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A neural network is a trainable mathematical function: it transforms input values through layers of weighted calculations to produce an output. During training, it measures prediction error, uses backpropagation to calculate how its parameters contributed to that error, and uses an optimizer to adjust them.
What is a neural network?
An artificial neural network is a computation made from adjustable parameters—primarily weights and biases—arranged in layers. The brain analogy can help visualize connected units, but artificial units are mathematical operations, not miniature biological neurons.
A single unit takes inputs, multiplies each by a weight, adds the results and a bias, then applies an activation function:
output = activation(weight₁ × input₁ + weight₂ × input₂ + … + bias)
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The weights control how strongly each input affects the result; the bias shifts the combined value. A layer applies this operation to many units, and a network composes layers so that the output of one becomes the input to the next.
Why activation functions matter
Activations introduce nonlinearity, allowing a network to model relationships that cannot be represented by a single linear transformation. Without nonlinear activations, stacking linear layers still produces one linear transformation; adding depth alone would not give the network nonlinear modeling capacity.
How does a neural network make a prediction?
In a forward pass, input values move through the network’s layers. Each layer combines values using its weights and biases, applies its activation, and passes the result onward. The final layer produces the prediction, whose form depends on the task—for example, scores for possible classes in a classifier.
At this point, the network has computed an output; it has not necessarily learned anything from that example. Learning requires comparing the output with a target and changing parameters based on that comparison.
How do neural networks learn?
Training repeats a cycle over examples or batches of data. The objective is to reduce a chosen loss, which measures the discrepancy between a prediction and its target for the specific task.
- Run a forward pass. Supply an input example or batch and calculate the network’s prediction.
- Calculate the loss. Compare the prediction with the target using the loss function selected for the training objective.
- Calculate gradients. Differentiate the loss with respect to the network’s parameters.
- Update parameters. An optimizer uses those gradients to adjust weights and biases.
- Repeat and evaluate. Continue across the training data, while monitoring performance on data not used to fit the model.
For basic gradient descent, a parameter update can be written as weight = weight - learning_rate × gradient. The learning rate controls the size of each step. Gradient calculation and parameter updating are separate operations: backpropagation provides gradients, while the optimizer applies an update rule.
A decreasing training loss by itself does not establish that the model generalizes to new data. Evaluation on data withheld from fitting helps assess performance beyond the examples used for parameter updates.
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What is backpropagation?
Backpropagation calculates how the loss changes with respect to each parameter by applying the chain rule through the network’s computation graph. Because a prediction depends on earlier layer calculations, the chain rule passes derivatives backward from the loss through those calculations to the weights and biases.
Computing these derivatives efficiently is the key contribution: rather than perturbing every parameter and rerunning the entire network to estimate its effect, backpropagation organizes the repeated chain-rule calculations across the graph. The University of Toronto’s CSC311 notes explain the method through computation graphs and the chain rule.
Backpropagation does not itself choose a parameter update or guarantee that a model will learn a useful solution. It computes gradients; an optimizer uses them to make updates, and the data, model, loss, and training choices determine what the process is trying to learn.
How do you implement a neural network training loop in Python?
With PyTorch, a beginner-friendly implementation separates the model definition from the training loop. A model commonly subclasses torch.nn.Module, defines learnable parameters and a forward(input) method, and returns its output from that method. The framework’s autograd system tracks operations so it can differentiate the computation graph.
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- Clear gradients left from the previous update with
optimizer.zero_grad(). - Compute the model output from the current input.
- Calculate the loss from the output and target.
- Call
loss.backward()to calculate gradients. - Call
optimizer.step()to update the parameters.
Gradients accumulate by default in PyTorch, so clearing them between updates is important when each update should use only the current batch’s gradients. The official PyTorch neural networks tutorial, last updated May 11, 2026, demonstrates the procedure with a feed-forward image classifier. Its sequence is representative of the training mechanics, not a claim about a particular model’s accuracy or speed.
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The implementation still needs suitable data, a model, a loss function, gradient handling, and parameter updates. A framework automates differentiation and common optimization operations; it does not decide whether the data, objective, architecture, or evaluation are appropriate for a particular task.
Independent reader supportYour contribution helps us test, update, and keep practical guides available for everyone.Should you build from scratch or use a framework?
Both routes teach useful things, but they serve different immediate goals.
- Build a tiny network from scratch when the priority is understanding weighted sums, derivatives, and how gradients affect parameters. Small arrays and explicit derivatives make the calculations visible.
- Use a framework when the priority is defining models and running experiments without manually implementing every derivative. Automatic differentiation handles the chain-rule calculations, while the training loop remains explicit.
A productive progression is to first trace a small forward pass and its derivatives, then use framework autograd for a practical model. The PyTorch examples resource contrasts manually implementing forward and backward passes with using framework autograd.
What should you learn next?
Start with the mechanics needed to interpret a training run: tensors and their shapes, matrix multiplication, weighted sums, derivatives, loss functions, and gradient descent. Then practice defining a model, preparing inputs and targets, selecting a loss, handling gradients, updating parameters, and evaluating on data held out from fitting.
Architecture depends on the problem and data shape. A basic feed-forward network is a useful starting point; image, sequence, and language tasks may motivate specialized architectures. There is no universal architecture ranking independent of the task.
For a hands-on longer course of study, the publisher listing for Deep Learning with Python, Third Edition describes examples using Keras, PyTorch, JAX, and TensorFlow. Simon & Schuster lists François Chollet and Matthew Watson as authors, publication on November 18, 2025, and 648 pages; the listing says intermediate Python is intended and prior machine-learning or linear-algebra experience is not required. It is an optional next step, not a prerequisite for understanding the training loop here.
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