While working on machine learning projects, I became fascinated with computational graph optimization. The challenge was clear: our models were performing countless individual scalar operations when they could be leveraging efficient vector operations instead. Here’s my journey into transforming these computations and the insights I gained along the way.
Let’s begin by examining our starting point: scalar operations. Consider how a typical ML framework represents a dot product operation:
v0 = input
v3 = input
v6 = v0 * v3 # multiply first pair
v10 = v6 + 0 # accumulator initialization
v1 = input
v4 = input
v7 = v1 * v4 # multiply second pair
v11 = v10 + v7 # accumulate result
This representation highlights three key issues I wanted to address: individual scalar operations create unnecessary overhead, the higher-level pattern (dot product) is lost in the details, and we’re not taking advantage of modern hardware’s vector capabilities.
To tackle these challenges, I developed a solution centered around using a union-find data structure to track equivalent computations. The core idea was to maintain a forwarding chain that could point to optimized versions of operations:
class Value:
def __init__(self, data, _children=(), _op=''):
self.data = data
self._prev = set(_children)
self._op = _op
self.grad = 0
self._backward = lambda: None
self.forwarded = None # Tracks optimized versions
def find(self):
"""Find the most optimized version of this operation"""
current = self
while isinstance(current, Value) and current.forwarded:
current = current.forwarded
return current
def make_equal_to(self, other):
"""Mark two operations as equivalent"""
self.find().forwarded = other
The optimization process I developed follows a three-phase strategy: flattening phase to combine nested additions into single operations, pattern detection to identify sequences that could benefit from vectorization, and transformation to convert identified patterns into vector operations.
Here’s my implementation of the flattening phase:
def optimize_additions(v):
"""Flatten nested addition chains"""
if v._op == "+":
args = v.args()
if any(arg._op == "+" for arg in args):
flattened = []
for arg in args:
if arg._op == "+":
flattened.extend(arg.args())
else:
flattened.append(arg)
return Value(0, tuple(flattened), "+")
return v
To support efficient vector computations, I created a specialized type:
class Array(Value):
def __init__(self, data):
super().__init__(0, data, 'array')
self._data = data
def dot(self, other):
"""Optimized dot product implementation"""
return sum(a * b for a, b in zip(self._data, other._data))
The heart of the optimization process combines these components into a complete pipeline:
def optimize(v):
"""Main optimization pipeline"""
# First pass: flatten additions
v = optimize_additions(v)
# Second pass: identify vector patterns
for node in v.topo():
if node._op == "+" and has_multiplication_chain(node):
vectors = extract_vectors(node)
node.make_equal_to(create_vector_operation(vectors))
return v
To validate this approach, I tested it on a neural network architecture similar to those used for MNIST:
input_size = 28 * 28
network = MLP(input_size, [50, 10])
sample_input = [Value(i) for i in range(input_size)]
output = network(sample_input)
# Measure before and after optimization
original_size = count_nodes(output)
optimized = optimize(output)
final_size = count_nodes(optimized)
The results were remarkable: the initial graph size of approximately 40,000 nodes was reduced to merely 50 nodes, representing over 99% reduction in graph size.
This work has opened several exciting research directions I’m eager to explore: automatic generation of efficient backward passes, extension to more complex mathematical patterns, hardware-specific optimization strategies, and dynamic pattern recognition for runtime optimization.
Through this exploration of computational graph optimization, I’ve found that vectorization isn’t just about performance—it’s about expressing computations in a way that better matches both our mental models and modern hardware capabilities. While this implementation focuses on basic patterns like dot products, the principles can extend to more complex operations and architectures.
The journey from scalar to vector operations has taught me valuable lessons about the intersection of mathematical patterns, hardware capabilities, and software optimization. I’m excited to see how these techniques evolve and find new applications in the machine learning landscape.