0Pricing
Machine Learning Academy · Lesson

Autograd: Automatic Differentiation for Backpropagation

Learners will define a scalar computation graph, call .backward(), and inspect .grad on leaf tensors to understand how gradients flow through the network.

Autograd: Automatic Differentiation for Backpropagation is a free Machine Learning Academy lesson on CoddyKit — lesson 2 of 4. You can read the complete lesson below for free — then practise it hands-on in the browser with a built-in code editor and a 24/7 AI tutor. It is part of the Machine Learning Academy learning path, one of 4 lessons in the course, and your progress syncs across the web and the CoddyKit app.

What Is Automatic Differentiation?

Automatic differentiation (autograd) is the engine that computes gradients in PyTorch without requiring the programmer to derive them by hand. Unlike numerical differentiation (finite differences) or symbolic differentiation (algebra), autograd works by recording operations on tensors at runtime and replaying them in reverse. This makes it possible to train models of any architecture efficiently and correctly.

import torch

# A simple differentiable computation
x = torch.tensor(3.0, requires_grad=True)
y = x ** 2 + 2 * x + 1   # y = (x+1)^2

# Compute gradient dy/dx
y.backward()
print(x.grad)   # tensor(8.) because dy/dx = 2x+2 = 2*3+2 = 8

requires_grad: Enabling Gradient Tracking

Gradient tracking is disabled by default. Setting requires_grad=True tells PyTorch to record all operations on that tensor so gradients can be computed later. Leaf tensors (parameters) require gradients; intermediate tensors inherit gradient-tracking from their parents automatically. Model parameters created by nn.Module have requires_grad=True set automatically.

import torch

# Leaf tensor with gradient tracking
w = torch.tensor(2.0, requires_grad=True)
b = torch.tensor(0.5, requires_grad=True)

# Forward pass
x = torch.tensor(3.0)   # input, no grad needed
y_pred = w * x + b      # y = wx + b = 6.5

print(y_pred.requires_grad)  # True (inherited)
print(w.is_leaf)              # True
print(y_pred.is_leaf)         # False

The Computation Graph

As you perform operations on tensors with requires_grad=True, PyTorch builds a dynamic computation graph (a directed acyclic graph of operations). Each node stores the operation and references to its inputs. When .backward() is called, PyTorch traverses this graph in reverse (from output to inputs) using the chain rule to compute partial derivatives for every leaf tensor. The graph is re-built fresh on each forward pass, enabling dynamic architectures.

import torch

a = torch.tensor(2.0, requires_grad=True)
b = torch.tensor(3.0, requires_grad=True)

# Build computation graph
c = a * b          # c = 6
d = c + a          # d = 8
e = d ** 2         # e = 64

# Inspect graph node
print(e.grad_fn)          # <PowBackward0 object>
print(e.grad_fn.next_functions)  # parents in the graph

Calling .backward(): Computing Gradients

Calling .backward() on a scalar tensor triggers backpropagation through the entire computation graph, depositing gradients into the .grad attribute of each leaf tensor. If the output is not scalar, you must provide a gradient tensor of the same shape (the upstream gradient). Gradients accumulate by default — always call optimizer.zero_grad() (or tensor.grad.zero_()) before the next backward pass to avoid incorrect updates.

import torch

x = torch.tensor(4.0, requires_grad=True)
loss = (x - 1) ** 2   # minimum at x=1

loss.backward()
print(x.grad)   # tensor(6.) = 2*(x-1) = 2*3 = 6

# Gradients accumulate! Reset before next pass
x.grad.zero_()
new_loss = (x - 2) ** 2
new_loss.backward()
print(x.grad)   # tensor(4.) = 2*(x-2) = 2*2 = 4

Gradient Flow Through Multiple Operations

The chain rule states that the derivative of a composition of functions is the product of derivatives. Autograd applies this mechanically through every operation in the graph. Understanding how gradients flow helps debug issues like vanishing gradients (products near zero) and exploding gradients (products greater than 1 repeated many times). Activation function choice (ReLU vs sigmoid) directly affects gradient magnitudes.

import torch

# Chain: z = sigmoid(wx + b), loss = (z - y_true)^2
def sigmoid(x):
    return 1 / (1 + torch.exp(-x))

w = torch.tensor(0.5, requires_grad=True)
b = torch.tensor(0.0, requires_grad=True)
x = torch.tensor(2.0)
y_true = torch.tensor(1.0)

z = sigmoid(w * x + b)
loss = (z - y_true) ** 2
loss.backward()

print(f'dL/dw = {w.grad:.4f}')  # gradient w.r.t. weight
print(f'dL/db = {b.grad:.4f}')  # gradient w.r.t. bias

torch.no_grad(): Disabling Gradient Tracking

During inference (prediction on new data), you do not need gradients — computing them wastes time and memory. Wrapping inference code with torch.no_grad() disables the computation graph entirely, speeding up forward passes and reducing memory usage by up to 50%. This is also used during evaluation loops to get validation metrics without influencing training.

import torch

x = torch.randn(100, requires_grad=True)

# With gradient tracking (training)
out = x ** 2
print(out.requires_grad)   # True

# Without gradient tracking (inference)
with torch.no_grad():
    out_no_grad = x ** 2
    print(out_no_grad.requires_grad)  # False

# Also used as a decorator
@torch.no_grad()
def predict(model, data):
    return model(data)

Gradient Accumulation Issue and zero_grad

PyTorch accumulates (adds) gradients into .grad each time .backward() is called. This is intentional for some advanced techniques, but during normal training it means you must zero gradients before each backward pass. The standard idiom uses the optimizer's zero_grad() method, which clears gradients for all parameters the optimizer manages. Forgetting this step leads to gradients growing unboundedly, corrupting updates.

import torch
import torch.nn as nn
import torch.optim as optim

model = nn.Linear(2, 1)
optimizer = optim.SGD(model.parameters(), lr=0.01)

for epoch in range(3):
    optimizer.zero_grad()         # clear old gradients
    x = torch.randn(4, 2)
    y_pred = model(x)
    loss = y_pred.sum()
    loss.backward()               # compute new gradients
    optimizer.step()              # update weights
    print(f'Epoch {epoch}: loss={loss.item():.3f}')

Inspecting Gradient Values for Debugging

Inspecting gradient values is crucial for diagnosing training problems. You can access them via tensor.grad after calling backward. Plotting the gradient norm across layers reveals whether gradients vanish (near zero) or explode (very large) in deep networks. A common debugging hook is to print gradient statistics after every N batches to detect instability early in training.

import torch
import torch.nn as nn

model = nn.Sequential(
    nn.Linear(4, 8),
    nn.ReLU(),
    nn.Linear(8, 1)
)

x = torch.randn(16, 4)
y_pred = model(x)
loss = y_pred.mean()
loss.backward()

# Inspect gradients of each parameter
for name, param in model.named_parameters():
    if param.grad is not None:
        norm = param.grad.norm().item()
        print(f'{name}: grad_norm={norm:.4f}')

Detaching Tensors from the Graph

Sometimes you want to use a computed tensor as a constant input to another computation — without letting gradients flow through it. .detach() returns a new tensor that shares data but is removed from the computation graph. This is used in target networks (reinforcement learning), stopping gradient flow in specific branches of a network, and converting tensors to NumPy arrays for plotting.

import torch

a = torch.tensor(3.0, requires_grad=True)
b = a * 2             # b depends on a
c = b.detach()        # c is a constant copy of b's value

d = c * 5             # gradient does NOT flow back to a
d.backward()

# a.grad is None because c severed the graph
print(a.grad)         # None

# Detach before converting to NumPy
np_val = b.detach().numpy()
print(np_val)         # [6.0] (as NumPy array)

Second-Order Gradients with create_graph

PyTorch supports higher-order differentiation. Passing create_graph=True to .backward() keeps the computation graph for the gradient computation itself, allowing you to differentiate through gradients. This is used in meta-learning (MAML), gradient penalty regularisation (Wasserstein GAN), and any technique that optimises with respect to gradients.

import torch

x = torch.tensor(2.0, requires_grad=True)
y = x ** 3           # y = x^3

# First derivative dy/dx = 3x^2
dy = torch.autograd.grad(y, x, create_graph=True)[0]
print(dy)            # tensor(12., grad_fn=...)

# Second derivative d^2y/dx^2 = 6x
d2y = torch.autograd.grad(dy, x)[0]
print(d2y)           # tensor(6.)

Autograd in the Training Pipeline

Autograd is the foundation of every neural network training loop. The standard four-step pattern is: (1) zero gradients with optimizer.zero_grad(), (2) forward pass to compute predictions and loss, (3) backward pass with loss.backward() to compute gradients, and (4) optimizer step with optimizer.step() to update parameters. Understanding that autograd is doing the heavy calculus work lets you focus on model architecture and hyperparameters.

import torch
import torch.nn as nn
import torch.optim as optim

model = nn.Linear(1, 1)
optimizer = optim.SGD(model.parameters(), lr=0.1)
criterion = nn.MSELoss()

# Synthetic data: y = 3x + 1
X = torch.randn(20, 1)
y = 3 * X + 1 + 0.1 * torch.randn(20, 1)

for epoch in range(5):
    optimizer.zero_grad()       # (1) zero grads
    y_pred = model(X)           # (2) forward
    loss = criterion(y_pred, y) # (2) loss
    loss.backward()             # (3) backward
    optimizer.step()            # (4) update
    print(f'Epoch {epoch}: loss={loss.item():.4f}')

Quick Check

Test your understanding of Machine Learning with Python concepts from this lesson.

Lesson Recap

In this lesson you learned: autograd builds a dynamic computation graph recording every operation on tensors with requires_grad=True, .backward() computes gradients via the chain rule and deposits them in .grad, and gradients accumulate so you must zero them before each training step. Next up we explore building a feedforward network with nn.Module.

Frequently asked questions

Is the “Autograd: Automatic Differentiation for Backpropagation” lesson free?

Yes — the full text of “Autograd: Automatic Differentiation for Backpropagation” is free to read here on the web, and the Machine Learning Academy course includes 4 lessons in total. To practise it interactively (a built-in code editor and a 24/7 AI tutor) and unlock the rest of the Machine Learning Academy course, upgrade to CoddyKit PRO.

What will I learn in “Autograd: Automatic Differentiation for Backpropagation”?

Learners will define a scalar computation graph, call .backward(), and inspect .grad on leaf tensors to understand how gradients flow through the network. You practise Machine Learning Academy with hands-on code you run directly in the browser, and a 24/7 AI tutor answers your questions as you work through the lesson.

Do I need any experience to start Machine Learning Academy?

No prior experience is required. Machine Learning Academy on CoddyKit is structured for beginners through advanced learners; this is — lesson 2 of 4, so you can start here or from the beginning and move at your own pace.

How long does the “Autograd: Automatic Differentiation for Backpropagation” lesson take?

Most CoddyKit lessons take about 5–10 minutes. Each one is bite-sized and interactive, so you make steady progress and pick up exactly where you left off across the web and the app.

Can I write and run code in this Machine Learning Academy lesson?

Yes. Every Machine Learning Academy lesson includes a built-in code editor, so you write and run real code right in your browser and get instant AI feedback — no local setup required.

All lessons in this course

  1. PyTorch Tensors: Creation, Operations, and GPU Transfer
  2. Autograd: Automatic Differentiation for Backpropagation
  3. Building a Feedforward Network with nn.Module
  4. Training Loop: Loss, Optimizer, and Epochs
← Back to Machine Learning Academy