Differentiation (Reverse mode).
Reference: https://en.wikipedia.org/wiki/Automatic_differentiation
Author: Poojan Smart
Email: smrtpoojan@gmail.com
"""
from __future__ import annotations
from collections import defaultdict
from enum import Enum
from types import TracebackType
from typing import Any
import numpy as np
from typing_extensions import Self # noqa: UP035
class OpType(Enum):
"""
Class represents list of supported operations on Variable for gradient calculation.
"""
ADD = 0
SUB = 1
MUL = 2
DIV = 3
MATMUL = 4
POWER = 5
NOOP = 6
class Variable:
"""
Class represents n-dimensional object which is used to wrap numpy array on which
operations will be performed and the gradient will be calculated.
Examples:
>>> Variable(5.0)
Variable(5.0)
>>> Variable([5.0, 2.9])
Variable([5. 2.9])
>>> Variable([5.0, 2.9]) + Variable([1.0, 5.5])
Variable([6. 8.4])
>>> Variable([[8.0, 10.0]])
Variable([[ 8. 10.]])
"""
def __init__(self, value: Any) -> None:
self.value = np.array(value)
# pointers to the operations to which the Variable is input
self.param_to: list[Operation] = []
# pointer to the operation of which the Variable is output of
self.result_of: Operation = Operation(OpType.NOOP)
def __repr__(self) -> str:
return f"Variable({self.value})"
def to_ndarray(self) -> np.ndarray:
return self.value
def __add__(self, other: Variable) -> Variable:
result = Variable(self.value + other.value)
with GradientTracker() as tracker:
# if tracker is enabled, computation graph will be updated
if tracker.enabled:
tracker.append(OpType.ADD, params=[self, other], output=result)
return result
def __sub__(self, other: Variable) -> Variable:
result = Variable(self.value - other.value)
with GradientTracker() as tracker:
# if tracker is enabled, computation graph will be updated
if tracker.enabled:
tracker.append(OpType.SUB, params=[self, other], output=result)
return result
def __mul__(self, other: Variable) -> Variable:
result = Variable(self.value * other.value)
with GradientTracker() as tracker:
# if tracker is enabled, computation graph will be updated
if tracker.enabled:
tracker.append(OpType.MUL, params=[self, other], output=result)
return result
def __truediv__(self, other: Variable) -> Variable:
result = Variable(self.value / other.value)
with GradientTracker() as tracker:
# if tracker is enabled, computation graph will be updated
if tracker.enabled:
tracker.append(OpType.DIV, params=[self, other], output=result)
return result
def __matmul__(self, other: Variable) -> Variable:
result = Variable(self.value @ other.value)
with GradientTracker() as tracker:
# if tracker is enabled, computation graph will be updated
if tracker.enabled:
tracker.append(OpType.MATMUL, params=[self, other], output=result)
return result
def __pow__(self, power: int) -> Variable:
result = Variable(self.value**power)
with GradientTracker() as tracker:
# if tracker is enabled, computation graph will be updated
if tracker.enabled:
tracker.append(
OpType.POWER,
params=[self],
output=result,
other_params={"power": power},
)
return result
def add_param_to(self, param_to: Operation) -> None:
self.param_to.append(param_to)
def add_result_of(self, result_of: Operation) -> None:
self.result_of = result_of
class Operation:
"""
Class represents operation between single or two Variable objects.
Operation objects contains type of operation, pointers to input Variable
objects and pointer to resulting Variable from the operation.
"""
def __init__(
self,
op_type: OpType,
other_params: dict | None = None,
) -> None:
self.op_type = op_type
self.other_params = {} if other_params is None else other_params
def add_params(self, params: list[Variable]) -> None:
self.params = params
def add_output(self, output: Variable) -> None:
self.output = output
def __eq__(self, value) -> bool:
return self.op_type == value if isinstance(value, OpType) else False
class GradientTracker:
"""
Class contains methods to compute partial derivatives of Variable
based on the computation graph.
Examples:
>>> with GradientTracker() as tracker:
... a = Variable([2.0, 5.0])
... b = Variable([1.0, 2.0])
... m = Variable([1.0, 2.0])
... c = a + b
... d = a * b
... e = c / d
>>> tracker.gradient(e, a)
array([-0.25, -0.04])
>>> tracker.gradient(e, b)
array([-1. , -0.25])
>>> tracker.gradient(e, m) is None
True
>>> with GradientTracker() as tracker:
... a = Variable([[2.0, 5.0]])
... b = Variable([[1.0], [2.0]])
... c = a @ b
>>> tracker.gradient(c, a)
array([[1., 2.]])
>>> tracker.gradient(c, b)
array([[2.],
[5.]])
>>> with GradientTracker() as tracker:
... a = Variable([[2.0, 5.0]])
... b = a ** 3
>>> tracker.gradient(b, a)
array([[12., 75.]])
"""
instance = None
def __new__(cls) -> Self:
"""
Executes at the creation of class object and returns if
object is already created. This class follows singleton
design pattern.
"""
if cls.instance is None:
cls.instance = super().__new__(cls)
return cls.instance
def __init__(self) -> None:
self.enabled = False
def __enter__(self) -> Self:
self.enabled = True
return self
def __exit__(
self,
exc_type: type[BaseException] | None,
exc: BaseException | None,
traceback: TracebackType | None,
) -> None:
self.enabled = False
def append(
self,
op_type: OpType,
params: list[Variable],
output: Variable,
other_params: dict | None = None,
) -> None:
"""
Adds Operation object to the related Variable objects for
creating computational graph for calculating gradients.
Args:
op_type: Operation type
params: Input parameters to the operation
output: Output variable of the operation
"""
operation = Operation(op_type, other_params=other_params)
param_nodes = []
for param in params:
param.add_param_to(operation)
param_nodes.append(param)
output.add_result_of(operation)
operation.add_params(param_nodes)
operation.add_output(output)
def gradient(self, target: Variable, source: Variable) -> np.ndarray | None:
"""
Reverse accumulation of partial derivatives to calculate gradients
of target variable with respect to source variable.
Args:
target: target variable for which gradients are calculated.
source: source variable with respect to which the gradients are
calculated.
Returns:
Gradient of the source variable with respect to the target variable
"""
# partial derivatives with respect to target
partial_deriv = defaultdict(lambda: 0)
partial_deriv[target] = np.ones_like(target.to_ndarray())
# iterating through each operations in the computation graph
operation_queue = [target.result_of]
while len(operation_queue) > 0:
operation = operation_queue.pop()
for param in operation.params:
# as per the chain rule, multiplying partial derivatives
# of variables with respect to the target
dparam_doutput = self.derivative(param, operation)
dparam_dtarget = dparam_doutput * partial_deriv[operation.output]
partial_deriv[param] += dparam_dtarget
if param.result_of and param.result_of != OpType.NOOP:
operation_queue.append(param.result_of)
return partial_deriv.get(source)
def derivative(self, param: Variable, operation: Operation) -> np.ndarray:
"""
Compute the derivative of given operation/function
Args:
param: variable to be differentiated
operation: function performed on the input variable
Returns:
Derivative of input variable with respect to the output of
the operation
"""
params = operation.params
if operation == OpType.ADD:
return np.ones_like(params[0].to_ndarray(), dtype=np.float64)
if operation == OpType.SUB:
if params[0] == param:
return np.ones_like(params[0].to_ndarray(), dtype=np.float64)
return -np.ones_like(params[1].to_ndarray(), dtype=np.float64)
if operation == OpType.MUL:
return (
params[1].to_ndarray().T
if params[0] == param
else params[0].to_ndarray().T
)
if operation == OpType.DIV:
if params[0] == param:
return 1 / params[1].to_ndarray()
return -params[0].to_ndarray() / (params[1].to_ndarray() ** 2)
if operation == OpType.MATMUL:
return (
params[1].to_ndarray().T
if params[0] == param
else params[0].to_ndarray().T
)
if operation == OpType.POWER:
power = operation.other_params["power"]
return power * (params[0].to_ndarray() ** (power - 1))
err_msg = f"invalid operation type: {operation.op_type}"
raise ValueError(err_msg)
if __name__ == "__main__":
import doctest
doctest.testmod()
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