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Neural Networks Fundamentals — Perceptrons to Deep Learning

Deep LearningNeural Networks🟢 Free Lesson

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Deep Learning

Neural Networks — The Foundation of Modern AI

Discover how neural networks form the backbone of modern AI systems, enabling machines to learn complex patterns from data.

  • Universal function approximation — learn any mapping from inputs to outputs
  • Backpropagation — efficient gradient computation for training
  • Deep architectures — stack layers for hierarchical feature learning

The brain is a computer made of meat, and it is very good at being a brain.

Neural Networks Fundamentals

Neural networks learn complex patterns by stacking simple computational units (neurons) in layers. At the mathematical core, a neural network is a parameterized nonlinear function that is optimized via gradient-based methods.


The Perceptron

The perceptron is the atomic unit of neural computation. Given input vector , weights , and bias :

x₁x₂x₃w₁w₂w₃Σ+ bσ(·)activationŷInputsSummationActivationOutput

Activation Functions

Activation functions introduce nonlinearity, enabling networks to approximate arbitrary functions. Without them, a multi-layer network collapses to a single linear transformation.

ReLU: f(x) = max(0, x)0Sigmoid: σ(x) = 1/(1+e⁻ˣ)0.5Tanh: tanh(x)GELU: x · Φ(x)0

Properties:

• ReLU: Range [0, ∞), gradient ∈ {0, 1}, dead neurons possible

• Sigmoid: Range (0, 1), gradient ∈ (0, 0.25], vanishing gradients

• Tanh: Range (-1, 1), zero-centered, still vanishing gradients

• GELU: Smooth approximation of ReLU, used in Transformers (BERT, GPT)

• Swish: f(x) = x · σ(x), self-gated, used in EfficientNet


Multi-Layer Perceptron (MLP)

An MLP stacks layers of neurons to form a deep network. Each layer computes an affine transformation followed by a nonlinear activation:

where and is the input.

Inputn = 3x1x2x3x4x5Hidden 164 neuronsHidden 232 neuronsOutput1 neuronσOutputŷW₍¹₎, b₍¹₎W₍²₎, b₍²₎W₍³₎, b₍³₎

Backpropagation

Backpropagation computes the gradient of the loss with respect to every parameter via the chain rule applied recursively from output to input.

Forward Pass →Input xBatch size BLinearz = Wx + bActivationa = σ(z)Loss LL(ŷ, y)← Backward Pass∂L/∂ŷ∂L/∂a · σ'(z)∂L/∂W= Δ · xᵀUpdate WW ← W - α∇LChain Rule (key insight):∂L/∂W₍⁻₎ = ∂L/∂a₍ᵈ₎ · ∏ σ'(z₍ᵏ₎) · W₍ᵏ⁺¹₎ · ∂a₍⁻₎/∂W₍⁻₎

Gradient Descent Variants

Batch GDUses entire dataset per updateStable, slowMini-Batch GDUses batch of B samplesNoisy, fast, generalizesSGD (B=1)Uses single sampleVery noisy, escapes local minConvergence ComparisonLossEpochsBatchMini-BSGD

Weight Initialization


PyTorch Implementation


Key Takeaways


What to Learn Next

-> Convolutional Neural Networks Learn how CNNs process visual data with parameter sharing.

-> RNNs and LSTMs Explore networks designed for sequential data.

-> Training Deep Networks Master optimizers, batch norm, and regularization.

-> Transformers Learn the architecture that replaced RNNs.

-> Weight Initialization Understand Xavier, He, and modern initialization.

-> Optimizers for Deep Learning SGD, Adam, AdamW, and beyond.

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